Bayesian_Monte_Carlo_Simulation_with_R.ipynb

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    "colab": {
      "name": "Bayesian Monte Carlo Simulation with R.ipynb",
      "provenance": [],
      "collapsed_sections": []
    }
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  "cells": [
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        "id": "tScs0JSjJJoS"
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      "source": [
        "# **Bayesian Inference via Monte Carlo Integration with $\\texttt{R}$**\n",
        "\n",
        "By **Sourish Das**\n",
        "\n",
        "**Chennai Mathematical Institute**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pT7yQBgzpM9g"
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      "source": [
        "### **Table of Contents**\n",
        "1. Introduction and Motivation\n",
        "\n",
        "2. Inverse Transform Method\n",
        "\n",
        "3. Acceptance Rejection Method\n",
        "\n",
        "4. Importance Sampling Method\n",
        "\n",
        "5. Markov Chain Monte Carlo Method\n",
        "  - Gibbs Sampler\n",
        "  - Metropolis-Hastings\n",
        "\n",
        "6. MCMC Diagnostics\n",
        "\n",
        "### **Introduction**\n",
        "    \n",
        "* Suppose $Y \\sim p(y|\\theta)$ and $p(\\theta)$ is the prior distribution over $\\theta$\n",
        "\n",
        "* Our objective is to make statistical inference about $\\theta$\n",
        "\n",
        "* The **posterior distribution** is:\n",
        "$$\n",
        "p(\\theta|y)=\\frac{p(y|\\theta)p(\\theta)}{\\int_{\\Theta}p(y|\\theta)p(\\theta)d\\theta}\n",
        "$$\n",
        "\n",
        "* **Posterior mean**: *integration problem*\n",
        "$$\n",
        "\\mathbb{E}(\\theta|y)=\\int_{\\Theta}\\theta p(\\theta|y)d\\theta =g(y)\n",
        "$$\n",
        "\n",
        "* **Posterior median**: *integration problem*\n",
        "$$\n",
        "\\int_{-\\infty}^{\\mu_0} p(\\theta|y)d\\theta =\\frac{1}{2},\n",
        "$$\n",
        "\n",
        "* **Posterior Mode**: *optimization problem*\n",
        "\\begin{eqnarray*}\n",
        "\\hat{\\theta}&=&\\text{argmax}_{\\Theta}~p(\\theta|y)\\\\\n",
        "&=&\\text{argmax}_{\\Theta}p(y|\\theta)p(\\theta)\n",
        "\\end{eqnarray*}\n",
        "\n",
        "* **Posterior predictive distribution**:\n",
        "$$\n",
        "p(\\tilde{y}|y)=\\int_{\\Theta}p(\\tilde{y}|\\theta,y)p(\\theta|y)d\\theta\n",
        "$$\n"
      ]
    },
    {
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      "source": [
        "- In order to get the posterior mean we have to solve this integration\n",
        "- Analytical solution does not exists for most of the sophisticated models\n",
        "- So we have to resort to simulation technique to solve this integration problem.\n",
        "- Typically it is known as **Monte Carlo Integration** method\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "75qt3gLsl0S0"
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      "source": [
        "**Monte Carlo methods** rely on\n",
        "  - The possibility of generation of endless flow of random variables\n",
        "  - For well-known or new distributions.\n",
        "  - Such a simulation is based on the generation of uniform random variables on the interval $(0, 1)$.\n",
        "  - We are not concerned with the details of producing uniform random numbers.\n",
        "  - We assume the existence of such a sequence"
      ]
    },
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      "source": [
        "As we want to estimate the **posterior mean**:\n",
        "$$\n",
        "\\mathbb{E}(\\theta|y)=\\int_{\\Theta}\\theta p(\\theta|y)d\\theta =g(y)\n",
        "$$\n",
        "-  we can do so by simulating random samples from $p(\\theta|y)$\n",
        "-  Suppose $(\\theta^1,...,\\theta^N)$ are random samples from $p(\\theta|y)$, then we can approximate $g(y)$ by\n",
        "$$\n",
        "\\hat{g}(y)=\\frac{1}{N}\\sum_{s=1}^N\\theta^s\n",
        "$$\n",
        "-  If we can ensure simple random sample then Laws of Large Number  ensures\n",
        "$$\n",
        "\\hat{g}(y)=\\frac{1}{N}\\sum_{s=1}^N\\theta^s \\rightarrow g(y)=\\mathbb{E}(\\theta|y)\n",
        "$$\n",
        "as $N \\rightarrow \\infty$, where $N$ is the simulation size.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4kSSAigXma0o"
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      "source": [
        "**Convergence**:\n",
        "$$\n",
        "\\bar{h}=\\frac{1}{n}\\sum_{i=1}^n h(\\theta^i)\\stackrel{a.s.}{\\longrightarrow} \\mathbb{E}(h(\\theta)|y)=\\int_{\\Theta}h(\\theta)p(\\theta|y)d\\theta\n",
        "$$\n",
        "- **Central Limit Theorem**: \n",
        "$$\n",
        "\\frac{\\bar{h}-\\mathbb{E}(\\theta|y)}{\\sqrt{\\sigma_n^2}}\\stackrel{L}{\\longrightarrow} N(0,1)\n",
        "$$\n",
        "where $\\sigma_n^2=\\frac{1}{n}\\sum_{i=1}^n[\\bar{h}-E(\\theta|y)]^2$, \n",
        "such that $\\int_{\\Theta}h(\\theta)^2p(\\theta|y)d\\theta< \\infty$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZVrupdGOmsE9"
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      "source": [
        "The advantage of CLT is we can evaluate the Monte Carlo error\n",
        "- It assumes $\\sigma_n^2$ is the proper estimate of the variance of $\\bar{h}_n$\n",
        "- If $\\sigma_n^2$ does not converge, converges too slowly, a CLT may not apply. In such cases we will not be able to estimate the Monte Carlo error."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Cz5bI8rfmz49"
      },
      "source": [
        "**Using the $\\texttt{R}$ Generator**\n",
        "\n",
        "- $\\texttt{R}$ has a large number of functions that will generate random samples from standard distributions.\n",
        "\n",
        "- If built-in function in $\\texttt{R}$ is available then we can use it directly.\n",
        "\n",
        "- However, if the built-in functions are not available and the posterior distribution is known upto its kernel then we have to use genric methods to draw samples from such distributions."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "M0Uh8PM_nQQx"
      },
      "source": [
        "**The Inverse Transform**\n",
        "\n",
        "- The probability integral transform allows us to transform a uniform into any random variable.\n",
        "\n",
        "- If $X$ has a density $f$ and cdf $F$, then we have the relation \n",
        "$$\n",
        "F(x)=\\int_{-\\infty}^x f(u)du\n",
        "$$\n",
        "we know $F(x) \\sim Unif(0,1)$ \n",
        "\n",
        "- so we set $U=F(x)$ and solve for $x$\n",
        "\n",
        "- If $X\\sim Exp(1)$ then $F(x)=1-e^{-x}$\n",
        "\n",
        "- solving for $x$ in $u=1-e^{-x}$ gives $x=-\\log(1-u)$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kqPzsKcpRy2q"
      },
      "source": [
        "**Generating Exponentially Distributed Random Samples**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "6F5Meds-pM9m",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 538
        },
        "outputId": "78ff452d-434a-4244-ea47-b5f8fc9aff97"
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      "source": [
        "# Number of random samples\n",
        "N=1000\n",
        "# Simulate from uniforms\n",
        "u=runif(N)\n",
        "# transforms of uniforms\n",
        "x= -log(1-u)\n",
        "# exponential in R\n",
        "y=rexp(N)\n",
        "par(mfrow=c(1,2)) # plot parameters\n",
        "hist(x,freq=F,main=\"Exp from Uniforms\",col=\"orange\")\n",
        "hist(y,freq=F,main=\"Exp from Built-in R\",col=\"orange\")\n",
        "ks.test(x,y)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "\n",
              "\tTwo-sample Kolmogorov-Smirnov test\n",
              "\n",
              "data:  x and y\n",
              "D = 0.047, p-value = 0.2193\n",
              "alternative hypothesis: two-sided\n"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
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R399NNP9+zZs7a2Nicnx/xAfycnp7Z9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",
            "text/plain": [
              "Plot with title “Exp from Built-in R”"
            ]
          },
          "metadata": {
            "tags": [],
            "image/png": {
              "width": 420,
              "height": 420
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Gra5LFqin4ax"
      },
      "source": [
        "**The Inverse Transform from Uniforms**\n",
        "\n",
        "This method is useful for other probability distributions ; ones obtained as a transformation of uniform random variables\n",
        "\n",
        "- **Logistic cdf**: $$F(x)=\\frac{1}{1+\\exp\\left(-\\frac{x-\\mu}{\\lambda}\\right)}$$\n",
        "\n",
        "- **Cauchy cdf**: $$F(x)=\\frac{1}{2}+\\frac{1}{\\pi}arctan((x-\\mu)/\\sigma)$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "6wYlZphyoI79"
      },
      "source": [
        "**General Transform Method**\n",
        "\n",
        "If $X_i \\stackrel{iid}{\\sim} Exp(1)$; three standard distribution can be derived as\n",
        "\n",
        "- $Y= 2\\sum_{i=1}^n X_i \\sim \\chi_{2n}^2$\n",
        "\n",
        "- $Y= \\beta \\sum_{i=1}^n X_i \\sim Gamma(n,\\beta)$\n",
        "\n",
        "- $Y= \\frac{\\sum_{i=1}^{n_1} X_i}{\\sum_{i=1}^{n_1+n_2} X_i}  \\sim Beta(n_1,n_2)$\n",
        "\n",
        "where $n \\in \\mathcal{N}=\\{1,2,...\\}$\n",
        "\n",
        "- These transformations are quite simple and we will use them quite often\n",
        "\n",
        "- However, there is a limit to their usefullness. Only when closed form CDF is available the method is available.\n",
        "\n",
        "- The method will not work for Gaussian distribution as the CDF is not in closed form.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4exa9zXPokzv"
      },
      "source": [
        "**Box-Muller transform**\n",
        "\n",
        "- Box-Muller transform to generate from Normal distribution\n",
        "\n",
        "- If $U_1$ and $U_2$ are iid from $Unif(0,1)$\n",
        "\n",
        "- The variable $X_1$ and $X_2$ \n",
        "$$\n",
        "X_1=\\sqrt{-2\\log(U_1)}\\cos(2\\pi U_2),~~~\\text{and}~~~ X_2=\\sqrt{-2\\log(U_1)}\\sin(2\\pi U_2)\n",
        "$$\n",
        "are iid $N(0,1)$ by virtue of a change of variable argument.\n",
        "\n",
        "- The Box-Muller algorithm is exact, not a crude CLT-based approximation\n",
        "\n",
        "- Note that this is not the generator implemented in R. It uses the probability inverse transform with a very accurate representation of the Gaussian cdf"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 118
        },
        "id": "JQngjLiODC2N",
        "outputId": "2554ae39-0a7f-4a07-ca6f-6c28db9fc15c"
      },
      "source": [
        "u1 = runif(N)\n",
        "u2 = runif(N)\n",
        "# Box-Muller transformation\n",
        "x1 = sqrt(-2*log(u1))*cos(2*pi*u2)\n",
        "\n",
        "# built-in function in R\n",
        "y = rnorm(N)\n",
        "\n",
        "ks.test(x1,y)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "\n",
              "\tTwo-sample Kolmogorov-Smirnov test\n",
              "\n",
              "data:  x1 and y\n",
              "D = 0.056, p-value = 0.08691\n",
              "alternative hypothesis: two-sided\n"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dPFi1I1OB6Mf"
      },
      "source": [
        "### Accept-Reject Method\n",
        "\n",
        "- There are many distributions where transform methods fail\n",
        "\n",
        "- For these cases, we must turn to indirect methods\n",
        "\n",
        "- We generate a candidate random variable\n",
        "\n",
        "- Only accept it subject to passing a test\n",
        "\n",
        "- This class of 'Accept-Reject' methods is extremely powerful. It will allow us to simulate from virtually any distribution.\n",
        "\n",
        "**Accept-Reject Methods**:\n",
        "\n",
        "-  Only require the functional form of the density $f$ of interest\n",
        "\n",
        "- $f$: **target density**, from where we want to simulate; but we don't know how !!!\n",
        "\n",
        "- $g$: **candidate density**; where it is simpler to simulate random variables from $g$\n",
        "\n",
        "- The only constraints we impose on this candidate density $g$\n",
        "\n",
        "- $f$ and $g$ have compatible supports (i.e., $g(x) > 0$ when $f(x) > 0$).\n",
        "\n",
        "- $X \\sim f$ can be simulated as follows:\n",
        "\n",
        "- Generate $Y \\sim g$ and independently generate $U \\sim unif(0,1)$\n",
        "\n",
        "- If $U\\leq \\frac{1}{M}\\frac{f(Y)}{g(Y)}$ set $X=Y$\n",
        "\n",
        "- If inequality is not satisfied then discard $Y$ and $U$ and start again.\n",
        "\n",
        "- Note $M=\\sup_x \\frac{f(x)}{g(x)}$\n",
        "\n",
        "- $\\mathbb{P}(\\text{Accept})=\\frac{1}{M}$ and expected waiting time $M$.\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1wmrO8HBEXBV"
      },
      "source": [
        "### **Why does this method work?**\n",
        "\n",
        "A straightforward probability calculation shows\n",
        "$$\n",
        "P(Y \\leq x| \\text{Accept})=P\\left(Y \\leq x| U \\leq \\frac{f(Y)}{Mg(Y)}\\right)= P(X \\leq x),\n",
        "$$\n",
        "where $Y \\sim g$ and $X \\sim f$.\n",
        "\n",
        "- Simulating from $g$, the output of this algorithm is exactly distributed from $f$.\n",
        "\n",
        "- The **Accept-Reject** method is applicable in any dimension.\n",
        "\n",
        "- As long as $g$ is a density over the same space as $f$.\n",
        "\n",
        "- Only need to know $f/g$ upto a constant\n",
        "\n",
        "- Only need an upper bound on $M$\n",
        "\n",
        "**Accept-Rejection Algorithm for Beta Distribution**\n",
        "\n",
        "- Generate $X \\sim Beta(a,b)$\n",
        "\n",
        "- No direct method available if $a$ and $b$ are not integers\n",
        "\n",
        "- Generate for $a=2.3$ and $b=7.9$\n",
        "\n",
        "- We can generate if $a$ and $b$ is integer\n",
        "\n",
        "1. **Candidate distribution** $Unif(0,1)$  \n",
        "2. **Target distribution** $Beta(2.3,7.9)$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "k7PEP3FLgriy"
      },
      "source": [
        "Ref : https://bookdown.org/rdpeng/advstatcomp/rejection-sampling.html"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 470
        },
        "id": "LfeWAUeWFLDl",
        "outputId": "f01af813-0cef-40a8-f299-5921ebf21a4a"
      },
      "source": [
        "N=5000\n",
        "sample.x = runif(N,0,1)\n",
        "accept = c()\n",
        "\n",
        "for(i in 1:length(sample.x)){\n",
        " U = runif(1, 0, 1)\n",
        " if(dunif(sample.x[i], 0, 1)*4*U <= dbeta(sample.x[i], 2.3, 7.9)) {\n",
        "   accept[i] = 'Yes'\n",
        " }\n",
        " else if(dunif(sample.x[i],0,1)*4*U > dbeta(sample.x[i], 2.3, 7.9)) {\n",
        "  accept[i] = 'No'\n",
        " }\n",
        "}\n",
        "\n",
        "T = data.frame(sample.x, accept = factor(accept, levels= c('Yes','No')))\n",
        "x = sort(T[,1][T$accept=='Yes'])\n",
        "par(mfrow=c(1,2))\n",
        "hist(x, breaks = seq(0,1,0.01), freq = FALSE, main = 'Histogram of X', xlab = 'X')\n",
        "lines(x, dbeta(x,2.3,7.9),col=\"red\",lwd=3)\n",
        "hist(sample.x,main= 'Histogram of candidate density',freq=FALSE)\n",
        "\n",
        "cat(\"Acceptance Rate :\",\"\\n\")\n",
        "(length(x)/length(sample.x))*100\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Acceptance Rate : \n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "[1] 24.04"
            ],
            "text/latex": "24.04",
            "text/markdown": "24.04",
            "text/html": [
              "24.04"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
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X7JKPPrLg4dOlRufP/992fOnFGmf/nl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",
            "text/plain": [
              "Plot with title “Histogram of candidate density”"
            ]
          },
          "metadata": {
            "tags": [],
            "image/png": {
              "width": 420,
              "height": 420
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "mnhcPZ5OFWnz"
      },
      "source": [
        "1. **Candidate distribution** $Beta(1,5)$ \n",
        "2. **Target distribution** $Beta(2.3,7.9)$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 470
        },
        "id": "g_SGoAOwFgre",
        "outputId": "4e54b645-09c9-44cb-a018-58f7b3eebbed"
      },
      "source": [
        "N = 5000\n",
        "sample.x = rbeta(N,1,5)\n",
        "accept = c()\n",
        "\n",
        "for(i in 1:length(sample.x)){\n",
        " U = runif(1, 0, 1)\n",
        " if(dbeta(sample.x[i], 1, 5)*3*U <= dbeta(sample.x[i], 2.3, 7.9)) {\n",
        "   accept[i] = 'Yes'\n",
        " }\n",
        " else if(dbeta(sample.x[i],1,5)*3*U > dbeta(sample.x[i], 2.3, 7.9)) {\n",
        "  accept[i] = 'No'\n",
        " }\n",
        "}\n",
        "\n",
        "T = data.frame(sample.x, accept = factor(accept, levels= c('Yes','No')))\n",
        "x = sort(T[,1][T$accept=='Yes'])\n",
        "par(mfrow=c(1,2))\n",
        "hist(x, breaks = seq(0,1,0.01), freq = FALSE, main = 'Histogram of X', xlab = 'X')\n",
        "lines(x, dbeta(x,2.3,7.9),col=\"red\",lwd=2)\n",
        "hist(sample.x,main= 'Histogram of candidate density',freq=FALSE)\n",
        "\n",
        "cat(\"Acceptance Rate :\",\"\\n\")\n",
        "(length(x)/length(sample.x))*100\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Acceptance Rate : \n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "[1] 33.22"
            ],
            "text/latex": "33.22",
            "text/markdown": "33.22",
            "text/html": [
              "33.22"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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AfBzhMoiqxYISLyhz9IVFTdbRIS5A9/EBH58EOx291XGwAAMAyCnSfYvl1OnBAR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\nDgAAwCD+H93VD65t4KSuAAAAAElFTkSuQmCC",
            "text/plain": [
              "Plot with title “Histogram of candidate density”"
            ]
          },
          "metadata": {
            "tags": [],
            "image/png": {
              "width": 420,
              "height": 420
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "KHZGA03BFspx"
      },
      "source": [
        "1. Candidate distribution $Beta(2,7)$ \n",
        "2. Target distribution $Beta(2.3,7.9)$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 470
        },
        "id": "52x3jZMLFsNh",
        "outputId": "c02e2b01-4734-4f38-d4fe-b075b7a4f6c1"
      },
      "source": [
        "N = 5000\n",
        "sample.x = rbeta(N,2,7)\n",
        "accept = c()\n",
        "\n",
        "for(i in 1:length(sample.x)){\n",
        " U = runif(1, 0, 1)\n",
        " if(dbeta(sample.x[i], 2, 7)*3*U <= dbeta(sample.x[i], 2.3, 7.9)) {\n",
        "   accept[i] = 'Yes'\n",
        " }\n",
        " else if(dbeta(sample.x[i],2,7)*3*U > dbeta(sample.x[i], 2.3, 7.9)) {\n",
        "  accept[i] = 'No'\n",
        " }\n",
        "}\n",
        "\n",
        "T = data.frame(sample.x, accept = factor(accept, levels= c('Yes','No')))\n",
        "x = sort(T[,1][T$accept=='Yes'])\n",
        "par(mfrow=c(1,2))\n",
        "hist(x, breaks = seq(0,1,0.01), freq = FALSE, main = 'Histogram of X', xlab = 'X')\n",
        "lines(x, dbeta(x,2.3,7.9),col=\"red\")\n",
        "hist(sample.x,main= 'Histogram of candidate density',freq=FALSE)\n",
        "\n",
        "cat(\"Acceptance Rate :\",\"\\n\")\n",
        "(length(x)/length(sample.x))*100\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Acceptance Rate : \n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "[1] 33.2"
            ],
            "text/latex": "33.2",
            "text/markdown": "33.2",
            "text/html": [
              "33.2"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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CguLq4yVa/Xh4WFHT16VI7SAACAchHsFMre3t7Nza22qXZ2du7u7tasBwAAKB/B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3Op3e2dL3MBERET6eQuB2u72zvY//WVX9\njCmKYrfb9+7d27lzZx9fXyeCZsVOpXs7tW7devny5Q1PdoqilJaWXnCmE5HMzMz169dXV1ef\nOHHCZrMlJCTk5uZ26NDh5MmTZrO5SZMmOTk5GRkZ3pMA6m96PJ5mzZp5B585c6ampiYpKSk3\nN7d9+/bl5eVVVVXNmzf3btrt9oqKiuTk5Ly8vHbt2jkcDqfTmZKSkpeX17ZtW5fLZbfbW7Ro\n4d2sqKg4e/ZsamrqoUOH0tPTq6urT58+3apVK+9mTU3NyZMn09LSDh8+3LJlS4/HU1JSkp6e\nfuTIkRYtWiiKUlxc3Lp16yNHjqSkpJhMpqKiotatWx89ejQ5OdlsNhcWFrZp0+bYsWPNmjWz\n2WxHjx5t165dQUFBkyZNwsLCvJvHjx+Pj4+PjIzMz89v3759UVFRbGxsVFSUd7O4uDg6Ojom\nJiYvL69Dhw4lJSWRkZGxsbHezdLS0vDw8Li4OO8mb6k+39KwsLDk5ORL+kWDH9nt9rVr117q\ns/w41/n4NzocjvLyct/n5MLCwsTERN8bg/dT6uPgqqoq7y+mj+NLSkoiIiLi4+N9HO+dWHys\nvG63+9ixY23btvXxxet+MX0cn5+fn5aW5mOxUxTFOzn4+OJnz56trKz0fQbwznW+V+RL+s/q\ncrnKyspatmzp4/jjx4/HxcX53u/DwsIyMzN9HKwfQbNiN3Xq1Ndff33BggWPPfbY+R8Rp9O5\nYMGC2bNnz5gx46WXXtIkIQAEQGVlZW5uroh07dpV6ywAdCdoit2ZM2eGDBmyc+fO2NjYBu7t\n9L//+7++l3EAAAAjCZpiJyLV1dXeezvt2bOHezsBMDZuxg7gMgRTsavDvZ0AGBg3Ywdw2YKy\n2AGAUdW/GXv//v3Pvxn78ePHu3fvzs3YAVxQ0FwVCwChgJuxA2gMVuwAQEdatGgxbNiwP//5\nzw2MGT169ObNm48ePRqwVACChU/3AAQABMapU6fat2/f8JjMzMySkpLA5AEQXCh2AKAjKt2M\nHUCIoNgBgI6MGDFixYoVCxcuvOC3IzqdzpkzZ65atWrUqFGBzwZA/zjHDgB0hJuxA2gMih0A\n6As3Ywdw2Sh2AKBT3IwdwKWi2AEAABgEF08AAAAYBMUOAADAICh2AAAABkGxAwAAMAiKHQAA\ngEFQ7AAAAAyCYgcAAGAQVq0DBL1t27b17t1b6xRAQG3duvXqq6/WOgUCirkOISgY5zqKXWOF\nh4eLyFdffRUXF6d1lp8ZMGDAvHnzrr32Wq2D/MzkyZOvvPLKSZMmaR3kZ5YsWbJnz54333xT\n6yA/s3HjxqeffnrDhg1aB/mZ8vLywYMHez/2CCm6net8sXLlymXLln3++edaB7kcs2bNEpGZ\nM2dqHeRy3HXXXffdd9+dd96pdZBLFrxzHcXOP7p3796kSROtU/yM2Wzu0KFDr169tA7yM3Fx\ncS1atNBbqhYtWhw5ckRvqYqLi81ms95SnT59WusI0JIO5zpfbN++PSIiQm+/TT5q2rSpiARp\n+IiIiPT09GAMH7xzHefYAQAAGATFDgAAwCAodgAAAAZBsQMAADAIih0AAIBBUOwAAAAMgmIH\nAABgEBQ7AAAAg6DYAQAAGATfPNFYYWFhJpPJZrNpHeRcYWFhYWFhWqc4V1hYmA7fK5vNps/3\nSoepbDabyWTSYTCoTbdznS/0+dvko+BNLsH8zgfvXGdSFEXrDEHv0KFD7dq10zrFuQ4fPpye\nnm4262tRtqSkJCYmJjo6WusgP+N0Oh0OR3JystZBfsbj8Rw9erRNmzZaBzmXPj/wCIDg/U9f\nXV1dWlraqlUrrYNcjrKyMhFJTEzUOsjlKCgoaN68eTDWIwnaDzzFDgAAwCD0tZwDAACAy0ax\nAwAAMAiKHQAAgEFQ7AAAAAyCYgcAAGAQFDsAAACDoNgBAAAYBMUOAADAICh2AAAABkGxAwAA\nMAiKHQAAgEFQ7AAAAAyCYgcAAGAQFDsAAACDoNgBAAAYBMXuF5w5c2bq1Klt2rQJCwtLTU0d\nP358UVGR358SmGBlZWVPPvlk69atw8PD27ZtO2LEiG+//VbzVPU98cQTJpNp/Pjxekj15Zdf\nDhw4MDY2NiEhYfDgwV9//bXmqfbv3z927NgWLVrYbLakpKQ77rhj69at/k3lVVNT8/TTT1ss\nlqysLF/GB+YzD1Xpdq7zhT7nQx/pc9r0kT5nVx/peRJuFAUXV1VV1bNnTxG566675syZk52d\nbbPZ2rZte/r0aT8+JTDBTp061aZNGxG55ZZbnnvuufvuu89qtUZERPzwww8apqpv27ZtFotF\nRB5++GF/RbrsVO+9956ItG/f/tlnn33yySeTkpLCwsI2bdqkYaoff/wxNja2SZMmzz///NKl\nS1988cWUlBSr1bp27Vp/pfLat29fz549Y2NjzWZzr169fnF8YD7zUJVu5zpf6HM+9JE+p00f\n6XN29ZGeJ+FGotg15NVXXxWR+fPn1+359NNPRWT69Ol+fEpggk2ZMkVE3njjjbo9n3/+uYgM\nGzZMw1R1ampqevTo0b17d7/PUJeRqqSkJCYm5qqrrnI4HN49OTk5MTExkydP1jDVvffeKyJf\nffVV3Z7vv/9eRK6//np/pVIU5ezZs5GRkVlZWTk5OeHh4b4Uu8B85qEq3c51vtDnfOgjfU6b\nPtLn7Ooj3U7CjUexa0iPHj1iY2MrKyvr7+zQoUPz5s09Ho+/nhKYYFOnTh0yZEh1dXXdHo/H\nExkZ2bp1aw1T1XnppZdMJtOXX37p9xnqMlK9/PLLIvJ///d/9Xf68T/f5aXq06ePiNT/L6go\nSlxcXJs2bfwY7NSpU9OnT/f+LT4Wu8B85qEq3c51vtDnfOgjfU6bPtLn7Ooj3U7Cjcc5dhdV\nWVm5Z8+e3r17h4eH199/7bXXlpaW5ufn++UpgQkmIq+99tqaNWtsNlvdnurqarfb3apVKw1T\neeXl5c2aNWvixIl9+/b1S5hGplqzZk1kZOTgwYNFpKqqqry8XERMJpO2qTp16iQiBw4cqNtz\n8uRJh8ORmZnpr2Ai0qRJk4ULF9b/nDQsMJ95qEq3c50v9Dkf+kif06aP9Dm7+kjPk3DjUewu\n6tixY7W1tWlpaefsb926tYgcOnTIL08JTLALevvtt2tqakaPHq15qgkTJiQkJMybN88vSRqf\nav/+/W3btv3xxx+vvfbayMjI+Pj4Dh06vP/++9qmmjFjRmJi4v33379x48bi4uJdu3aNHj06\nIiJi5syZ/gp2GQLzmYeqdDvX+UKf86GP9Dlt+kifs6uPjDQJn49id1F2u11EoqOjz9kfExNT\n92jjnxKYYOdbv379U089de21106cOFHbVO+///7atWvfeOON+Ph4vyRpfKrTp087nc5bbrml\nb9++K1as+MMf/lBTU/PQQw999NFHGqbKzMzcsmVLTU3NgAEDWrRo0bNnz5ycnDVr1niPDmgl\nMJ95qEq3c50v9Dkf+kif06aP9Dm7+shIk/D5rFoH0Lvzl4gVRbng/sY8JTDB6nz88ccPPfRQ\n165dV61aZbX68zNwqalKS0unT59+66233nXXXX6M0chU1dXVR44c+eCDDx544AHvnpEjR3bs\n2HH69OmjRo3yXoMW+FQ//fTTLbfc4na7X3nllY4dO5aWlr766qs333zzZ599NnToUL9EumyB\n+cxDVbqd63yhz/nQR/qcNn2kz9nVR0aahOuj2F1UXFycXKi5e88JiI2N9ctTAhOsjqIoL7zw\nwuzZs2+66ably5f7K9Jlp3r88cerq6sXL17srxh+SRUTE+N2u+++++66PS1atLj55ptXrFix\nb9++K6+8UpNU2dnZJSUlBw8ebNmypXfP6NGjO3bsOG7cuPz8fN/PivOvwHzmoSrdznW+0Od8\n6CN9Tps+0ufs6iMjTcLn41DsRaWnp1ut1iNHjpyzPy8vT0QyMjL88pTABPNSFGX8+PGzZ89+\n7LHHvvjiC//OYpeR6ssvv/zkk0+mTZtmNpsLCgoKCgqOHz8uIi6Xq6CgwPsLFvhUIuK9wdU5\nv6VJSUnip2NMl5HK4XB89913ffr0qZtQRCQqKmrIkCGFhYUHDx5sfKrLE5jPPFSl27nOF/qc\nD32kz2nTR/qcXX1kpEn4AgJ+HW4w6dOnT1RUlNPprNtTW1ubmpqalpbmx6cEJpiiKI8//riI\nzJ07149JGpNq+vTpDXwyZ8yYoUkqRVEeffRREfn222/r77zxxhtF5OjRo5qkKi0tFZFrrrnm\nnP333HOPiGzfvt0vqc7h4+1OAvOZh6p0O9f5Qp/zoY/0OW36SJ+zq4+CYhK+PBS7hrzzzjsi\n8sILL9TtWbJkiYjMmjXLu1lRUbFr167c3Fzfn6JVMO/tNx9//HE/xmhkqn379v3z5z755BMR\nufHGG//5z3/+9NNPmqRSFGX79u0mk2nw4MF1tzjatm2b2Wzu1q2bXyJdXqq2bdvabLYDBw7U\n7SkrK2vSpElcXNw5t2LylwsWO60+81CVbuc6X+hzPvSRPqdNlcIrAZldfRQUk/Dlodg1xO12\nDxgwQERuv/32WbNmjR492mQyXXnllXUdf8+ePSIyZMgQ35+iVbD27duLyGOPPTbjPP76CqDL\nSHWOsrIy8fedNi8v1dSpU0WkR48es2bNeuSRRyIjI8PCwtatW6dhqpUrV5rN5qZNm/7ud797\n77335syZ07ZtWxFZvHixv1IpivL111/XfTAsFktKSkrd5smTJy8YLDCfeahKt3OdSuEDMB+q\nF/4cakybPtLn7Kpe+MBMwo1HsfsFdrvd+0XRNputZcuWU6ZMOXXqVN2jF/zUNvwUrYI1sHqf\nn5+vVapzqDRDXUYqj8fz1ltvde/ePSIiIj4+ftiwYVu3bkthqXIAAApHSURBVNU81ebNm0eM\nGJGUlGS1WhMTE4cOHbp69Wr/pmrgtlg5OTkXCxaYzzxUpdu5zhf6nA9VCn8ODYudotfZ1Uf6\nnIQbz6Q0+BEHAABAsOCqWAAAAIOg2AEAABgExQ4AAMAgKHYAAAAGQbEDAAAwCIodAACAQVDs\nAAAADIJiBwAAYBAUOwAAAIOg2AEAABgExQ4AAMAgKHYAAAAGQbEDAAAwCIodAACAQVDsAAAA\nDIJiBwAAYBAUOwAAAIOg2AEAABgExQ4AAMAgKHYAAAAGQbEDAAAwCIodAACAQVDsAAAADIJi\nBwAAYBAUOwAAAIOg2AEAABgExQ4AAMAgKHYAAAAGQbEDAAAwCIodAACAQVDsAAAADIJiBwAA\nYBAUOwAAgtjo0aNNJlNBQYHWQaALFDsEDbfbfc0115jN5q+++uqchyoqKq644orw8PDvv/9e\nk2wAAOgBxQ5Bw2q1Llu2LDY2dty4cWfOnKn/0NNPP33w4ME5c+Z0795dq3gAAGiOYodg0q5d\nu8WLFx87dmzKlCl1O7/55pvXX3990KBB06dP1zAbAACao9ghyNx///333XffRx99tHz5chFx\nOBzjxo1LSEhYunSpyWTSOh0Ag6uqqnr55Ze7d+8eHx8fGxvbrVu3l19+2ePx1A3YunXrHXfc\n0axZs7CwsDZt2owdO/bw4cN1j957770mk+nMmTMTJkxITk6Oiorq27fv1q1bXS7X1KlTW7Zs\nGRMT069fv507d9Y95Y477jCZTEVFRePHj09OTg4PD+/UqdOSJUsulrCkpGTKlCmtW7cOCwtL\nSkoaMWLEtm3bLjZ4zZo1ZrP53nvvrb9z2LBhFotl48aNl/cWQWMKEGzOnj3brl27xMTEgoKC\niRMnisjy5cu1DgUgJDz00EMicu+99y5ZsuStt9664447RGTKlCneR7dv3x4REZGamjp79ux3\n3nnnt7/9bWxsbPPmzU+ePOkd8OCDD4rI0KFDZ82atXPnzvfffz8iIiI9Pf3WW2+dMWPG9u3b\nP/vss4SEhOTk5Orqau9TRo0aJSK9e/eeMWPGpk2bNmzYcMMNN4jIu+++W3/AsWPHFEUpLS1t\n3bp1fHz8jBkzPvzww7lz57Zq1So8PPzrr7++2E/knUX/85//eDc/++wzEZk2bZpKbyDURrFD\nUNqyZYvVau3SpYuIPPjgg1rHARAqoqKirrnmmvp7pk2bdtddd7ndbkVR3nzzzZ49e65bt67u\n0TfeeENE3njjDe/mww8/LCKTJk2qG3DPPfeIyN1331235/HHHxeRTZs2eTe9vW3MmDF1A86c\nORMeHt6mTZv6A7zFbtKkSVarddu2bXWDjx49Ghsbm5WVdbGfyG63t2nTJiMjo7Ky0uFwpKWl\ndezY0eVyXeo7A52wBn6NEGi8vn37Tp8+ff78+TExMa+//rrWcQCECpvNduTIkdLS0ubNm3v3\nvPrqq3WPTpo0adKkSd4/19TU1NbWdu7cWUTqH40VkTvvvLPuzxkZGSJy++231+254oorRKSo\nqKj+U0aPHl335/j4+AEDBqxZs6aoqKhFixZ1+xVFWbFiRbdu3Vq1alVcXFwXuF+/fv/6178c\nDkdMTMz5P1FMTMx77703ZMiQefPmOZ3OwsLCjRs3RkZGXsKbAj2h2CEoVVZWfvHFF2az2eFw\nfPLJJ7/+9a+1TgQgJMyePfvxxx/PyMi4/fbbBw0adOONN7Zs2bL+gA8//PBPf/rTDz/8UP/i\nfbfbXX9M/adYrdZz9thsNhGpqamp/5SOHTue/wrFxcX1i11paenJkydPnjxZf2edo0ePelvm\n+QYNGjRp0qSXXnrJ4/FMnz79mmuuudiPD/2j2CEoPfHEE3v37l2xYsXs2bOfeOKJIUOGtG/f\nXutQAIzvN7/5TdeuXd94442VK1d++OGHJpPp5ptvfvPNN1u3bi0izzzzzLx587Kysl577bW2\nbduGh4fv3bt3/Pjx57yIt7o1vOccUVFR9Tejo6NF5JwbP9ntdhHp0aPHvHnzzn+F1NTUBl7/\n4YcffvPNN0XkgQceaDgJdI5ih+Dzj3/8Y8mSJdnZ2XfffXf79u379OkzduzYDRs2WCwWraMB\nML7BgwcPHjy4qqpqw4YNf/3rX5cuXTp06NC9e/d6PJ5FixalpaWtW7eu7qDn2bNn/fKXOp3O\n+pvel23atGn9nbGxsd4/3HTTTZf04h6P59FHH01OTna73ZMnT16/fj03GQhe3O4EQeb48ePZ\n2dlt2rRZtGiRiFx11VXPPPPMli1bLvgvVABQSXh4+NChQ99///2JEyfm5ubu3r27uLi4oqIi\nKyur/qls69ev98tf99NPP9XfzMnJEZFzDrkmJyc3a9Zs//7956zknThxouEXf/XVV7ds2fKH\nP/xh4cKFGzZs4MTloEaxQzDxeDwPPPBAWVnZBx98UPdv09/97ndXXXXV7Nmzd+zYoW08AMb2\n7bfftmzZcunSpfV3ms1mEbHZbMnJySaTqf51Ert37/YOrqysbORf/d5779X9+eDBg9u2bbvi\niiuSkpLOGTZy5MjKysqXX365bs+JEye6det22223eTcrKyt3796dl5dX/9Wef/75YcOGjRo1\naty4cYMGDXrmmWe8xRHBiEOxCCYLFixYu3bt9OnTr7vuurqdNpvtgw8+yMrKuv/++3fu3MnF\nXABUkpWV1aRJk0ceeWTjxo09evQwmUzbt29///33r732Wu/mLbfc8sUXX0ycOPH666/ft2/f\nH//4x2XLlg0fPnz16tUff/zx8OHDL/uvrqqquu2222699VaPx7NgwQJFUZ5//vnzh73wwgur\nV6+eO3duUVHRwIEDjx8//tZbb506deo3v/mNd0Bubu5VV101ZMiQNWvWiIjH4xk3bpzZbPae\nYCcib731Vrdu3R566KFvvvnG21kRZLS+3wrgq61bt9psti5dulRWVp7/6Jw5c0TkscceC3ww\nAKHj1KlTU6dObd++fVRUVHx8fPfu3efOnWu3272PlpaW3nvvvUlJSfHx8YMHD96wYYOiKLNm\nzYqJiUlJSSkqKvLexy4nJ6fuBWfOnCki3pFe7777roh8/PHH3k3vbepycnKmTp2ampoaFhbW\nuXPn999/v258/fvYKYpSVFQ0adKktLQ0q9WakJAwfPjw7777rm7wnj17RGTIkCHezYULF4rI\nq6++Wv9nnD17toi88sorfnzfEDAmRVE0K5UAAKBBo0eP/vTTT48dO9aqVSutsyAIsMoKAABg\nEBQ7AAAAg6DYAQAAGATn2AEAABgEK3YAAAAGQbEDAAAwCIodAACAQVDsAAAADIJiBwAAYBAU\nOwAAAIOg2AEAABgExQ4AAMAgKHYAAAAGQbEDAAAwCIodAACAQVDsAAAADIJiBwAAYBAUOwAA\nAIOg2AEAABgExQ4AAMAgKHYAAAAGQbEDAAAwCIodAACAQVDsAAAADIJiBwAAYBAUOwAAAIOg\n2AEAABgExQ4AAMAgKHYAAAAGQbEDAAAwiP8HOMoAKd/l2OEAAAAASUVORK5CYII=",
            "text/plain": [
              "Plot with title “Histogram of candidate density”"
            ]
          },
          "metadata": {
            "tags": [],
            "image/png": {
              "width": 420,
              "height": 420
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dQYw9ftrSVei"
      },
      "source": [
        "## Importance Sampling\n",
        "\n",
        "Importance sampling is based on an alternative formulation of the SLLN\n",
        "\n",
        "- For notational convinience we assume $p(\\theta|y)=f(\\theta)$\n",
        "\n",
        "\\begin{eqnarray*}\n",
        "E_f(h(\\theta))&=&\\int_{\\Theta}h(\\theta)f(\\theta)d\\theta \\\\\n",
        "&=&\\int_{\\Theta}h(\\theta)\\frac{f(\\theta)}{g(\\theta)}g(\\theta)d\\theta \\\\ \n",
        "&=& E_g\\left[\\frac{h(\\theta)f(\\theta)}{g(\\theta)}\\right]\n",
        "\\end{eqnarray*}\n",
        "\n",
        "1. $f$ is the target density\n",
        "2. $g$ is the cadidate density\n",
        "\n",
        "So in '**Importance Sampling**' all you do is first you generate samples from candidate distribution $g$\n",
        "\n",
        "- Suppose $(\\theta^1,\\theta^2,...,\\theta^N)$ are the random samples generated from $g$\n",
        "\n",
        "- You estimates \n",
        "$$\n",
        "\\bar{h}=\\frac{1}{N}\\sum_{i=1}^N\\frac{f(\\theta^i)}{g(\\theta^i)}h(\\theta^i)\n",
        "$$\n",
        "and by virtue of SLLN we have\n",
        "$$\n",
        "\\bar{h}=\\frac{1}{N}\\sum_{i=1}^N\\frac{f(\\theta^i)}{g(\\theta^i)}h(\\theta^i) \\longrightarrow E_g\\left[\\frac{h(\\theta)f(\\theta)}{g(\\theta)}\\right]=E_f(h(\\theta))\n",
        "$$\n",
        "\n",
        "The logic underlying importance sampling lies in a simple rearrangement of terms in the target integral and multiplying by 1:\n",
        "$$\\int h(\\theta)f(\\theta)d\\theta=\\int h(\\theta)\\frac{f(\\theta)}{g(\\theta)}g(\\theta)d\\theta=\\int h(\\theta)\\omega(\\theta)g(\\theta)d\\theta$$\n",
        "here $g()$ is another density whose support is same as $f()$.\n",
        "\n",
        "- $\\omega()$ is called importance function\n",
        "\n",
        "- a good importance function will be large when the integrand is large and small otherwise."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AV3ZuO4uUwdB"
      },
      "source": [
        "### **Objectives of Importance sampling**\n",
        "\n",
        "1. It reduces the variance of an integral approximation.\n",
        "\n",
        "2. Another objective of IS when you cannot generate from the density $p$\n",
        "\n",
        "3. A third objective of IS is to draw inference or generate sample when the target density is unnormalized \n",
        "\n",
        "### **Ex: Importance Sampling to improve integral approximation**\n",
        "\n",
        "Consider the function $h(x)=10\\exp(-2|x-5|)$. Suppose that we want to calculate $E(h(X))$, where $X\\sim  Uniform(0,10)$. That is we want to calculate\n",
        "$$\n",
        "\\int_0^{10} \\exp(-2|x-5|)dx.\n",
        "$$\n",
        "The true value for this integral is about 1. \n",
        "\n",
        "- The simple way to do this is to generate $x_i$ from the $Uniform(0,10)$ and look at the sample mean of $h(x_i)$\n",
        "\n",
        "- Notice this is equivalent to importance sampling with importance function $\\omega(x)=f(x)$, where $f(x)=\\frac{1}{10}I(0<x<10)$\n",
        "\n",
        "- Note that $h(x)=10\\exp(-2|x-5|)$ where $x \\sim Unif(0,10)$\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "9Xojd18iVwEs",
        "outputId": "23b40372-cf89-46f2-b4e7-63fc9a62fbbb"
      },
      "source": [
        "set.seed(7831)\n",
        "sim.size = 50000\n",
        "X = runif(sim.size,0,10)\n",
        "Y = 10*exp(-2*abs(X-5))\n",
        "cat('Estimate = ',round(mean(Y),3),'\\n')\n",
        "cat('Monte Carlo Error = ',round(var(Y),3),'\\n')\n",
        "\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Estimate =  0.999 \n",
            "Monte Carlo Error =  3.993 \n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 437
        },
        "id": "3TWvBQv2WZn3",
        "outputId": "84e2d9b6-1444-4a4c-b48a-aeb390f83cff"
      },
      "source": [
        "x = seq(0,10,0.1)\n",
        "h = 10*exp(-2*abs(x-5))\n",
        "plot(x,h,ylab=\"h(x)\",type=\"l\",lwd=2)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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itejf1kJ7GJmZC2jlgd8XKGFL8/gMnxXfMuYJPsLBYxE9IPmBvxcoYUv0+AV+K7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\nB6JwKn006AW3AAAAAElFTkSuQmCC",
            "text/plain": [
              "plot without title"
            ]
          },
          "metadata": {
            "tags": [],
            "image/png": {
              "width": 420,
              "height": 420
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "NSIqlES3WkbT"
      },
      "source": [
        "- Note that $h(x)=10\\exp(-2|x-5|)$ where $x \\sim Unif(0,10)$\n",
        "\n",
        "- The function $h()$ in this case is peaked at 5, and decays quickly elsewhere, therefore, under the uniform distribution, many of the points are contributing very little to this expectation.  \n",
        "\n",
        "- Something more like a gaussian function $ce^{-x^2}$ with a peak at 5 and small variance, say, 1, would provide greater precision\n",
        "\n",
        "- We can re-write the integral as\n",
        "$$\n",
        "\\int_{0}^{10}10\\exp(-2|x-5|)\\frac{1/10}{\\frac{1}{\\sqrt{2\\pi}}e^{-(x-5)^2/2}}\\frac{1}{\\sqrt{2\\pi}}e^{-(x-5)^2/2}dx\n",
        "$$\n",
        "\n",
        "- We want to estimate $E[h(X)\\omega(X)]$ where $X\\sim N(5,1)$\n",
        "\n",
        "- So in this case $f(x)\\sim Uniform(0,10)$, $g(x) \\sim N(5,1)$\n",
        "\n",
        "- The weight function is \n",
        "$$\n",
        "\\omega(x)=\\frac{\\sqrt{2\\pi}e^{(x-5)^2/2}}{10}\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "mMeOp3ahXDzg",
        "outputId": "8836f4f7-19c1-46ea-b029-e404b2252c8f"
      },
      "source": [
        "w = function(x)dunif(x,0,10)/dnorm(x,mean=5,sd=1)\n",
        "h = function(x)10*exp(-2*abs(x-5))\n",
        "set.seed(7831)\n",
        "sim.size = 50000\n",
        "x = rnorm(sim.size,mean=5,sd=1)\n",
        "y = h(x)*w(x)\n",
        "cat('Estimate = ',round(mean(y),3),'\\n')\n",
        "cat('Monte Carlo Error = ',round(var(y),3),'\\n')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Estimate =  1 \n",
            "Monte Carlo Error =  0.361 \n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3NInU1vBXbS_"
      },
      "source": [
        "Note that here Monte Carlo variance is much smaller\n",
        "\n",
        "\n",
        "## **Importance Sampling - Application**\n",
        "\n",
        "Suppose daily return of a stock is defined as \n",
        "$$\n",
        "r_t=\\frac{P_t-P_{t-1}}{P_{t-1}}\\times 100\n",
        "$$\n",
        "where $P_t$ is the price of the stock on $t^{th}$ day. Suppose it is believed that return of the stock follows $r_t \\sim N(0,1)$.\n",
        "\n",
        "- **Objective**: We want to estimate $P(r_t<-5)$, i.e., we want to estimate the probability that the price of the stock will drop more than $5\\%$ in one day?\n",
        "\n",
        "As we are interested in $P(r_t<-5)$ where $r_t\\sim N(0,1)$, the probability in log-scales, i.e., $\\ln(P(r_t<-5))$\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "id": "bkQT8KPWX344",
        "outputId": "27347566-651d-4f16-8af7-c3ec2cb9332c"
      },
      "source": [
        "log(pnorm(-5,mean=0,sd=1))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "[1] -15.065"
            ],
            "text/latex": "-15.0649983939887",
            "text/markdown": "-15.0649983939887",
            "text/html": [
              "-15.0649983939887"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DCGr3VuXX7Tw"
      },
      "source": [
        "- Simulating $Z^{(i)}$ from $N(0,1)$ only produces a hit once in about 3 million iterations !\n",
        "\n",
        "- This is a very **rare event** for standard normal distribution\n",
        "\n",
        "- Estimates from Importance sampling"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "FwJ0cOb1YJ51",
        "outputId": "f9ff071b-6e9e-4c69-a90d-fcff356f19ee"
      },
      "source": [
        "set.seed(5472)\n",
        "a = -5\n",
        "x = rnorm(10000,mean=a,sd=2)\n",
        "y = rep(0,length(x))\n",
        "y[x<a] = 1\n",
        "#print(length(y))\n",
        "g = dnorm(x,mean=a,sd=2)\n",
        "f = dnorm(x)\n",
        "w = f/g\n",
        "print(log(mean(w*y)))\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[1] -15.0732\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 242
        },
        "id": "ePpgv4LgZw14",
        "outputId": "5f12d432-6add-4a2f-d0a8-0bfce5d9777a"
      },
      "source": [
        "result = data.frame(matrix(NA,nrow=5,ncol=3))\n",
        "colnames(result) = c('Target','Built-in R','Importance Sampling')\n",
        "result[1,1] = 'ln(P(r < -1)'\n",
        "result[1,2] = -1.841\n",
        "result[1,3] = -1.865\n",
        "\n",
        "result[2,1] = 'ln(P(r < -2)'\n",
        "result[2,2] = -3.783 \n",
        "result[2,3] = -3.803\n",
        "\n",
        "result[3,1] = 'ln(P(r < -3)'\n",
        "result[3,2] = -6.608\n",
        "result[3,3] = -6.624\n",
        "\n",
        "result[4,1] = 'ln(P(r < -4)'\n",
        "result[4,2] = -10.360\n",
        "result[4,3] = -10.372\n",
        "\n",
        "result[5,1] = 'ln(P(r < -5)'\n",
        "result[5,2] = -15.065 \n",
        "result[5,3] = -15.073\n",
        "\n",
        "result"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  Target       Built-in R Importance Sampling\n",
              "1 ln(P(r < -1)  -1.841     -1.865            \n",
              "2 ln(P(r < -2)  -3.783     -3.803            \n",
              "3 ln(P(r < -3)  -6.608     -6.624            \n",
              "4 ln(P(r < -4) -10.360    -10.372            \n",
              "5 ln(P(r < -5) -15.065    -15.073            "
            ],
            "text/latex": "A data.frame: 5 × 3\n\\begin{tabular}{lll}\n Target & Built-in R & Importance Sampling\\\\\n <chr> & <dbl> & <dbl>\\\\\n\\hline\n\t ln(P(r < -1) &  -1.841 &  -1.865\\\\\n\t ln(P(r < -2) &  -3.783 &  -3.803\\\\\n\t ln(P(r < -3) &  -6.608 &  -6.624\\\\\n\t ln(P(r < -4) & -10.360 & -10.372\\\\\n\t ln(P(r < -5) & -15.065 & -15.073\\\\\n\\end{tabular}\n",
            "text/markdown": "\nA data.frame: 5 × 3\n\n| Target &lt;chr&gt; | Built-in R &lt;dbl&gt; | Importance Sampling &lt;dbl&gt; |\n|---|---|---|\n| ln(P(r &lt; -1) |  -1.841 |  -1.865 |\n| ln(P(r &lt; -2) |  -3.783 |  -3.803 |\n| ln(P(r &lt; -3) |  -6.608 |  -6.624 |\n| ln(P(r &lt; -4) | -10.360 | -10.372 |\n| ln(P(r &lt; -5) | -15.065 | -15.073 |\n\n",
            "text/html": [
              "<table class=\"dataframe\">\n",
              "<caption>A data.frame: 5 × 3</caption>\n",
              "<thead>\n",
              "\t<tr><th scope=col>Target</th><th scope=col>Built-in R</th><th scope=col>Importance Sampling</th></tr>\n",
              "\t<tr><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
              "</thead>\n",
              "<tbody>\n",
              "\t<tr><td>ln(P(r &lt; -1)</td><td> -1.841</td><td> -1.865</td></tr>\n",
              "\t<tr><td>ln(P(r &lt; -2)</td><td> -3.783</td><td> -3.803</td></tr>\n",
              "\t<tr><td>ln(P(r &lt; -3)</td><td> -6.608</td><td> -6.624</td></tr>\n",
              "\t<tr><td>ln(P(r &lt; -4)</td><td>-10.360</td><td>-10.372</td></tr>\n",
              "\t<tr><td>ln(P(r &lt; -5)</td><td>-15.065</td><td>-15.073</td></tr>\n",
              "</tbody>\n",
              "</table>\n"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "T7Xrkt2vcDBt"
      },
      "source": [
        "## **Importance Sampling for  Bayesian Inference**\n",
        "\n",
        "In Bayesian inference is the primary example where you want to determine the properties of a probability distribution given upto its kernel function.\n",
        "\n",
        "- Suppose $k(\\theta)$ is the kernel function of the posterior distribution $p(\\theta|y)$, i.e., $$f(\\theta)=p(\\theta|y)=\\frac{k(\\theta)}{C},$$ where $C=\\int k(\\theta)d\\theta=\\int p(y|\\theta)p(\\theta)d\\theta$\n",
        "\n",
        "- Typically we do not know $C$ and we know the posterior distribution in its unnormalized form, i.e.,\n",
        "$$\n",
        "p(\\theta|y)\\propto p(y|\\theta)p(\\theta)=k(\\theta)\n",
        "$$\n",
        "\n",
        "- If you want to calculate $E[h(\\theta)]$ where unnormalized density of $f(\\theta)$ is $k(\\theta)$\n",
        "\n",
        "- We can rewrite this as\n",
        "\\begin{eqnarray*}\n",
        "E[h(\\theta)]&=&\\int h(\\theta)\\frac{k(\\theta)}{C}d\\theta=\\frac{1}{C}\\int h(\\theta)\\frac{k(\\theta)}{g(\\theta)}g(\\theta)d\\theta\\\\ \n",
        "&=& \\frac{1}{C}\\int h(\\theta)\\tilde{w}(\\theta)g(\\theta)d\\theta\n",
        "\\end{eqnarray*}\n",
        "where $\\omega(\\theta)=\\frac{f(\\theta)}{g(\\theta)}=\\frac{1}{C}\\frac{k(\\theta)}{g(\\theta)}=\\frac{1}{C}\\tilde{\\omega}(\\theta)$ and $f(\\theta)=p(\\theta|y)$ is the normalized posterior density"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MmJyggSUcc04"
      },
      "source": [
        "- By LLN if $\\theta^1,\\theta^2,...,\\theta^N$ are iid random samples from $g$, then \n",
        "$$\n",
        "\\frac{1}{N}\\sum_{i=1}^N h(\\theta^i)\\tilde{\\omega}(\\theta^i)\\longrightarrow C E[h(\\theta)]\n",
        "$$\n",
        "as $N \\longrightarrow \\infty$.\n",
        "\n",
        "- Also by LLN,\n",
        "$$\n",
        "\\frac{1}{N}\\sum_{i=1}^N \\tilde{\\omega}(\\theta^i) \\longrightarrow C\n",
        "$$\n",
        "\n",
        "- Therefore Monte Carlo estimator for $E[h(\\theta)]$ using Importance Sampling Scheme is \n",
        "$$\n",
        "\\bar{h}_{IS}=\\frac{\\sum_{i=1}^N h(\\theta^i)\\tilde{\\omega}(\\theta^i)}{\\sum_{i=1}^N\\tilde{\\omega}(\\theta^i)}\n",
        "$$\n",
        "\n",
        "- The method is only reliable when the weights are not too variable.\n",
        "\n",
        "- As a rule of thumb, when\n",
        "$$\n",
        "V=\\sqrt{\\frac{1}{N}\\sum_{i=1}^N\\left(\\frac{\\tilde{\\omega}(\\theta^i)}{\\bar{\\omega}}-1\\right)^2}\n",
        "$$\n",
        "is less than 5, the IS method is reasonable. Note that $\\bar{\\omega}=\\frac{1}{N}\\sum_{i=1}^N \\tilde{\\omega}(\\theta^i)$\n",
        "\n",
        "- You will know you chose a bad $g$ when $V$ is large."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pjPdPlr9eCM3"
      },
      "source": [
        "When $V<5$ the variance $\\bar{h}_{IS}$ can be estimated as \n",
        "$$\n",
        "\\sigma_{IS}^2=\\frac{1}{N}\\sum_{i=1}^N(\\zeta^i-\\bar{h}_{IS})^2\n",
        "$$\n",
        "where $$\\zeta^i=\\frac{\\tilde{\\omega}(\\theta^i)h(\\theta^i)}{\\bar{\\omega}}$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cTuSpbaPeUmD"
      },
      "source": [
        "## **Modeling Road Accident**\n",
        "\n",
        "- Suppose following data presents number of accidents every month for last 12 months in a particular stretch on natitonal highway 34 (NH34)\n",
        "\n",
        "- $6, 2, 2, 1, 2, 1, 1, 2, 3, 5, 2, 1$\n",
        "\n",
        "- We define $y_i$ as the number of accident on $i^{th}$ month, i.e., $y_1=6$, $y_2=2$,... $y_{12}=1$.\n",
        "\n",
        "- We assume $y_i\\stackrel{iid}{\\sim} \\text{Poisson}(\\theta)$ and $\\theta \\sim \\text{log-Normal}(\\mu=2,\\sigma=1)$\n",
        "\n",
        "- In this problem since a non-conjugate prior is being selected the posterior distribution is known upto normalizing constant.\n",
        "\n",
        "- Simulating random samples from the unnormalized posterior distribution is difficult\n",
        "\n",
        "- So we simulate from a candidate distribution and using the importance sampling method we estimate the $\\theta$ as"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "id": "juEkj646e2jx",
        "outputId": "a09c31fc-7649-4eb4-8779-eb8a6c922482"
      },
      "source": [
        "y = c(6, 2, 2, 1, 2, 1, 1, 2, 3, 5, 2,1)\n",
        "\n",
        "IS=function(y,C,N){\n",
        "  ## y : data\n",
        "  ## C : C is the tuning parameter for the g distribution\n",
        "  ## N : the number of monte carlo samples\n",
        "  \n",
        "  # sample size\n",
        "  n = length(y)\n",
        "  \n",
        "  # log-post\n",
        "  log.posterior = function(t)sum(dpois(y,t,log=TRUE))+dlnorm(t,2,1,log=TRUE)\n",
        "  \n",
        "  # parameters for the trial distribution\n",
        "  a = C*mean(y); b = C;\n",
        "  \n",
        "  # log trial density, g\n",
        "  log.g = function(t){\n",
        "      dgamma(t,a,b,log=TRUE)\n",
        "  } \n",
        "  \n",
        "  # log importance function\n",
        "  log.w = function(t) log.posterior(t) - log.g(t)\n",
        "  \n",
        "  # generate from the trial distribution\n",
        "  U = rgamma(N, a, b)\n",
        "  \n",
        "  # calculate the list of log.w values\n",
        "  LP = log.w(U)\n",
        "  \n",
        "  # factor out the largest value to prevent numerical underflow\n",
        "  w = max(LP)\n",
        "  LP = LP - w\n",
        "  \n",
        "  # importance sampling estimate\n",
        "  I = mean( exp(LP)*U )/mean(exp(LP))\n",
        "  \n",
        "  # calculate V\n",
        "  V = mean( ( (exp(LP)/mean(exp(LP))) - 1)^2 )\n",
        "  \n",
        "  # calculate s.e. of the IS estimate\n",
        "  Z = exp(LP)/mean(exp(LP))\n",
        "  sig.sq = (1/N)*sum( (Z-I)^2 )\n",
        "  se = sqrt( sig.sq/N )\n",
        "  # return the 95 percent confidence interval for the estimate\n",
        "  return(c(I - 1.96*se, I + 1.96*se, V, I))\n",
        "}\n",
        "set.seed(5818)\n",
        "\n",
        "\n",
        "## calculate V, E(theta|X), and var(E(theta|X))\n",
        "## for a grid of values for c\n",
        "const = seq(.05, 20, length=500)\n",
        "A = matrix(0, 500, 2)\n",
        "\n",
        "for(j in 1:500){\n",
        "  OUT = IS(y, const[j], 1000)\n",
        "  V = OUT[3]\n",
        "  Var = (OUT[2]-OUT[1])/3.92\n",
        "  A[j,] = c(V, Var)\n",
        "}\n",
        "\n",
        "# final Estimate\n",
        "IS(y, const[which.min(A[,2])], 1000)[4]\n",
        "\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "[1] 2.737056"
            ],
            "text/latex": "2.73705619939327",
            "text/markdown": "2.73705619939327",
            "text/html": [
              "2.73705619939327"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Kod2V6lIfaec"
      },
      "source": [
        "**Importance Sampling**\n",
        "\n",
        "- Note that simulated ($\\theta^1,...,\\theta^N$) is not from the target density ... it is from the candidate density $g$\n",
        "\n",
        "- IS is an estimation method\n",
        "\n",
        "- Later we will see that simulation from target can be done by adding step to IS "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "vvyT3lkRdyNQ"
      },
      "source": [
        "\n",
        "REf : Lecture Notes of Prof Art Owen at: https://statweb.stanford.edu/~owen/mc/Ch-var-is.pdf"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "33_wish0MLkI"
      },
      "source": [
        "## **Markov Chain Monte Carlo**\n",
        "\n",
        "Ref: http://patricklam.org/teaching/mcmc_print.pdf\n",
        "\n",
        "- **Markov Chain** a stochastic process in which future states are independent of past states given the present state\n",
        "- **Monte Carlo** simulation technique\n",
        "- Up until now, we have done Monte Carlo simulation to find integrals instead of doing it analytically. This process is called **Monte Carlo Integration**.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zcXh9WUXOpaQ"
      },
      "source": [
        "- Suppose we have a distribution $f(\\theta)$ (perhaps a posterior) from which we want to draw samples.\n",
        "\n",
        "- To derive it analytically, we need to take integrals:\n",
        "$$\n",
        "I=\\int_{\\Theta}h(\\theta)f(\\theta)d\\theta\n",
        "$$\n",
        "where $h(\\theta)$ is some function of $\\theta$ ($h(\\theta) = \\theta$ for the mean and $h(\\theta) = (\\theta - E(\\theta))^2$ for the variance).\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xCmHtMgdOyKh"
      },
      "source": [
        "- We can approximate the integrals via Monte Carlo Integration by\n",
        "simulating $M$ values from $p(\\theta)$ and calculating\n",
        "$$\n",
        "\\hat{I}=\\frac{1}{M}\\sum_{i=1}^Mh(\\theta^i)\n",
        "$$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "oRSYNzt7QoIQ"
      },
      "source": [
        "- For example, we can compute the expected value of the $Beta(3,4)$\n",
        "distribution analytically:\n",
        "$$\n",
        "E(\\theta)=\\int_{0}^1 \\theta f(\\theta)d\\theta=\\int_{0}^1 \\theta \\theta^{3-1}(1-\\theta)^{4-1}d\\theta=\\frac{3}{7}=0.4286\n",
        "$$\n",
        "or via Monte Carlo methods:\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "id": "e_d4lkQbQtEK",
        "outputId": "78943be0-9c23-4f4d-a776-d9b901d65a45"
      },
      "source": [
        "set.seed(88732)\n",
        "M <- 50000\n",
        "beta.sims <- rbeta(M, 3, 4)\n",
        "sum(beta.sims)/M"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "[1] 0.4288105"
            ],
            "text/latex": "0.428810523264182",
            "text/markdown": "0.428810523264182",
            "text/html": [
              "0.428810523264182"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sofjKjdAQzYZ"
      },
      "source": [
        "- Our Monte Carlo approximation $\\hat{I}$ is a simulation estimator such that\n",
        "$\n",
        "\\hat{I}\\longrightarrow I\n",
        "$\n",
        "as \n",
        "$\n",
        "M \\longrightarrow \\infty\n",
        "$\n",
        "- This follows from LLN.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "myWp6b8_Q6KV"
      },
      "source": [
        "- Let $\\theta_1,\\theta_2, \\cdots$ be a sequence of **independent** and identically distributed random variables, each having a finite mean $\\mu =E(\\theta_i)$.\n",
        "- Then with probability 1,\n",
        "$$\n",
        "\\frac{\\theta_1+\\theta_2+...\\theta_M}{M}\\longrightarrow \\mu ~~~ as ~~ M \\longrightarrow \\infty\n",
        "$$\n",
        "- In the example, each simulated random sample was independent and distributed from the same $Beta(3,4)$ distribution.\n",
        "- But what if we cannot generate draws that are **independent**?\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xI-KPEpYRifp"
      },
      "source": [
        "### **Fail to Draw Independent Random Samples**\n",
        "\n",
        "- Suppose we want to draw from our posterior distribution $p(\\theta|y)$,\n",
        "but we cannot draw independent sample from it.\n",
        "- For example, we often do not know the normalizing constant.\n",
        "- We may be able to sample draws from $p(\\theta|y)$ that are slightly dependent.\n",
        "- If we can draw slightly dependent samples using a **Markov chain**, then can we solve the integration?\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bWxLm1BxXpgC"
      },
      "source": [
        "## **What is a Markov Chain?**\n",
        "\n",
        "- **Definition**: Markov Chain is a stochastic process in which future states are independent of past states given the present state\n",
        "- **Stochastic process**: a consecutive set of random (not deterministic) quantities defined on some known state space $\\Theta$.\n",
        "- Think of $\\Theta$ as our parameter space\n",
        "- Consecutive implies a time component, indexed by $t$."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jNgPzZIMYEyq"
      },
      "source": [
        "- Consider a draw of $\\theta^{(t)}$ to be a state at iteration $t$.\n",
        "- The next draw $\\theta^{(t+1)}$ is dependent only on the current draw $\\theta^{(t)}$, and not on any past draws.\n",
        "- **Markov Property**:\n",
        "$$\n",
        "p(\\theta^{(t+1)}|\\theta^{(t)},\\theta^{(t-1)},...,\\theta^{(1)})=p(\\theta^{(t+1)}|\\theta^{(t)})\n",
        "$$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "erxV5O9vYWuz"
      },
      "source": [
        "- So our Markov chain is a bunch of draws of $\\theta$ that are each dependent on the previous draw. \n",
        "- The chain wanders around the parameter space, remembering only where it has been in the last time.\n",
        "- What are the rules that govern how the chain jumps from one state to another at each time?\n",
        "- The rule that govern the jump are known as **transition kernel**."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BeIBybWdYn_u"
      },
      "source": [
        "## **Transition Kernel**\n",
        "\n",
        "- For discrete state space ($k$ possible states): a $k \\times k$ matrix of\n",
        "transition probabilities.\n",
        "- **Example**: Suppose $k = 3$. The $3 \\times 3$ transition matrix \\textbf{P} would be\n",
        "$$\n",
        "\\begin{table}\n",
        "\\begin{tabular}{|c|c|c|}\\hline\n",
        "$p(\\theta_1^{(t=1)}|\\theta_1^{(t)})$ & $p(\\theta_2^{(t=1)}|\\theta_1^{(t)})$ & $p(\\theta_3^{(t=1)}|\\theta_1^{(t)})$ \\\\ \\hline\n",
        "$p(\\theta_1^{(t=1)}|\\theta_2^{(t)})$ & $p(\\theta_2^{(t=1)}|\\theta_2^{(t)})$ & $p(\\theta_3^{(t=1)}|\\theta_2^{(t)})$ \\\\ \\hline\n",
        "$p(\\theta_1^{(t=1)}|\\theta_3^{(t)})$ & $p(\\theta_2^{(t=1)}|\\theta_3^{(t)})$ & $p(\\theta_3^{(t=1)}|\\theta_3^{(t)})$ \\\\ \\hline\n",
        "\\end{tabular}\n",
        "\\end{table}\n",
        "$$\n",
        "where the subscripts index the 3 possible values that $\\theta$ can take.\n",
        "- The rows sum to one and define a conditional probability mass function, conditional on the current state.\n",
        "- The columns are the marginal probabilities of being in a certain state in the next time.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Tomfs3BFZDqa"
      },
      "source": [
        "1. Define a starting distribution $\\Pi^{(0)}$ (a $1 \\times k$ vector of probabilities that sum to one).\n",
        "- At iteration 1, our distribution $\\Pi^{(1)}$ (from which $\\theta^{(1)}$ is drawn) is\n",
        "$$\n",
        "\\Pi_{(1 \\times k) }^{(1)} = \\Pi_{(1 \\times k) }^{(0)} \\times \\textbf{P} \n",
        "$$\n",
        "- At iteration 2, our distribution $\\Pi^{(2)}$ (from which $\\theta^{(2)}$ is drawn) is\n",
        "$$\n",
        "\\Pi_{(1 \\times k) }^{(2)} = \\Pi_{(1 \\times k) }^{(1)} \\times \\textbf{P} \n",
        "$$\n",
        "- At iteration t, our distribution $\\Pi^{(t)}$ (from which $\\theta^{(t)}$ is drawn) is\n",
        "$$\n",
        "\\Pi_{(1 \\times k) }^{(t)} = \\Pi_{(1 \\times k) }^{(t-1)} \\times \\textbf{P}=\\Pi_{(1\\times k)}^{(0)} \\times \\textbf{P}^t \n",
        "$$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bO2iPjJ7ZW3N"
      },
      "source": [
        "## **Stationarity of Markov Chain**\n",
        "\n",
        "- Define a stationary distribution $\\pi$ to be some distribution $\\Pi$ such that $\\pi=\\pi\\textbf{P}$\n",
        "\n",
        "- Here we blur the distinction between the Markov chain and its transtion matrix $\\textbf{P}$.\n",
        "\n",
        "- **If posterior distribution is proper then the Markov Chain of the MCMC algorithm will typically converge to $p(\\theta|y)$ regardless of our starting points.**\n",
        "\n",
        "- **So if we can devise a Markov chain whose stationary distribution $\\Pi$ is our desired posterior distribution $p(\\theta|y)$ then we can run this chain to get draws that approximately from $p(\\theta|y)$ once the chain has converged.**\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Hw7jQEoOZt_Z"
      },
      "source": [
        "## **Fundamental Theorem of Markov Chain**\n",
        "\n",
        "- If a Markov chain $\\textbf{P}$ is $\\textit{irreducible}$, $\\textit{aperiodic}$ and $\\textit{positive recurret}$ then it has an unique stationary distribution $\\pi$.  \n",
        "\n",
        "- This is the unique (normalized such that the entries sum to 1) left eigenvector of $\\textbf{P}$ with eigenvalue 1. \n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "F7j0Zq_waEhQ"
      },
      "source": [
        "## **Burn-In**\n",
        "\n",
        "- In light of the Fundamental theorem of MC, we shall refer to an irreducible, aperiodic Markov chain as ergodic.\n",
        "- So as long as we can simulate an ergodic Markov chain, irrespective of the starting point the chain will converge to target stationary distribution.\n",
        "- However, the time it takes for the chain to converge may vary; depending on the starting point.\n",
        "- As a best practice, it is advisable to throw out a certain number\n",
        "of the first draws, known as the $\\textbf{burn-in}$.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9mpI5hJaaUa_"
      },
      "source": [
        "## **Markov Chain Monte Carlo Integration**\n",
        "\n",
        "- Once we have a Markov chain that has converged to the stationary distribution, then the draws in our chain appear to be like draws from $p(\\theta|y)$, so it seems like we should be able to use Monte Carlo Integration methods to find quantities of interest.\n",
        "\n",
        "- $\\textbf{One Problem}$: *MC draws are not independent, which we required for Monte Carlo Integration to work (required condition for SLLN to work).*\n",
        "\n",
        "- The answer is $\\textbf{Ergodic Theorem}$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "G33iRTn7au4p"
      },
      "source": [
        "### **Ergodic Theorem**\n",
        "\n",
        "Let $\\theta^{(1)},\\theta^{(2)},...,\\theta^{(T)}$ be $T$ values from Markov chain that is aperiodic, irreducible and positive recurrent (that is the chain is ergodic) and $\\mathbb{E}[g(\\theta)] < \\infty$\n",
        "Then with probability 1\n",
        "$$\n",
        "\\frac{1}{T}\\sum_{t=1}^Tg(\\theta^{(t)})\\longrightarrow \\int_{\\Theta}g(\\theta)\\pi(\\theta)d\\theta\n",
        "$$\n",
        "as $T \\longrightarrow \\infty$ where $\\pi$ is the stationary distribution.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "J2eIJDRWa9Y2"
      },
      "source": [
        "1. This is the Markov chain analog to the LLN.\n",
        "2. But what does it mean for a chain to be aperiodic, irreducible, and positive recurrent?\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "fLwlFjrDbH2t"
      },
      "source": [
        "### **Aperiodicity**\n",
        "\n",
        "- A Markov chain $P$ is aperiodic if for all $x, y \\in \\Omega$ we have $gcd\\{t : P^t(x, y) > 0\\} = 1$.\n",
        "- If the only length of time for which the chain repeats some cycle of values is the trivial case with cycle length equal to one then the chain is aperiodic.\n",
        "- Intuitively we can say repeatation is allowed but as long as the chain is not repeating itself in an identical cycle, then the chain is aperiodic\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8xBauXGDbTV7"
      },
      "source": [
        "### **Irreducibility**\n",
        "\n",
        "- A Markov chain is \\textit{irreducible} if it is possible go from any state to any other state (not necessarily in one step).\n",
        "- A Markov chain $\\textbf{P}$ is \\textit{irreducible} if for all $x, y$, there exists some $t$ such that $P^t(x, y) > 0$.\n",
        "- We can show if $\\textbf{P}$ is \\textit{irreducible}, then $\\textbf{P}$ is {aperiodic} $\\iff$ there exists $t$ such that $P^{t}(x,y)>0$ for all $x,y \\in \\Omega$.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XSYghw7Zc2Fy"
      },
      "source": [
        "### **Positive Recurrence**\n",
        "\n",
        "- A Markov chain is \\textit{recurrent} if for any given state $i$ , if the chain starts at $i$, it will eventually return to $i$ with probability $1$.\n",
        "- A Markov chain is \\textit{positive recurrent} if the expected return time\n",
        "to state \\textit{i} is finite; otherwise it is null recurrent.\n",
        "- So if our Markov chain is aperiodic, irreducible, and positive\n",
        "recurrent then it is ergodic and the ergodic theorem allows us to do Monte\n",
        "Carlo Integration by calculating $E[g(\\theta)]$ from draws, ignoring the dependence between draws.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GSF9T0PndES3"
      },
      "source": [
        "### **What is MCMC?**\n",
        "\n",
        "- **MCMC** is a class of Monte Carlo methods in which we can simulate dependent sample that are approximately from a posterior probability distribution.\n",
        "- We then take the draws and calculate quantities of interest for the posterior distribution.\n",
        "- In Bayesian statistics, there are generally two MCMC algorithms that we use: the Gibbs Sampler and the Metropolis-Hastings algorithm.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tGRLx-Q8deC7"
      },
      "source": [
        "## **Two MCMC Samplers**\n",
        "\n",
        "1. Gibbs Sampling\n",
        "2. Metropolis Hastings Algorithms\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "nbaGxYIA-QSo"
      },
      "source": [
        "### **Gibbs Sampler**\n",
        "\n",
        "- Suppose we have a joint posterior distribution $p(\\theta_1,...,\\theta_k| data )$ that we want to sample from.\n",
        "- We can use the Gibbs sampler to sample from the target distribution\n",
        "if we know the \\textbf{full conditional posterior} distributions for each parameter.\n",
        "- For each parameter, the \\textcolor{red}{full conditional posterior} distribution is the distribution of the parameter conditional on the known information and all the other parameters: $p(\\theta_j|\\theta_{-j},y)$\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "U6tWiIMs_lfV"
      },
      "source": [
        "### **The Hammersley-Clifford Theorem**\n",
        "- For two blocks - Suppose we have a joint density $f(x,y)$.  The theorem proves that we can write out the joint density in terms of the conditional densities.$f(x|y)$ and $f(y|x)$\n",
        "$$\n",
        "f(x,y)=\\frac{f(y|x)}{\\int \\frac{f(y|x)}{f(x|y)}dy}\n",
        "$$\n",
        "- We can write the denominator as\n",
        "\\begin{eqnarray*}\n",
        "\\int \\frac{f(y|x)}{f(x|y)}dy &=& \\int \\frac{\\frac{f(x,y)}{f(x)}}{\\frac{f(x,y)}{f(y)}}dy\\\\\n",
        "&=&\\int \\frac{f(y)}{f(x)}dy\\\\\n",
        "&=&\\frac{1}{f(x)}\n",
        "\\end{eqnarray*}\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "b1Z3r4u9AUth"
      },
      "source": [
        "### **Gibbs Sampler Steps**\n",
        "\n",
        "- Suppose that we are interested in sampling from the posterior\n",
        "$p(\\theta|y)$, where $\\theta$ is a vector of three parameters,\n",
        "$\\theta_1$, $\\theta_2$, $\\theta_3$. \n",
        "      \n",
        "- The steps to a Gibbs Sampler are \n",
        "    1. Pick a vector of starting values $\\theta^{(0)}$(*Defining a starting distribution$\\Pi^{(0)}$ and drawing $\\theta^{(0)}$ from it.*) \n",
        "\n",
        "    2. Draw a value $\\theta_1^{(1)}$ from the full conditional $p(\\theta_1|\\theta_2^{(0)}, \\theta_3^{(0)},y)$. \n",
        "\n",
        "    3. Draw a value $\\theta_2^{(1)}$ from the full conditional $p(\\theta_2|\\theta_1^{(1)}, \\theta_3^{(0)},y)$. (*Note that we must use the updated value of $\\theta_1^{(1)}$.*) \n",
        "\n",
        "    4. Draw a value $\\theta_3^{(1)}$ from the full conditional $p(\\theta_3|\\theta_1^{(1)}, \\theta_2^{(1)}, y)$. \n",
        "\n",
        "- Steps 2-4 are analogous to multiplying $\\Pi^{(0)}$ and **P** to get $\\Pi^{(1)}$ and then drawing $\\theta^{(1)}$ \n",
        "\n",
        "    5. Draw $\\theta^{(2)}$ using $\\theta^{(1)}$ and continually using the most updated. \n",
        "\n",
        "    6. Repeat untill you have $M$ draws with each draw being a vector $\\theta^{(t)}$.\n",
        "values. \n",
        "    7. Optional burn-in and/or thinning.\n",
        "     \n",
        "- 'Gibbs sample' is a Markov chain with a bunch of draws of $\\theta$ that are approximately from our posterior. \n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "z5DyEqQvRLNm"
      },
      "source": [
        "### **The Metropolis-Hastings Sampler**\n",
        "\n",
        "- Suppose we have a posterior $p(\\theta|y)$ that we want to sample from, but \n",
        "  - the posterior does not look like any known distribution that we know \n",
        "  - If the posterior consists of more than 2 parameters then grid\n",
        "approximations is not possible. \n",
        "  - some (or all) of the full conditionals do not follow known distributions. Therefore Gibbs sampling is not possible for those\n",
        "whose full conditionals are not known. \n",
        "- In such cases, we can use the **Metropolis-Hastings** algorithm."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7qjDSzbSXtHT"
      },
      "source": [
        "**The Metropolis-Hastings Algorithm** :\n",
        "1. Choose starting value $\\theta^{(0)}$\n",
        "\n",
        "2. At iteration $t$, draw a candidate $\\theta^{*}$ from a proposal distribution $q_t(\\theta^{*}|\\theta^{t-1}))$.\n",
        "\n",
        "3. Compute the acceptance ratio\n",
        "$$\n",
        "r=\\frac{p(\\theta^*|y)/q_t(\\theta^*|\\theta^{(t-1)})}{p(\\theta^{(t-1)}|y)/q_t(\\theta^{(t-1)}|\\theta^{*})}\n",
        "$$\n",
        "\n",
        "4. Accept $\\theta^{(t)}=\\theta^{*}$ if $U<\\min(1,r)$ where $U\\sim uniform(0,1)$\n",
        "   \n",
        "   Otherwise $\\theta^{(t)}=\\theta^{(t-1)}$\n",
        "\n",
        "5. Repeat steps 2-4 $M$ times to get $M$ draws from $p(\\theta|y)$ with optional burn-in and/or thinning\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lgr02DufZ0jt"
      },
      "source": [
        "**Step 1: Choose a starting value $\\theta^{(0)}$**\n",
        "\n",
        "  1. This is similar to draw sample from initial stationary distribution.\n",
        "  2. Note that $\\theta^{(0)}$ must have positive probability.\n",
        "$$\n",
        "p(\\theta^{(0)}|y)>0\n",
        "$$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "WvSnsbdiaxZV"
      },
      "source": [
        "**Step 2: Draw $\\theta^{(*)}$ from $q_t(\\theta^{*}|\\theta^{(t-1)})$**\n",
        "\n",
        "1. The proposal distribution $q_t(\\theta^{*}|\\theta^{(t-1)})$ determines where it moves in the next iteration of the Markov chain.\n",
        "2. The proposal distribution is analogous to transition kernel.\n",
        "3. The support of proposal density must contain the support of the posterior\n",
        "4. If a symmetric proposal distribution that is dependent on $\\theta^{(t-1)}$, then it is known as **random walk Metropolis sampling**.\n",
        "5. If the proposal distribution does not depend on $\\theta^{(t-1)}$\n",
        "$$\n",
        "q_t(\\theta^*|\\theta^{(t-1)})=q_t(\\theta^*)\n",
        "$$\n",
        "then it is known as **independent Metropolis-Hasting sampling**.\n",
        "6. In this case all candidate draws $\\theta^{*}$ are drawn from the same distribution, regardless of where the previous draw was.\n",
        "7. This can be very efficient or very inefficient, depending how close the proposal density with respect to posterior density.\n",
        "8. Typically, if proposal density has heavier tails than the posterior, then chain will behave nicely\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bVxM2jEebNWO"
      },
      "source": [
        "**Step 3: Compute acceptance ratio $r$ .**\n",
        "\n",
        "1. The acceptance ratios:\n",
        "$$\n",
        "r=\\frac{p(\\theta^*|y)/q_t(\\theta^*|\\theta^{(t-1)})}{p(\\theta^{(t-1)}|y)/q_t(\\theta^{(t-1)}|\\theta^{*})}\n",
        "$$\n",
        "2. In the case where our proposal density is symmetric\n",
        "$$\n",
        "r=\\frac{p(\\theta^{*}|y)}{p(\\theta^{(t-1)}|y)}\n",
        "$$\n",
        "3. If the candidate sample has higher probability than the current sample, then the candidate is better so we definitely accept it.\n",
        "4. Otherwise, the candidate is accepted according to the ratio of the probabilities of the candidate and current samples.\n",
        "5. Since $r$ is a ratio, we only need to know the $p(\\theta|y)$ upto a constant.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "okN6R7t8eaWp"
      },
      "source": [
        "**Step 4: Decide to whether accept $\\theta^{*}$**\n",
        "\n",
        "1. Accept $\\theta^{*}$ as $\\theta^{(t)}$ with probability $\\min(r,1)$\n",
        "\n",
        "2. For each $\\theta^{*}$, draw a value $u$ from the $Uniform(0,1)$ distribution.\n",
        "\n",
        "3. If $u \\leq r$ accept $\\theta^{*}$ as $\\theta^{(t)}$. Otherwise $\\theta^{(t-1)}=\\theta^{(t)}$\n",
        "\n",
        "4. Candidate sample with higher density than the current samples are always accepted\n",
        "\n",
        "5. Unlike in rejection sampling, each iteration always produces a sample, either $\\theta^{*}$ or $\\theta^{(t-1)}$.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BXWTWgMNGO2W"
      },
      "source": [
        "### **How MCMC works?**\n",
        "\n",
        "- In the cartoon below the **red density** is our **target density**, which is **stationary density**. It does not move around. It is static in one place.\n",
        "\n",
        "- But we know it only upto its kernel. We don't know how to simulate from it.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dwcyWOGgF0wZ"
      },
      "source": [
        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IG8Sf8AYza9/wCl8tev0AFFFFAH/9b9/KKKKACiiigAooooAK+ZPhh/ycp8bv8Arl4Y/wDSOavpuvmT4Yf8nKfG7/rl4Y/9I5qAPpuoLb/Vn7/33+/1+8ent/d9sVPUFt/qz9/77/f6/ePT2/u+2KAJ6KKKACiiigDF8R+H9J8V6Df+Gtdh+0WGpQvBMmSpKOMZVhgqw6qwIKkAgggV8r6J431b4W/EcaL8SLjzDeW9vZXeoPhI7mKGQx2Oq9MDeZRa3wBxDN5L4EMgavsKvPviP8N9B+JWiJpmsRqtxauZbS4KBzDIVKMGXI8yKVGaOaInbLGzI3XI4MbhpytUpfEvx8v68+59lwpnmHouWDzBN4epo7bxb+1H7k2uvLFtS5UjhfF3gD4Y/E9vEontDZazpTfY765t1ijuHAiS4RZkcPFcxFXVkW4R0zkgAjNeSR+J/i3+zzFAfFls/inwOETNzG5MtknoXmdmiwP4J5GiP8NzF8kB86i8U+K/g94+g0XWIDPe28S2SwSy7v7U0nzHW3tmmm2iSWIsfsNySQTIbO6KTSRyyfaFxbeCvjV4NguYJ5pbJ5GkgngeS0u7O6h3RNj7skM0RLxyRuP70cikFlPk0nGu5Sp+7VX9a9/Pz+R+j5jCrlEKNLHL6xgKltX73LpvB3Ti3ZuNnblbWrUrfI3xBtfFnxL8TeMvGvwj0GPxTpPifw/4UsVuWWwc2tzoOt3tzf272WoyxFb5La9WW1WePyfNjxMy4UNgfDj4CfEef4l2niPx54cu7rSD46n8QvJrk2kTXTW7+FTpi3FxDp222Ev2xQNkUZ2gq3O0vR4k8A/En9n3xT/wlng+XzkvGSBLmJAljdF5Bsgu7YMkcEjElYxuS2Z2/cvZOxjm+u/hB8atC+KmnGF4To/iG13i60ydiJV8pgkkkQcI7xBztJZFdG+SVEcFa7cFmnPL2NZcs/wfp/Xpc+S4r8P1h6H9p5VU9thX1XxR8pL/ANustdJKL0Pzmg8PXXijxl4J+C2iWFrDrGman8RIn8T2t9aT3En9oWepwG4aGGVruF1nniF19oijCXARIy4II7bQvgF8WF8A6wJPDWujVbDQtC06Kx1O98OIl3JpWow3bW9imlwQIYo0icQTXlxG370qY1yzD9OoNI0m1v7jVLWyghvbsATTpGqyyhegdwNzY7ZNaNewfmZ83fGbw74p1fWfhr8QdI8Kz+Jrfwpf3N1e6Estml3i8sJrZJYvtM8dq81s8mCDMBsZyjMwUH4htf2afi7pXhzQ3m8L6vbx3Wg+IdFn0PQbrw66WP23Wbu+himk1eO4jFrcQTxq8ltmVPKTdGxwF/W+igD8/n/Z98Yx6T47K6J9t1m/8U+A7yyvZ57eW5uLPw+mii5m+0EQn901tdEbkjZ8MUj+dQfOPEXwJ+M1zZePvCvhTwndQaTqOieKrW3g1S40e7hjutVuBNAmiagjR6gkN2w824ivgiIQoByq1+oEHWX7/wB8/f8Aw+7/ALPpVigD83Pi1pHjb4cWPjHxjpOmR2Xiyx8e2us+EZLi4thDrZ1qzg0ma0iVJGn3qssu+OSJQWEbLuALL93fDjwVZfDjwD4e8B6dIZoNBsbezErffmaFAryue7yNl2J5LEk11Vzp9heS2893bRzy2j+ZCzorNE5UruQkZVtpIyOcEirdABRRRQAUUUUAFFFFABRRRQAUUUUAFQRf62b7/wB4fe+790fd9vX3zU9QRf62b7/3h977v3R9329ffNAE9FFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAFdP+PmX7/3U6/c7/d9/X8KsVXT/AI+Zfv8A3U6/c7/d9/X8KsUAeQfBn/kDeJP+xm17/wBL5a9fryD4M/8AIG8Sf9jNr3/pfLXr9ABRRRQB/9f9/KKKKACiiigAooooAK+ZPhh/ycp8bv8Arl4Y/wDSOavpuvmT4Yf8nKfG7/rl4Y/9I5qAPpuoLb/Vn7/33+/1+8ent/d9sVPUFt/qz9/77/f6/ePT2/u+2KAJ6KKKACiiigAooooA86+Jvwv8LfFbw82g+JYfmTc1tcoFM1tI6lSybwylWUlXjcMkiko6spIr4XtNV+IH7Mni9LfxPcCa21GYg3B3my1S2hRfnld8+VcxRghZmd5FRdt15sCC8j/S2sHxN4X0Dxjo0/h/xNZJf2FxjdG+QQynKujKQyOhAZHUhlYAqQQDXk4/LFUftaT5ZrZ/5/1+Gh+jcHceywNN5fj4e1wk/ih1V+sL7O+ttm0ndStJc94N8ceEfinoE8um4mQr5N9p92iieAypzFcQncMOpypG6ORDuRnQhj8ZfGT4EXfw/wBTtvHXgW7lsNPsmDQ3SM/m6Q0akRrI6pI7WWP3aSlHa0B2SLNZF4UzfFvwi8cfADUo/F3g25utU8PWCusNxblVvtOhYs7JKMeXJbhjkq6m3Yne62zh7tvfvh/+0bomvaRv8aiKCGNVE2p26ObBfMXK/a423S2JdTkGbdAwwYriVSDXl1a0K69ji1yzWz/X0/rR7foWAy3E5PU/tXhup9Ywk9JQ6/4XGz95J6aX1acZQu5aPwQ+OX/Cd7vB3jaOPS/GVjvV4flRbsQhd8kShnCuoZWkiDuArJJG8kMkcjfR9fA/ir4Q6c3ia1074e3EElwtpJrHheWC4VD5NlIi3OmidS21ITcLJYzciDzZISDbb4pPfPgn8XJfHdl/YPiYCDxHaRvJnyzALyCKTyXlWIljFLDKPJuoMkwzDGWjeN37suxs0/YV9+j7/wDB/NW76/I8b8KYWpTebZQrU3Zzhs4Xvqlr7t001d8slJJtRbXvlFFFe4fkpXg6y/f++fv/AIfd/wBn0qxVeDrL9/75+/8Ah93/AGfSrFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABUEX+tm+/94fe+790fd9vX3zU9QRf62b7/wB4fe+790fd9vX3zQBPRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBXT/AI+Zfv8A3U6/c7/d9/X8KsVXT/j5l+/91Ov3O/3ff1/CrFAHkHwZ/wCQN4k/7GbXv/S+WvX68g+DP/IG8Sf9jNr3/pfLXr9ABRRRQB//0P38ooooAKKKKACiiigAr5k+GH/Jynxu/wCuXhj/ANI5q+m6+ZPhh/ycp8bv+uXhj/0jmoA+m6gtv9Wfv/ff7/X7x6e3932xU9QW3+rP3/vv9/r949Pb+77YoAnooooAKKKKACiiigAooooAK+YPHP7OdnLqq+MvhTd/8Ivr8JZxFE7Q2spcln2FAxtzITl9qSQSN801vM2CPp+iubE4SnWjaa/zXoe5kXEeMy2q6uEna+jW8ZLtJPRr126WZ82+GNd8E6Tq9todzoCeCPHeqhYZmezt7d750Ay8dzAhjukLfOERtwwPMWPpV/xT8K9YuvFWt+IvCbR2F2UtdU02ZjhY9aiEkM4YKd3k3dsIorkYwVAZfn5X23W9B0TxLpsuj+IbCDU7GfG+C5jWWNsdMq4IyOx7Vwcfw0u9IVIPCHizV9FtE4Fq0kV/CB6Kb6OeVAP4VWQKOm3GAOSeEduWSuvLR/5ed7rXofTYbiam6jr0ansqklaSnecGrp2TV5KOiioOM1y3TlZtHWeDfE1v4y8L6b4mt4XtRfxB3gkx5kEo+WWF8cb4pAyNjjKnFdNXM+D/AAtY+DPD1t4esJprlIWmlknuH3zTz3MrzzyuQAN0ksjuQoVVzhVVQAOmr0KPNyLn3tr6nxWZew+s1Pq38PmfLfflvp+BXg6y/f8Avn7/AOH3f9n0qxVeDrL9/wC+fv8A4fd/2fSrFaHEFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFQRf62b7/wB4fe+790fd9vX3zU9QRf62b7/3h977v3R9329ffNAE9FFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAFdP+PmX7/3U6/c7/d9/X8KsVXT/AI+Zfv8A3U6/c7/d9/X8KsUAeQfBn/kDeJP+xm17/wBL5a9fryD4M/8AIG8Sf9jNr3/pfLXr9ABRRRQB/9H9/KKKKACiiigAooooAK+ZPhh/ycp8bv8Arl4Y/wDSOavpuvmT4Yf8nKfG7/rl4Y/9I5qAPpuoLb/Vn7/33+/1+8ent/d9sVPUFt/qz9/77/f6/ePT2/u+2KAJ6KKKACiiigAooooAKKKKACiiigAooooAKKKKAK8HWX7/AN8/f/D7v+z6VYqvB1l+/wDfP3/w+7/s+lWKACiiigAooooAKKKKACiiigAooooAKKKKACoIv9bN9/7w+9937o+77evvmp6gi/1s33/vD733fuj7vt6++aAJ6KKKACiiigAooooAKKKKACiiigAooooAKKKKAK6f8fMv3/up1+53+77+v4VYqun/AB8y/f8Aup1+53+77+v4VYoA8g+DP/IG8Sf9jNr3/pfLXr9eQfBn/kDeJP8AsZte/wDS+WvX6ACiiigD/9L9/KKKKACiiigD4SvfDfxq1/xkfB3xA0vxT/whlpf+KdT/ALR0TWorRrxZbkXGjW4ltb6G/jWGDzYhDsWMyGIFiikV9Q/Byz8a2Hwz0K0+ITzvrscTib7XJFNdJGZHMCXMsJaOSdIdizOhKvIGYEg5r0yigDifFOk+O9RuIH8I+IrXRYUQiRLjTjel2zwQ3nw7QBxjB+tfPnwTsvENn8efjbaeINTi1LVNnhwm6htvs0e1rObYPJ8yT7vOTv8Am9q+ua+ZPhh/ycp8bv8Arl4Y/wDSOagD6UkS4bzPLlC7kwmVztfn5jzz2446dearxJcNHN5crLu3hPMXJV9zfN1GV6bRxwBzzV+oLb/Vn7/33+/1+8ent/d9sUAEiXDeZ5coXcmEyudr8/Meee3HHTrzRIlw3meXKF3JhMrna/PzHnntxx0681PRQBBIlw3meXKF3JhMrna/PzHnntxx0680SJcN5nlyhdyYTK52vz8x557ccdOvNT0UAQSJcN5nlyhdyYTK52vz8x557ccdOvNEiXDeZ5coXcmEyudr8/Meee3HHTrzU9FAEEiXDeZ5coXcmEyudr8/Meee3HHTrzRIlw3meXKF3JhMrna/PzHnntxx0681PRQBBIlw3meXKF3JhMrna/PzHnntxx0680SJcN5nlyhdyYTK52vz8x557ccdOvNT0UAQSJcN5nlyhdyYTK52vz8x557ccdOvNEiXDeZ5coXcmEyudr8/Meee3HHTrzU9FAEEiXDeZ5coXcmEyudr8/Meee3HHTrzRIlw3meXKF3JhMrna/PzHnntxx0681PRQBQRLhjc+XKy7shN652vz8w5GV6YHseeeLEiXDeZ5coXcmEyudr8/Meee3HHTrzSQdZfv/fP3/w+7/s+lWKAIJEuG8zy5Qu5MJlc7X5+Y889uOOnXmiRLhvM8uULuTCZXO1+fmPPPbjjp15qeigCCRLhvM8uULuTCZXO1+fmPPPbjjp15okS4bzPLlC7kwmVztfn5jzz2446deanooAgkS4bzPLlC7kwmVztfn5jzz2446deaJEuG8zy5Qu5MJlc7X5+Y889uOOnXmp6KAIJEuG8zy5Qu5MJlc7X5+Y889uOOnXmiRLhvM8uULuTCZXO1+fmPPPbjjp15qeigCCRLhvM8uULuTCZXO1+fmPPPbjjp15okS4bzPLlC7kwmVztfn5jzz2446deanooAgkS4bzPLlC7kwmVztfn5jzz2446deaJEuG8zy5Qu5MJlc7X5+Y889uOOnXmp6KAIJEuG8zy5Qu5MJlc7X5+Y889uOOnXmq4S4aW78uVl3BQm9cqr7fvLyMjpx6g881fqCL/AFs33/vD733fuj7vt6++aACRLhvM8uULuTCZXO1+fmPPPbjjp15okS4bzPLlC7kwmVztfn5jzz2446deanooAgkS4bzPLlC7kwmVztfn5jzz2446deaJEuG8zy5Qu5MJlc7X5+Y889uOOnXmp6KAIJEuG8zy5Qu5MJlc7X5+Y889uOOnXmiRLhvM8uULuTCZXO1+fmPPPbjjp15qeigCCRLhvM8uULuTCZXO1+fmPPPbjjp15okS4bzPLlC7kwmVztfn5jzz2446deanooAgkS4bzPLlC7kwmVztfn5jzz2446deaJEuG8zy5Qu5MJlc7X5+Y889uOOnXmp6KAIJEuG8zy5Qu5MJlc7X5+Y889uOOnXmiRLhvM8uULuTCZXO1+fmPPPbjjp15qeigCCRLhvM8uULuTCZXO1+fmPPPbjjp15okS4bzPLlC7kwmVztfn5jzz2446deanooAoFLhri5EcrLuiQJuXKK/wA3zDnntkcdB61YkS4bzPLlC7kwmVztfn5jzz2446deaRP+PmX7/wB1Ov3O/wB339fwqxQB4/8ABj/kDeJP+xm17/0vlr2CvIPgz/yBvEn/AGM2vf8ApfLXr9ABRRRQB//T/fyiiigAooooAKKKKACvmT4Yf8nKfG7/AK5eGP8A0jmr6br5k+GH/Jynxu/65eGP/SOagD6bqC2/1Z+/99/v9fvHp7f3fbFT1Bbf6s/f++/3+v3j09v7vtigCeiiigAooooAKKKKACiiigAooooAKKKKACiiigCvB1l+/wDfP3/w+7/s+lWKrwdZfv8A3z9/8Pu/7PpVigAooooAKKKKACiiigAooooAKKKKACiiigAqCL/Wzff+8Pvfd+6Pu+3r75qeoIv9bN9/7w+9937o+77evvmgCeiiigAooooAKKKKACiiigAooooAKKKKACiiigCun/HzL9/7qdfud/u+/r+FWKrp/wAfMv3/ALqdfud/u+/r+FWKAPIPgz/yBvEn/Yza9/6Xy16/XkHwZ/5A3iT/ALGbXv8A0vlr1+gAooooA//U/fyiiigAooooAKKKKACvmT4Yf8nKfG7/AK5eGP8A0jmr6br5k+GH/Jynxu/65eGP/SOagD6bqC2/1Z+/99/v9fvHp7f3fbFT1Bbf6s/f++/3+v3j09v7vtigCeiiigAooooAKKKKACiiigAooooAKKzdQ1jS9KC/2jdRwNJ9xWPzufRVHzMfYA1mDXdQumxpekXEq9pbgi2j/J8y/wDkOtY0JtXtp935mE8TCL5W9ey1f3I6WiudVPFc4Jkms7POOFSS4I9fmLRf+g0DStbbBl1uQHHPlwQqP/Hlf+dV7FdZL8f0RP1iXSD/AA/Vm1B1l+/98/f/AA+7/s+lWK5iLSdUPmbNbvAQxH7yO3I/D90OKl+w+JIyTDqsUg7Ca23frG8dHso/zr8f8g9vL+R/h/mdFRXNm68U2ozNYW96oxzbzGNz64SRdv8A5Epw8T6bEwj1NZNMc9rpNiZ9BKMxE+wc0fV5dNfTUPrcPtaeun47HRUU1WV1DoQysMgjkEGnVgdIUUUUAFFFFABRRRQAUUUUAFQRf62b7/3h977v3R9329ffNT1BF/rZvv8A3h977v3R9329ffNAE9FFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAFdP+PmX7/3U6/c7/d9/X8KsVXT/j5l+/8AdTr9zv8Ad9/X8KsUAeQfBn/kDeJP+xm17/0vlr1+vIPgz/yBvEn/AGM2vf8ApfLXr9ABRRRQB//V/dvxb4lsPBnhbVvFuqJJJZ6NazXcqxbd7JChdgu9lQEgdWZVHVmUZI4Hwn8Y9F8U+LYPAcunXekeIvsV3e3Njd+T5ltFayW0asWhkkSSOf7UrQyxM6MEkUsJEdF8Z/aKsPH97qsaWdr40vNHc2QhHhC5soI1Tzx9sW6WWSK5aRotwQgtDjaMK24t7F8GtG1m18NxXfiaxvoLmAvb6c2tCyfWIdNKxkQ3Mljuh5lViqqxPliPzD5gbAB7HRRRQAUUUUAFfMnww/5OU+N3/XLwx/6RzV9N18yfDD/k5T43f9cvDH/pHNQB9N1Bbf6s/f8Avv8Af6/ePT2/u+2KnqC2/wBWfv8A33+/1+8ent/d9sUAT0UUUAFFFFABRRRQAUUVzeq6xdi6Gi6FEtxqDKGdnz5Nsh6PKRySf4UHLY6gAsLp03J2RlVrRgryNDVNZ0/R40e9kO+Y7YokBeWVv7qIuWY/QcdTgVjrF4j1obrlzoto3SKMq90w/wBuT5kj9wgY+jir+kaBb6ZI97NI17qMwxLdS4LsP7qgcIg7IuB35OSd2tnUjDSGr7v9F/n+BgqU6mtR2XZfq/8ALT1MnTtD0rSSz2NuFlk+/KxLyv8A78jku34mtaq91d2tjA91ezJbwxjLPIwVR9SeK5//AISK5vsf2Dp0t4p6TS/6NB/3043sPdEYe9SoTqe8/vf+bLdSlS9xaeS/yR1FFc4LXxPc83F9BZqf4YIS7D/tpI2D/wB+xVqz0q4tp1nn1O6uyoxtlMSoeP7scaCk6SS+Jfj/AJDjWk38D/D/ADv+BoW7qXlQMxYOeH/D7v8As+lWazreKQ3Ukx3YBZfmzyDg8DoAPXvVGfS9W86Saz1iWPeSRHLFFJGuc8DCo+Pq/apjBPd2/ryLnUkto3+79WjfprIrqUcBlYYIPIINc29z4rsTma0t9TiHU27mCX8I5SyH8ZRVmy8R6ZeXC2Ls1peMMi3uFMUpx12g8OB6oWHvVuhK11qvL+rr5maxUG+WWj89P+A/lcqP4XgtWM3h6dtJl5OyIbrdif70B+XnuU2t/tU1PEFzpsi23iiAWm44W6iJa1c/7THmIn0f5ewZjXVUySNJUaKVQ6OCGUjIIPUEU1XvpUV/z+//ADJeF5daTt5dPu/ysPBzyKK4w6df+GGE2hq1zpef3ljyXhXu1seuB3iPH9zB+Vuosb6z1K2S8sJlngkzh0ORkHBH1BGCOx4qalKy5ou6/rcujX5nyyVpdv1Xdf07FuiiisToCiiigAooooAKgi/1s33/ALw+9937o+77evvmp6gi/wBbN9/7w+9937o+77evvmgCeiiigAooooAKKKKACiiigAooooAKKKKACiiigCun/HzL9/7qdfud/u+/r+FWKrp/x8y/f+6nX7nf7vv6/hVigDyD4M/8gbxJ/wBjNr3/AKXy16/XkHwZ/wCQN4k/7GbXv/S+WvX6ACiiigD/1v38ooooAKKKKACiiigAr5k+GH/Jynxu/wCuXhj/ANI5q+m6+ZPhh/ycp8bv+uXhj/0jmoA+m6gtv9Wfv/ff7/X7x6e3932xU9QW3+rP3/vv9/r949Pb+77YoAnooooAKKKKACiisvWdUj0fT5L10MrgqkUSnDSyudqIvuzED26niqhByaityKk1GLlLZGbrOq3X2yLQNGwb+4XfJJwVtoM4MrA5BY8iNT95uvyq1aul6Xa6Ra/ZbXc25i7vIxeSR2+87seST+gwBgACqWgaRJpltJNeuJtRvW826lHRpCMBVzyEQYVB2A55JJ3q2qzSXJDb8/66HPQpyb9pU36Lsu3r3/yQVzF7r0813JpHh6Jbu9i4lkckW9uT/wA9GHJfuI1+b1Kgg1He395q99Noehy+StsVF5dAZMW4Z8qLjBlK8kk4QEEhiQK3tO06y0mzjsNOhEEEQwqr78kknkknkk5JPJOaagoK81d9v1f6L+mnOVR8tN2S3f6L9X8lrtkWfhu3SdNQ1eVtUv0OVlmA2Rn/AKZRD5Y/qMt6sa6SiisalSUneTN6VGMFaK/rz7hRRTHkjiXfIwRfUnAqDVsig6y/f++fv/h93/Z9KsVnWl9ZSvIkVwHbeeCwPp932rRptNbiUk9gqnfafY6nbtaajAlzC3VXUMM9iM9COxHIq5RRGTTuglFNWa0OQe11vw9+80xn1WwX71tK+biNf+mMrH58f3JDk9n6A7+mapY6xaC80+XzIySpyCrI6/eV1OCrDupAIrQrldZ0m/juRrnh0ql8n+tgYlYrtAMbH7K4wNkmCRjByuRXSpKppLR9/wDP/P7/AC45QlS1hrHt29P8vu7Pqq46902bQb+bxDoyO8M7F7+0QFvN+UDzol7SqANwH3xngvtroNK1S21eyS8tsrnKvG3DxSD70bjnDKeCK0azhOVOTT9GjSpTjVipJ+af9f09iC2uYLy3iu7VxJDModGHQqwyDU9cbAv/AAjOti0X5dL1eRjEO0F2csyD0Wblh6OCP4wK7KlWp8r02exWHrOSalutH/XZ7hRRRWRuFFFFABUEX+tm+/8AeH3vu/dH3fb1981PUEX+tm+/94fe+790fd9vX3zQBPRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBXT/j5l+/91Ov3O/3ff1/CrFV0/wCPmX7/AN1Ov3O/3ff1/CrFAHkHwZ/5A3iT/sZte/8AS+WvX68g+DP/ACBvEn/Yza9/6Xy16/QAUUUUAf/X/fyiiigAooooAKKKKACvmT4Yf8nKfG7/AK5eGP8A0jmr6br5k+GH/Jynxu/65eGP/SOagD6bqC2/1Z+/99/v9fvHp7f3fbFT1Bbf6s/f++/3+v3j09v7vtigCeiiigAooooAK5Tb/bHiYuwJtdEGF/utdyryf+2cZwPdz3FdJdXMVnbTXc5xHAjOx9FUZP6VjeF7Wa20S3e7XbdXW65nHpLOfMYf8BJ2j2Arope7GU/l9+/4afM5ay5pxh03fy2/HX5HQVg6/f3VtbxWWmFf7Qv28qDdghOMvKQeqxrlsdzhe9b1cvpSHUdav9ck5SEmytvQJGf3zD3aUbT7RilQSu5PZf1/XkPEybShHd/gur/rq0a+k6Xa6NYRafaA7I8ksxy7uxyzue7MxJY9ya0aKKylJybb3N4QUUoxWiCsHU9fgsbhdOtYmvtRkXctvFjcFPG+Rj8saZ/iY89FBPFM13U7q2MGl6SFfUr7cItwykSLjfM4HVUyOP4mKrxnItaPo1to1u0cRM08zb553x5k0h6s5GPoAOAMAAAYraEIxjzz+S/roc06kpSdOn03fbyXn+XmZ8el63qH7zWr8wIf+XeyJjUezTH94x912fSp4vCnhyJvMOnQzSc/PMvnSc9fnk3N+tdBRSeJn0dvTQpYSn9pXfnqc+vh7w/Ossc+mQSIWYYlgQjBAyFBH3aiPhXToudKkm0xwcg20hVM/wDXJt0R/FK3YOsv3/vn7/4fd/2fSrFJYia+0OWFpPeKORk1HXNC+bWIhqFiOtzbIRLGPWWEZJA7tGT/ALgHNdPa3Vte28d3ZyrPBMoZHQhlZT0II4IqeuJvrZvClzLrmmr/AMSyZt99bgcR5+9cRAdCOsqjhhlh8wO7SKjU0taX4P8Ayf4GMnKjre8fxX+a9dfXY7aimo6yKHQhlYAgg5BB7inVyncchqqL4f1L/hJIQVtbkpFfqPugfdS4x6pwrn+5yfuCuvqKeCG6gktrhBJFKpR1PIZWGCD7EVg+GHmjsZNJunMk+lyG2LN950UBomPqWjK5P97NdMnzwv1X5dPu2+45IL2dTlW0tfn1+/f7zQ1jTItY02fTpWKeaBtdfvRyKdyOv+0jAMPcVDoGpSappcdxcKEuYy0U6Dos0RKSAe24Ej1BBqbRtWs9e0mz1rT1mW2vokmjFxBLazBHGRvhnRJY2x1V1Vh0IBrz6Dxv4a0nx1d+Hbi4kt59WmdLeKW3njae8tLYT3Ig3oBMot/LbfHuQsHXdvDKFB80HHtqv1/ryHUXLUjNddH+a/y+Z6pRVDStSttZ0uz1iyEq299DHPGJ4ZLeUJKoZfMhmVJI2wfmR1VlPDAEEVmeGfFegeMLK61Hw5dfa7eyvb3TpW2PHtutPuHtbmPEiqTsmjddwBVsZUlSCec6joqKKKACoIv9bN9/7w+9937o+77evvmp6gi/1s33/vD733fuj7vt6++aAJ6KK53TfFegaxr2s+GNOuvO1Pw+bcX0Wx18k3UfmxfMyhW3Jz8pOOhweKAOiorntR8V6BpPiDR/C2oXXlaprwuWsodjt5otFV5vmVSq7VYH5iM9s10NABRRXJ33jjwzpmqLo2pXT2l1LdW9lF50E0cc9zcxtLHHDIyCOViiMW8tm2YIbBoA6yiiigAooooAKKKKACiiigCun/HzL9/7qdfud/u+/r+FWKrp/wAfMv3/ALqdfud/u+/r+FWKAPIPgz/yBvEn/Yza9/6Xy16/XkHwZ/5A3iT/ALGbXv8A0vlr1+gAooooA//Q/fyiiigAooooAKK+bYfiv8YPFupazL8LvAmkanoWialeaU9zq/iCXTbq4udPlaC48q2g069CoJVYI0sqF1AbaFZSfV/hp48sfiZ4I0zxrp9tJZR34lV4JWR2imt5XgmTfGWRwskbBXQlXADKcEUAd3XzJ8MP+TlPjd/1y8Mf+kc1ey+KviBoXg64gtdXtdWuHuELqdO0bUtTQAHHzvY206ofZiCeuMV8+fBLxHp/iP48/G3xBp8V5Fasnhxdl5Y3Vjc5is5t3+jXUUU+Dn5T5eG/hzg0AfXNQW3+rP3/AL7/AH+v3j09v7vtiiS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyMwRXEccc24THyt7ncjEkbm4XA+bpwBk428cjIBeoqCS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyMklxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZAJ6KgkuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjJJcRx+ZuDnyk3nCM2Rz93AO48fdGT045GQDC8WL52iSWXUXskNsR1yk8qxv/wCOsa6SuW8STxq+nI4chLuJ22oxyArsAMD5jlfujJ6ccjPRSXEcfmbg58pN5wjNkc/dwDuPH3Rk9OORnef8OK9f6/A5qetWT9F+b/Uh1K8XT9OutQcZW2ieUj2RS39KqeH7BtM0SxsZCTJFEnmE9WkIy7H3LEk1n+L5l/4R3U4SrNmEBhtYgrIdpAOCCcZyBk9OORXQyXEcfmbg58pN5wjNkc/dwDuPH3Rk9OORkelL1f5f8OC1rPyX5t/5InoqCS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyM1dRvfstleTIGL28DS/dYg4DEYIHJ+XoMnpxyM4pXdjolKyuzF8OIb6e+8RzD5ryRooM/wANtAxRMf77bpP+BD0FdXWBoJisfD9jb4ci2s4ScI7ZATsQDubg8DLdOORnYkuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjO2Jleb/AK0Rz4SFqcb7vV+r1f4k9FQSXEcfmbg58pN5wjNkc/dwDuPH3Rk9OORkkuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjOB0iQdZfv/fP3/w+7/s+lWKox3EcZudwmPlZc5Rjx6JgfN06DJ9uRU8lxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZAJ6QgMCrDIPUVDJcRx+ZuDnyk3nCM2Rz93AO48fdGT045GSS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyMgHOeGQ1h9t8Ot9zTJAIP8Ar2lG6IfRPmjHsldVXKyTLb+LZGw2H07e+1WbIhlO0AAHcfnbgZPtyK6OS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyM9GJd5c3fX/P8TlwitHk7Nr5dPwsT1zSqbXxcxHC6jZ5PpvtXxn6lZh+C+1b0lxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZ5zV544de0mQqxMYn3YRm+Voy2FwDuOU+6MnpxyMqhu15P8r/AKDxS0jLs1+Lt+TPym0j4HeJdV+EHiDW/EPg/VZvFvhz4Z+Ex4eM1rci6tNbtF1CSX7GhXcL2NxCJNg81QVU4DkN6f4z+H90fjdbeNPEnhO7v7eD4haqtncnTJrsxx3/AIWs4bWeMpG5SD7dCq+eMRpImWZSpI/S2S4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyM874pnjitLdyHJgu7NztRjw0wX5cD5jjPyjJ6ccjM0Y3kl3LxEuWDl2PyD8TeDPiBc+FPCIfwNfQeKfDPh/wOthcN4e1bU9Wd7SG2lvDaXyyJa6T9nbzUuISjTTMpDKd6A/YWq6XrOg/shfHiLVbS4064kf4lXcSzRvC7Q3F5qU0EyhgCUkjZXRhwykMpIINfbklxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZzNe03RvEGjan4d1+0F/pl/aywXdu8bPHNbzoySRkAfPuXIKjJwenIzkbH5Yr8MoPF1umm/D3wXruk+AtUl8BQa1Z3Vje2T3uoRazv1Odo5Ujkk22bAXt4oMcwIPmNsYjrLz4b3Gj/ABBudD0bwhq0HjvTvG2hv4b1e3sbk6bY+D7Y2SzwRX6p9lhtVso7mGazMivJMc7GLo1fpir29pEYI0ZY7aMEKqMQEAIAXAO4jb90ZPTjkZkkuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjIB+M3gLwr4q174cQXX7P+gazpPj5tF8aR61rElvcWiX3nrcR6ZFHqEyLDcyfaPJa08t5PsyoysE5Fd/q3w60bXLTxf/wrXwn4m8K/Dy7sfDX22zn8PXc8dxrNvfvJPLPol0Ibi/iSHYuotEGa4yMNI0ZI/UPRdI0LwnpKaFoFkLDT7BGdIIIm2KHZnYIADkltxKjJyenIzoC4SOW63CY+UA5yjFcbekeB83TkDJz25FAH5Ia74T8ReJPh54W8Ja18MY7TTra18SrYO/h7XdXsnmlvU+z/AGPRpJ0fSZbhcyW0l64W0jykW2PcB9Z/slaT40s49W1Xxtp2oWV7qOg+DvNk1GCaGWW6h0eNLoN5yqxkjlysoPzK+Q2DX2HJcRx+ZuDnyk3nCM2Rz93AO48fdGT045GSS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyMgH49W3w08cSvpkfgnwrruj/F+Dw141tvEGvyWF3Bb3Gu3dvts5vt8iLBcNLKGa1ljkcRRkISPlWtfwt8JNW1Hw7d6Lo2jX6eG9W1fwMuoafaeE9T8L25aDVwb+Ro7y+ubiWYWxIv7hEWJ0CEzOyvt/W6S4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyMklxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZAPyr+IPwdTSrLWvDcHhq7tfC+heL9Ul0XTZfDN54g8PfZbnSbBwp06xZLmNGuWuPstzbKUgl80cMwBtyeBNd1zVvDXi74ifDu/jgsNY8DXc9lLZ3WtPZBdGmguVSQpNNOLe4eNJpfmYEb5TkE1+pMlxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZJLiOPzNwc+Um84RmyOfu4B3Hj7oyenHIyAfEH7O3hvxBD8VNc8J6xHJ/YnwTiu/D2jyM25JhrUy6jEOpy1npgsYATzl5BgA8/c1cn4Y8K+E/Aem3GkeFNLTTLQySXsqQRMfMmnYtJITgmSRiOeS2No6bRXSyXEcfmbg58pN5wjNkc/dwDuPH3Rk9OORkAnoqCS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyMklxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZAJ6KgkuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjJJcRx+ZuDnyk3nCM2Rz93AO48fdGT045GQCeioJLiOPzNwc+Um84RmyOfu4B3Hj7oyenHIySXEcfmbg58pN5wjNkc/dwDuPH3Rk9OORkARP+PmX7/3U6/c7/d9/X8KsVSM6Rz3BYSny41c/IzLj5vuYB3NxyBk9OORmaS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyMgHk/wAGf+QN4k/7GbXv/S+WvX68f+DH/IG8Sf8AYza9/wCl8tewUAFFFFAH/9H9/KKKKACiiigDxrxV+zz8EfG2t3HiTxT4L02/1O82/abhoQj3O0bR5+zb52FAH7zdwAOgFeraVpWl6FptrouiWcOn6fYxJDb21vGsUMMUY2okcaAKqqBgAAADpV+igAr5k+GH/Jynxu/65eGP/SOavpuvmT4Yf8nKfG7/AK5eGP8A0jmoA+m6gtv9Wfv/AH3+/wBfvHp7f3fbFT1Bbf6s/f8Avv8Af6/ePT2/u+2KAJ6KKKACiiigDnPEh8uLT7jtFfW2f+2jiL+b10dYXiaCa40G9W2XdPHGZYh0zJF+8Qf99KK1rW5hvbWG8t23RTosiH1VhkH8q3lrTT83+hzQ0qyXkn+f/AMbxWjv4a1Tyl3SJbyOo9WRdwH4kVuxyJLGssZyrgMD6g8iiRElRo5BuVwQQehB6iue8JySf2LFYTnM+ms1pJ7mA7Vb/gSbW+ho3p+j/P8A4YL2req/L/h/wOkqOaJJ4nhk5WRSp+hGDUlFYJnQ1c57wpO0/h6xEhzLBH9nlx/z1gPlSD/vpTXQ1yMcn9g+IXt5BtsdaffE38KXYXDofTzVAZf9oN3Irrq6MSve5ls9f69Njmwcvc5HvHR/157hRRRXOdRXg6y/f++fv/h93/Z9KsVXg6y/f++fv/h93/Z9KsUAFFFZmr6rb6NYSX9wC+3CpGvLySMcJGg7szEAVUYuTSW5M5qKcpbIzbVhdeKr6VcFbG3hgz3EkhaRxn2Xyz+NdLWJ4f06fT9P/wBNIa9unae5K8jzZOSAe4QYRf8AZUVt1pXa5rLZaGOFi1C8t3r9/wDlsFc3f/vfE2kwgA+VFdTk+mPLjH5+ZXSVzGmn7d4i1TUBgx2qx2UZ/wBpcySkH6uqn3U06GnNLsn+On6k4nXlj3a/DX9Dp65zxGSyabbr1mvrftn/AFbeaf0Sujrm75vtPiTTLMYItUmu25+6cCFOPcSP+Row/wAV+13+A8X8Fu9l97OkooorA6QooooAKgi/1s33/vD733fuj7vt6++anqCL/Wzff+8Pvfd+6Pu+3r75oAnooooAKKKKACiiigAooooAKKKKACiiigAooooArp/x8y/f+6nX7nf7vv6/hViq6f8AHzL9/wC6nX7nf7vv6/hVigDyD4M/8gbxJ/2M2vf+l8tev15B8Gf+QN4k/wCxm17/ANL5a9foAKKKKAP/0v38ooooAKKKKACiiigAr5k+GH/Jynxu/wCuXhj/ANI5q+m6+ZPhh/ycp8bv+uXhj/0jmoA+m6gtv9Wfv/ff7/X7x6e3932xU9QW3+rP3/vv9/r949Pb+77YoAnooooAKKKKACuV8K/6FDdeHW4bSpSkY9baT54SPYKdn1Q11VcprxOkXlv4nT/UwjyL3j/l3Y5En/bJvmP+yX9q6KHvJ0++3r/V18zkxPutVe2/o9/u0fojq65O4YaJ4kS7b5bTWtsMh7JdIMRsf+uifJn1VB3rqwQRkcg1S1PTrXVrCbTr1S0M64ODhgeoZT2ZTgg9iAamjNRdpbPR/wBfiXiKblG8d1qv689i9RXMaDql0ZZNB1ph/adoud+MC5hzhZkHv0dR91uOhUnp6mrTcHZl0aynHmX9eRR1LTbPVrKXT7+PzIZQMjJBBByGUjkMpAII5BAI5rmrXVb3w8407xPIZLfdtg1AgBHBPyrPgARyDpu4R+oIY7a7OmuiSI0cihlYEEEZBB6girp1rLlkrr+tv6/Qzq0Ly54O0vz8n/Wn33UEEAg5Bpa5hPDSWB3aBdyaav8AzxXElt+ETfcHtGUrZsF1JYmXVHhkkDfK0KsilfdWLEH8TUzhG14v/P8Ar5lU6k27Sjb8V/n+BNB1l+/98/f/AA+7/s+lWKyoWvkspjApmugzbRcMY1J7fMqthfoprPaz8TXny3OoRWMZPItYt8mP+ukuV/8AIdEaV92kOdZraLf9eehoarrWn6PGjXjkySkiKFBvllYfwxoOWP6DqcDmsjTtLv8AUL5Ne8Qp5csWfstoGDLbAjBZiOGmYEgkcKPlXPLNqadoWm6XK9zbxl7mUYeeVjJM49C7EnHoowB2ArYrR1YxVqf3/wCXYyVGU2pVdlsv1ffy6LzdmFFFRyyxQRPNM4jjjBZmY4VVHJJJ4AFcyR1tmdrWpf2Vp73MaCW4YiOCIkDzJn4RMn1PU9hk9qNE0z+yNLgsS/myKC0sneSWQl5HP+85J/GsTS45tf1NfEV5G0VpbblsInBVjuG17hlPILj5UB5CZPVyB2FdNX3I+z69f8vl+Zx0P3kva9Nl6dX8/wAku4Vyvh7GoXupeIeqXUgt4D/0wtSyg/RpDIwPcEVN4kvLhYItH05yl/qhMUbDrFGB+9m/4AvT1YqO9bdnaW9haQ2Nogjgt0WNFHQKowB+VJe7Tv1l+X/D/kxv36qXSP5v/gfmizRRRXOdYUUUUAFQRf62b7/3h977v3R9329ffNT1BF/rZvv/AHh977v3R9329ffNAE9FFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAFdP8Aj5l+/wDdTr9zv9339fwqxVdP+PmX7/3U6/c7/d9/X8KsUAeQfBn/AJA3iT/sZte/9L5a9fryD4M/8gbxJ/2M2vf+l8tev0AFFFFAH//T/fyiiigAooooAKKKKACvmT4Yf8nKfG7/AK5eGP8A0jmr6br5k+GH/Jynxu/65eGP/SOagD6bqC2/1Z+/99/v9fvHp7f3fbFT1Bbf6s/f++/3+v3j09v7vtigCeiiigAooooAKa6JIjRyKHRwQQRkEHqCKdRQBxmjSv4evU8LXrE2z5OnSsc7o1GTbsT/ABxj7v8AeT1Ksa7Os3VtKtNZsXsL0HYxDKyHa8bqcq6N1VlPIPrWVZazLaX0eha6Ql3JkwThdsNyATwpycSBcFkOM8lcjOOqa9p7636/5/5/fttw037F8kvh6P8AR/p92++lq2lJqcICSta3UXMNxGBvjbg8Z4KnA3KeCOvrWZpuvTLdJo3iGNbTUTny2XPkXIHVoWPfHJjPzL7rhj1FU7/T7LVLV7PUIVnhfqrDPI6EdwQeQRyDyKinVVuSa0/Ff12/I1q0Zc3PTev4P1/z/PYuUVya2viHROLGT+17MdIp323KD0WU/LJ7CTB9XNXLXxPpE8q2txIbG6bpBdL5MhP+zu4f6oWHvQ8O94ary/yFHFR2n7r8/wBHs/zOgooorA6ivB1l+/8AfP3/AMPu/wCz6VYqvB1l+/8AfP3/AMPu/wCz6VYoAKKp3uoWGmwm51G5jtYh1eVwi/mcVgnX73URs8OWL3Ab/l5uAYLce43DzH9tq7T/AHhWsKEpK6WnfoYVMTCLs3r23f3G/fX1nptrJe38ywQRDLO5wB/9c9h1Nc1HBdeKXiudSt2tNLhkEkVvKMSXDLyjzIR8iqfmVD8xIUttI21bs/Di/a49U1uc6lfRnMbMu2GE/wDTGLJCn/aJZ+27HFdLWnPGHwavv/l/n+Bl7OdT+IrR7d/X/JfNvYKpajqNppNlLqF9J5cEIyx6nngAAckk4AA5JIA5o1DUbLSrOS/1CYQQRD5mb34AAHJJPAAySeAM1hWFnd6vejWtYjMUMTZsrVxzGMY86Qf89WycA/cXAwGLVNKkrc8tvz8l/Whdas0+SHxP8PN/1r97T9Asbx5ZvEGsJ5d9egKsRwfs0A5SLI/i/ikI6scchVrp6KKirUc3dmlGkoR5V/w/mFFFFZmoUUUUAFQRf62b7/3h977v3R9329ffNT1BF/rZvv8A3h977v3R9329ffNAE9FFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAFdP+PmX7/3U6/c7/d9/X8KsVXT/j5l+/8AdTr9zv8Ad9/X8KsUAeQfBn/kDeJP+xm17/0vlr1+vIPgz/yBvEn/AGM2vf8ApfLXr9ABRRRQB//U/fyiiigAooooAKKKKACvmT4Yf8nKfG7/AK5eGP8A0jmr6br5k+GH/Jynxu/65eGP/SOagD6bqC2/1Z+/99/v9fvHp7f3fbFT1Bbf6s/f++/3+v3j09v7vtigCeiiigAooooAKKKKACqWoadZaraPY6hCJoZMZU9iOQQRyCDyCMEHkHNXaKcZNO6JlFSTTV0catxrXhr93fiTVtMX7twi7rqFfSVFGZQP76Dd/eU8tXUWV9Z6lbJeWE6XEEgyrxsGU/iKtVz134ctJbl9Q06V9NvX5aWDAEh/6aRkFH+rDd6EV0OcJ/Fo+/T5r/L7jlVOpT+DVdnv8n+j+86GoLm1tb2Fre8hSeJuqSKGU/UHiueF94k04hNRsV1GIf8ALazO1/q0Ejcf8Bdj7VZt/FGhXEwtTdrBcH/ljODBL+CSBWP4Cl7Ca1jr6f1p8yvrVN+7PTyen/D/ACuRf8ItpcXOntPYdMC3meNBj0jyY/8Ax2nf2Pq0YxDrlweMDzY4Hx+Uan8zXRdeRQSAMml9Yn1d/XX8x/VKa2VvS6/I5eHTdcZpM6zKoDEcwRg9vu5B49KlPh+ebIvNXvZ1P8KukI/AwojfrW1bOriRkZmBc8t0/wCA/wCzVmj6xLy+5f5B9Vh1u/m3+piWnhzQ7KcXcFmhuRwJpMyzf9/HLP8ArW3VS8v7DTojPqFzHaxDq8rqi/mxArGPiSG4GNGtZ9SY9GjTZF/39k2oR/ulj7UclSp7zu/P/ghz0qXuqy8l/kjpK5/UfENvaXB06xjbUNRwD9nhxlAejSsfljX3bk/wgniqp03X9V/5C94LGA9beyY7iPRrggN/3wqH3Nbmn6bYaVbi006BLeIEnCDGSepJ6knuTye9Vywj8Wr7dPv/AMvvJc6k/hXKu73+S/z+4xbLQp57uLWPEUi3V7FzDGmfs9sT/wA81PLPjgyNz6BQSK6iiisqlVyd2bUqMYK0f+HCiiiszUKKKKACiiigAqCL/Wzff+8Pvfd+6Pu+3r75qeoIv9bN9/7w+9937o+77evvmgCeiiigAooooAKKKKACiiigAooooAKKKKACiiigCun/AB8y/f8Aup1+53+77+v4VYqun/HzL9/7qdfud/u+/r+FWKAPIPgz/wAgbxJ/2M2vf+l8tev15B8Gf+QN4k/7GbXv/S+WvX6ACiiigD//1f38ooooAKKKKACiiigAr5k+GH/Jynxu/wCuXhj/ANI5q+m6+ZPhh/ycp8bv+uXhj/0jmoA+m6gtv9Wfv/ff7/X7x6e3932xU9QW3+rP3/vv9/r949Pb+77YoAnooooAKKKKACiiigAooooAKKKKACobi2t7uMw3USTRnqrqGB/A1NRTTtsJpNWZzo8J6DH/AMelubP2tZZLcflEyitG105LS1ktVmlnV88zSNIwyMY3E7sfjWjRWkq83o3cyhhqcXeMUjLitpJba4t2mmiLu2GDfMo4+4zDp74qj/wjVm5Jubq8nz2a6lVf++UZR+lbUHWX7/3z9/8AD7v+z6VYpRrSj8LsE8PCXxK5jWnh7QrGTz7WwhSbj95sBkOOmXOWP51s0UVM5yk7ydy6dOMVaKsFFFFSWFFFFABRRRQAUUUUAFFFFABUEX+tm+/94fe+790fd9vX3zU9QRf62b7/AN4fe+790fd9vX3zQBPRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBXT/j5l+/91Ov3O/3ff1/CrFV0/4+Zfv/AHU6/c7/AHff1/CrFAHkHwZ/5A3iT/sZte/9L5a9fryD4M/8gbxJ/wBjNr3/AKXy16/QAUUUUAf/1v38ooooAKKKKACiiigAr5k+GH/Jynxu/wCuXhj/ANI5q+m6+ZPhh/ycp8bv+uXhj/0jmoA+m6gtv9Wfv/ff7/X7x6e3932xU9QW3+rP3/vv9/r949Pb+77YoAnooooAKKKKACiiigAooooAKKKKACiiigAooooArwdZfv8A3z9/8Pu/7PpViq8HWX7/AN8/f/D7v+z6VYoAKKKKACiiigAooooAKKKKACiiigAooooAKgi/1s33/vD733fuj7vt6++anqCL/Wzff+8Pvfd+6Pu+3r75oAnooooAKKKKACiiigAooooAKKKKACiiigAooooArp/x8y/f+6nX7nf7vv6/hViq6f8AHzL9/wC6nX7nf7vv6/hVigDyD4M/8gbxJ/2M2vf+l8tev15B8Gf+QN4k/wCxm17/ANL5a9foAKKKKAP/1/38ooooAKKKKAPHvEPx5+F/hqwiv73Uri7+0X97psUGn6fe6jdyXWmyNFeKltaQyzssDqRI4Qoo53YIz6J4Z8TaD4y0Cw8U+F72PUdK1OJZre4iJ2ujexwQQeCpAKkEEAgivlnxb8J7L4Q6+fi/4a8Qa9NLdalNYx6UgsZrW2l8aaraQ3dzFvtDcBYrho7pkaZkxFjAUmvpPwF4L074e+E7DwlplxPeRWfmu9xclDPcTzyNNPPKY1RN8sru7bEVcsdqgYFAHYV8yfDD/k5T43f9cvDH/pHNXsvinwDo/i+4gudTvdWtWt0KKNO1a/05CCc5dLSeJXPuwJ7ZxXk/wp8E6L4T+J/xOs9LkvZTero7SzXl9c3ty2LaRR+/uZJJcKPujdhe2KAPo6oLb/Vn7/33+/1+8ent/d9sUSW6yeblnHmpsO12GBzyuDweeo56egqvFbrJHNl5h5u9DuZlIAZuVx068Ec4288CgC/RUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAJB1l+/8AfP3/AMPu/wCz6VYqglushucvMPNyhyzLgDPKY6deo59+BViS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9QRf62b7/wB4fe+790fd9vX3zRJbrJ5uWceamw7XYYHPK4PB56jnp6Cq4t1klu8vMPNAQ/MVUDb1jx0PPJHOe/AoAv0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FACJ/x8y/f+6nX7nf7vv6/hViqJgWWe5UtMPMjRDhiqgfNyhHRueSOenoKnkt1k83LOPNTYdrsMDnlcHg89Rz09BQB5P8Gf+QN4k/7GbXv/AEvlr1+vH/gx/wAgbxJ/2M2vf+l8tewUAFFFFAH/0P38ooooAKKKKACiiigAr55XxVZ+B/iz4zl17TNZeDVYNKktprHRdS1GCQRRyo482ztpowysOULBgCDjBBP0NRQB5L/wurwd/wBA/wASf+Errv8A8g1y9l+0n8M7mOY2Vp4rvUhuLiBpIfB3iO4QSwTPFLH5kWnMuY5FZCucqVKthgRX0FXmvwp0LVfDvhe+0/WYDbXEuu+IrxVLK2YL3WLy5gfKkj54ZUbHUZwQDkAA5P8A4aL+H/8A0CfF/wD4RHif/wCVtH/DRfw//wCgT4v/APCI8T//ACtr3aigDwn/AIaL+H//AECfF/8A4RHif/5W0f8ADRfw/wD+gT4v/wDCI8T/APytr3aigDwn/hov4f8A/QJ8X/8AhEeJ/wD5W0f8NF/D/wD6BPi//wAIjxP/APK2vdqKAPCf+Gi/h/8A9Anxf/4RHif/AOVtH/DRfw//AOgT4v8A/CI8T/8Aytr3aigDwn/hov4f/wDQJ8X/APhEeJ//AJW0f8NF/D//AKBPi/8A8IjxP/8AK2vdqKAPCf8Ahov4f/8AQJ8X/wDhEeJ//lbR/wANF/D/AP6BPi//AMIjxP8A/K2vdqKAPCf+Gi/h/wD9Anxf/wCER4n/APlbR/w0X8P/APoE+L//AAiPE/8A8ra92ooA+ebD9pj4Y3bXS2Vp4sujbzNFKE8G+JJPKlABMbbNOOxgCDtbDDIJHIrQ/wCGi/h//wBAnxf/AOER4n/+Vtdb8O9C1XRLvxhJqkBgXU9duLy3JZW8yB4IEV/lJxlkYYODx0r0mgDwn/hov4f/APQJ8X/+ER4n/wDlbR/w0X8P/wDoE+L/APwiPE//AMra92ooA8J/4aL+H/8A0CfF/wD4RHif/wCVtH/DRfw//wCgT4v/APCI8T//ACtr3aigDwn/AIaL+H//AECfF/8A4RHif/5W0f8ADRfw/wD+gT4v/wDCI8T/APytr3aigDwn/hov4f8A/QJ8X/8AhEeJ/wD5W0f8NF/D/wD6BPi//wAIjxP/APK2vdqKAPCf+Gi/h/8A9Anxf/4RHif/AOVtH/DRfw//AOgT4v8A/CI8T/8Aytr3aigDwn/hov4f/wDQJ8X/APhEeJ//AJW0f8NF/D//AKBPi/8A8IjxP/8AK2vdqKAPCf8Ahov4f/8AQJ8X/wDhEeJ//lbWfaftMfDG5ur6C1tPFlxLaSrHPGvg3xI7QSGNJAjqunExko6vtYAkMGA2sCfoavNvBOharpXi34g6lqEBhttZ1i2ubR9ynzYU0qxt2YAEkYlidcNg8ZxggkA5L/hov4f/APQJ8X/+ER4n/wDlbR/w0X8P/wDoE+L/APwiPE//AMra92ooA8J/4aL+H/8A0CfF/wD4RHif/wCVtH/DRfw//wCgT4v/APCI8T//ACtr3aigDwn/AIaL+H//AECfF/8A4RHif/5W0f8ADRfw/wD+gT4v/wDCI8T/APytr3aigDwn/hov4f8A/QJ8X/8AhEeJ/wD5W0f8NF/D/wD6BPi//wAIjxP/APK2vdqKAPCf+Gi/h/8A9Anxf/4RHif/AOVtH/DRfw//AOgT4v8A/CI8T/8Aytr3aigDwn/hov4f/wDQJ8X/APhEeJ//AJW0f8NF/D//AKBPi/8A8IjxP/8AK2vdqKAPCf8Ahov4f/8AQJ8X/wDhEeJ//lbR/wANF/D/AP6BPi//AMIjxP8A/K2vdqKAPB9N/aG+HOp6leWFpbeJjd2scMktvJ4U1+OSOOYyCNzG9grgOUcBiMMVIB4ON7/hdXg7/oH+JP8Awldd/wDkGtHR9C1W1+LPirxHPAV07UNI0W2gl3KQ8trNqDTLtB3DaJozkgA7uM4OPSaAPHfgj9om8MatqM1nd2Meo69rV1Al7azWU5gmvZWjdoLhI5UDr8y7lBIIPQivYqKKACiiigD/0f38ooooAKKKKACiiigAooooAKK4Lxx401bwf/Zv9l+DtY8W/b5jFJ/ZP2L/AERRj95P9surX5Of+We9uD8vTPxL8NPD8Hgf42eH/F06+EfFsvxG8T+LLVdQsdJaPX9Olie/ujv1GWZmkit44DYzJ5MXlsY1BbkuAfovRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRXnvxbvde0z4U+M9R8LbxrVrouoy2Pl53/aktpGh2453bwMY70AehUV8R/AjR/B/hv4raPZfC9IIdF1jwFY6nqgtCClzdvcItneT7SQ9xPGbjdM2XkC/MzbRj7coAKKKKACiiigAooooAKKKKACiiigAooooAKKCcDNfnR+zT4eg+HnxE8J28i+EfFNx8Q9A1fVv7f0PSWtNVDwXdtJO11eSTSSXME73QBLJCUlRVKc4QA/ReiiigAooooAKKKKACiiigAooooAKKKKACiivOvix4Zh8XfD7WtFudQvtOgkt5HlfT7hrWaRI1LGIzJiREfGH8tlYjIDDNAHovXkUV+bnhPRfBPi7R/gz4a+LUFtd+FIPhN/aMEN8R9mF1BHpyz3Sh8KJ7aBgUkHzxB2Klckn7C/Z5v/ABHqvwC+GuqeMJJJtdvPDWjzX8kxJle6ks4mmaTPO8uSWz3zQB7DRRRQAUUUUAf/0v38ooooAKKKKACiiigAooooAK4zTfhx8PdG8VXvjnSPDGmWPiPUlKXWpwWUMd7OrEFhJOqCRwSATljnAz0FdnRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAcd4R+HngH4fpex+BPDWm+HF1KXz7oadZw2gnl5+eTyVXe3J5bJ5rsaKKACiiigAooooAKKKKACiiigAooooAKKKKACuM8O/Dj4e+ENY1LxB4U8MaZo2qayc313ZWUNvPdHcW/fSRorSfMS3zE8knqTXZ0UAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFRzQxXETwToskUilWVgCrKRggg8EEdRUlFAHBa98K/hj4q0bSvDnibwjpGraTofl/2fZ3dhbz29n5SeWnkROhSLanyjYBheOnFd4AAAAMAUtFABRRRQAUUUUAf/0/38ooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAr5ttP2kvDd38QLfwasVt9nu/EV14XilF6pvBf2lpLcO72fl5WAvDJCH8wnftO3a2V+kq+erb4CwQeLbTVG1VW0PT/E154sgshbkTjUr20mtnDXHmkGEPcTTBREG3lBu2oQ4B9C0V5P8ABnwXdeAfCd74duYJYEGs6zcwedcNdyvb3V/NPDI8zvI7s6OGJdi+T83zZr1igAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooA//9T9/KKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooA//Z)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7XI7orJPHesR"
      },
      "source": [
        "- We choose a simple initial proposal density which maybe far from target density.\n",
        "\n",
        "- But we know how to simulate from it"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2hT_qSjPF5FN"
      },
      "source": [
        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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "khvrTSThIawc"
      },
      "source": [
        "- We simulate from proposal density and accept a value with certain probability\n",
        "\n",
        "- Once we accept a value; we update our proposal density and simulate from new proposal density"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "m9IhucjoF_Rs"
      },
      "source": [
        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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cbIstJO9KcjO"
      },
      "source": [
        "- Since the updation of proposal density was made using the transition kernel density; it creates a **Markov Chain**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "5Iqz5_LUGHzP"
      },
      "source": [
        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IG8Sf8AYza9/wCl8tev0AFFFFAH/9b9/KKKKACiiigAooooAK+ZPhh/ycp8bv8Arl4Y/wDSOavpuvmT4Yf8nKfG7/rl4Y/9I5qAPpuoLb/Vn7/33+/1+8ent/d9sVPUFt/qz9/77/f6/ePT2/u+2KAJ6KKKACiiigDF8R+H9J8V6Df+Gtdh+0WGpQvBMmSpKOMZVhgqw6qwIKkAgggV8r6J431b4W/EcaL8SLjzDeW9vZXeoPhI7mKGQx2Oq9MDeZRa3wBxDN5L4EMgavsKvPviP8N9B+JWiJpmsRqtxauZbS4KBzDIVKMGXI8yKVGaOaInbLGzI3XI4MbhpytUpfEvx8v68+59lwpnmHouWDzBN4epo7bxb+1H7k2uvLFtS5UjhfF3gD4Y/E9vEontDZazpTfY765t1ijuHAiS4RZkcPFcxFXVkW4R0zkgAjNeSR+J/i3+zzFAfFls/inwOETNzG5MtknoXmdmiwP4J5GiP8NzF8kB86i8U+K/g94+g0XWIDPe28S2SwSy7v7U0nzHW3tmmm2iSWIsfsNySQTIbO6KTSRyyfaFxbeCvjV4NguYJ5pbJ5GkgngeS0u7O6h3RNj7skM0RLxyRuP70cikFlPk0nGu5Sp+7VX9a9/Pz+R+j5jCrlEKNLHL6xgKltX73LpvB3Ti3ZuNnblbWrUrfI3xBtfFnxL8TeMvGvwj0GPxTpPifw/4UsVuWWwc2tzoOt3tzf272WoyxFb5La9WW1WePyfNjxMy4UNgfDj4CfEef4l2niPx54cu7rSD46n8QvJrk2kTXTW7+FTpi3FxDp222Ev2xQNkUZ2gq3O0vR4k8A/En9n3xT/wlng+XzkvGSBLmJAljdF5Bsgu7YMkcEjElYxuS2Z2/cvZOxjm+u/hB8atC+KmnGF4To/iG13i60ydiJV8pgkkkQcI7xBztJZFdG+SVEcFa7cFmnPL2NZcs/wfp/Xpc+S4r8P1h6H9p5VU9thX1XxR8pL/ANustdJKL0Pzmg8PXXijxl4J+C2iWFrDrGman8RIn8T2t9aT3En9oWepwG4aGGVruF1nniF19oijCXARIy4II7bQvgF8WF8A6wJPDWujVbDQtC06Kx1O98OIl3JpWow3bW9imlwQIYo0icQTXlxG370qY1yzD9OoNI0m1v7jVLWyghvbsATTpGqyyhegdwNzY7ZNaNewfmZ83fGbw74p1fWfhr8QdI8Kz+Jrfwpf3N1e6Estml3i8sJrZJYvtM8dq81s8mCDMBsZyjMwUH4htf2afi7pXhzQ3m8L6vbx3Wg+IdFn0PQbrw66WP23Wbu+himk1eO4jFrcQTxq8ltmVPKTdGxwF/W+igD8/n/Z98Yx6T47K6J9t1m/8U+A7yyvZ57eW5uLPw+mii5m+0EQn901tdEbkjZ8MUj+dQfOPEXwJ+M1zZePvCvhTwndQaTqOieKrW3g1S40e7hjutVuBNAmiagjR6gkN2w824ivgiIQoByq1+oEHWX7/wB8/f8Aw+7/ALPpVigD83Pi1pHjb4cWPjHxjpOmR2Xiyx8e2us+EZLi4thDrZ1qzg0ma0iVJGn3qssu+OSJQWEbLuALL93fDjwVZfDjwD4e8B6dIZoNBsbezErffmaFAryue7yNl2J5LEk11Vzp9heS2893bRzy2j+ZCzorNE5UruQkZVtpIyOcEirdABRRRQAUUUUAFFFFABRRRQAUUUUAFQRf62b7/wB4fe+790fd9vX3zU9QRf62b7/3h977v3R9329ffNAE9FFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAFdP+PmX7/3U6/c7/d9/X8KsVXT/AI+Zfv8A3U6/c7/d9/X8KsUAeQfBn/kDeJP+xm17/wBL5a9fryD4M/8AIG8Sf9jNr3/pfLXr9ABRRRQB/9f9/KKKKACiiigAooooAK+ZPhh/ycp8bv8Arl4Y/wDSOavpuvmT4Yf8nKfG7/rl4Y/9I5qAPpuoLb/Vn7/33+/1+8ent/d9sVPUFt/qz9/77/f6/ePT2/u+2KAJ6KKKACiiigAooprOqDLkKPfigDzz4m/C/wALfFbw82g+JYfmTc1tcoFM1tI6lSybwylWUlXjcMkiko6spIr4XtNV+IH7Mni9LfxPcCa21GYg3B3my1S2hRfnld8+VcxRghZmd5FRdt15sCC8j/S2sHxN4X0Dxjo0/h/xNZJf2FxjdG+QQynKujKQyOhAZHUhlYAqQQDXk4/LFUftaT5ZrZ/5/wBfhofo3B3HssDTeX4+HtcJP4odVfrC+zvrbZtJ3UrSXPeDfHHhH4p6BPLpuJkK+TfafdoongMqcxXEJ3DDqcqRujkQ7kZ0IY/GXxk+BF38P9TtvHXgW7lsNPsmDQ3SM/m6Q0akRrI6pI7WWP3aSlHa0B2SLNZF4UzfFvwi8cfADUo/F3g25utU8PWCusNxblVvtOhYs7JKMeXJbhjkq6m3Yne62zh7tvfvh/8AtG6Jr2kb/GoighjVRNqdujmwXzFyv2uNt0tiXU5Bm3QMMGK4lUg15dWtCuvY4tcs1s/19P60e36FgMtxOT1P7V4bqfWMJPSUOv8AhcbP3knppfVpxlC7lo/BD45f8J3u8HeNo49L8ZWO9Xh+VFuxCF3yRKGcK6hlaSIO4CskkbyQyRyN9H18D+KvhDpzeJrXTvh7cQSXC2kmseF5YLhUPk2UiLc6aJ1LbUhNwsljNyIPNkhINtvik98+Cfxcl8d2X9g+JgIPEdpG8mfLMAvIIpPJeVYiWMUsMo8m6gyTDMMZaN43fuy7GzT9hX36Pv8A8H81bvr8jxvwphalN5tlCtTdnOGzhe+qWvu3TTV3yyUkm1Fte+UUUV7h+SleDrL9/wC+fv8A4fd/2fSrFV4Osv3/AL5+/wDh93/Z9KsUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFQRf62b7/3h977v3R9329ffNT1BF/rZvv8A3h977v3R9329ffNAE9FFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAFdP+PmX7/3U6/c7/d9/X8KsVXT/j5l+/8AdTr9zv8Ad9/X8KsUAeQfBn/kDeJP+xm17/0vlr1+vIPgz/yBvEn/AGM2vf8ApfLXr9ABRRRQB//Q/fyiiigAooooAKKKKACvmT4Yf8nKfG7/AK5eGP8A0jmr6br5k+GH/Jynxu/65eGP/SOagD6bqC2/1Z+/99/v9fvHp7f3fbFT1Bbf6s/f++/3+v3j09v7vtigCeiiigAooooAydfmvLfQtSuNPbbdRW0zREAMRIqEqcHg8447023t0n0qzW2mNzGsSFJZWJdxswHLDqSDk8c+1bFchocg0W+fwpP8sShprBj0a3z80Q94ScY/uFT646aesGlutfl/wP8AM46vu1E5bPT0fT7/ANEja1GLWWWKXSZ4lePO+OZCUlzjHzKQUI9QGHPQ1nQ+JYoZ0stdgbS7mQ7UMhDQSH0jmHyknsrbWP8AdrF0LQdF1KC8k1OD7TqCXdwlxJIxMqtvJUK2cqvllSgXACkVo3Hh2+ggeHS7w3Fu4Ia01DNzC6ntvbMi59y4/wBmujkpJ+zk9utrfjr+K+aOX2lZr2sFv0vf8Ha3ye/RnX18weOf2c7OXVV8ZfCm7/4RfX4SziKJ2htZS5LPsKBjbmQnL7Ukgkb5preZsEemWt7qOhXMdlaRtasxwunXcmYZPazuuRn0jf8A75Qc13mla3ZauHWHdDcQ4823lGyaIn+8voezDKnqCRXBmGUxnH31ddH1X9d1p5n0nDPGOJwNZzws+VvRxesZLs09GvJq63stzwTwxrvgnSdXttDudATwR471ULDMz2dvbvfOgGXjuYEMd0hb5wiNuGB5ix9Kv+KfhXrF14q1vxF4TaOwuylrqmmzMcLHrUQkhnDBTu8m7thFFcjGCoDL8/K+263oOieJdNl0fxDYQanYz43wXMayxtjplXBGR2PauDj+Gl3pCpB4Q8WavotonAtWkiv4QPRTfRzyoB/CqyBR024wB5M8I7csldeWj/y873WvQ+0w3E1N1HXo1PZVJK0lO84NXTsmryUdFFQcZrlunKzaOs8G+Jrfxl4X03xNbwvai/iDvBJjzIJR8ssL443xSBkbHGVOK6auZ8H+FrHwZ4etvD1hNNcpC00sk9w++aee5leeeVyABuklkdyFCqucKqqAB01ehR5uRc+9tfU+KzL2H1mp9W/h8z5b78t9PwK8HWX7/wB8/f8Aw+7/ALPpViq8HWX7/wB8/f8Aw+7/ALPpVitDiCiiigAooooAKKKKACiiigAooooAKKKKACoIv9bN9/7w+9937o+77evvmp6gi/1s33/vD733fuj7vt6++aAJ6KKKACiiigAooooAKKKKACiiigAooooAKKKKAK6f8fMv3/up1+53+77+v4VYqun/AB8y/f8Aup1+53+77+v4VYoA8g+DP/IG8Sf9jNr3/pfLXr9eQfBn/kDeJP8AsZte/wDS+WvX6ACiiigD/9H9/KKKKACiiigAooooAK+ZPhh/ycp8bv8Arl4Y/wDSOavpuvmT4Yf8nKfG7/rl4Y/9I5qAPpuoLb/Vn7/33+/1+8ent/d9sVPUFt/qz9/77/f6/ePT2/u+2KAJ6KKKACiiigArH1vSE1i0ESyG3uYGEtvOoy0Uq9GA7jkhl6MpIPBrYoqoTcXzIipTU4uMtmeYRavNZX02tzw+RdWypFrFsmWwgz5d3F3ZAM84yUyD80e2vQL7Ubew06XVJMyQQp5pKYb5AMlhyBgDnr0rO1nRDfy2+pWMn2fUrPPlS9nQ/fik65jfHPBwcMOQK47TtQh0F57J4v8AiUbttxbHBbTmlOOVPW2fkggYTnHyfc9JwjWSlHddP0/y+7dK/kKpPDycZvR7Pz7/AOf/AIFs3y+hXsFtfIthdwJc29wDvVwGXAx2IIPOK8+1Pw/cWsvmaZNJfx2J2r5cgOoWW4BsRSNnzUIIJilzkdC3C1taPct4fu4vDeoPvtpMjTrljkPH1+zs3/PRB93++gzywatfVPD9tfzjULaRrHUUXalzDgPgdFcH5ZE/2WBHpg81NKo6MrX0f3P+tnbXprsXWpKvDmt7y37ry8+6vo9HpozC0TxgkiwwavIjLM/lQ3kYKQyyA48uRW+aCbsY36n7pJ4HeV5TqFm9xeG01RIdP1i6XyxLsL6fqagf6uVCfvY/hJ3r/AzqDmTRdcv9EkexvopTaW2PNilYyT2angOH/wCW9t6SffTow4IXSvg4zXNT37f5f1vpo7IxwuYSg+Stt37eT/q9tdVeR6lRUUM0NzClxbyLLFKoZHUhlZTyCCOCDUteS0e4nfUrwdZfv/fP3/w+7/s+lWKrwdZfv/fP3/w+7/s+lWKBhRRXPeHvFegeK11FtAuvtQ0m9n066+R08u6tiBLH86ru2k/eXKnsTQB0NFY0HiHRbnxDeeFILtH1fT7W2vbi2GfMjtrx5o4JG7bZHt5gvuhrZoAKK57wt4r0Dxpo41/wzdfbLBp7q2Emx4/31lPJbTrtkVW+SWJ1zjBxlSVIJ6GgAooooAKKKKACiud0fxXoGv6truh6TdeffeGrqKz1CPY6+RPNbRXaJuZQr5hnjfKFgN2CdwIC3fivQLDxRpngu7utms6za3l7aQbHPmW9g0CXD7wpRdjXMQwzAtu+UEBsAHQ1BF/rZvv/AHh977v3R9329ffNT1BF/rZvv/eH3vu/dH3fb1980AT0Vyc/jfwzaazD4fvbp7W+ur8aZbpPBNEtzdmze/8ALt3dAk2LaN3LRsygo6Eh1ZR1lABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAFdP+PmX7/3U6/c7/d9/X8KsVXT/j5l+/8AdTr9zv8Ad9/X8KsUAeQfBn/kDeJP+xm17/0vlr1+vIPgz/yBvEn/AGM2vf8ApfLXr9ABRRRQB//S/fyiiigAooooA+Er3w38atf8ZHwd8QNL8U/8IZaX/inU/wC0dE1qK0a8WW5Fxo1uJbW+hv41hg82IQ7FjMhiBYopFfUPwcs/Gth8M9CtPiE8767HE4m+1yRTXSRmRzAlzLCWjknSHYszoSryBmBIOa9MooA4nxTpPjvUbiB/CPiK10WFEIkS4043pds8EN58O0AcYwfrXz58E7LxDZ/Hn422niDU4tS1TZ4cJuobb7NHtazm2DyfMk+7zk7/AJvavrmvmT4Yf8nKfG7/AK5eGP8A0jmoA+lJEuG8zy5Qu5MJlc7X5+Y889uOOnXmq8SXDRzeXKy7t4TzFyVfc3zdRlem0ccAc81fqC2/1Z+/99/v9fvHp7f3fbFABIlw3meXKF3JhMrna/PzHnntxx0680SJcN5nlyhdyYTK52vz8x557ccdOvNT0UAQSJcN5nlyhdyYTK52vz8x557ccdOvNEiXDeZ5coXcmEyudr8/Meee3HHTrzU9FAEEiXDeZ5coXcmEyudr8/Meee3HHTrzRIlw3meXKF3JhMrna/PzHnntxx0681PRQBBIlw3meXKF3JhMrna/PzHnntxx0681ha1oMmpS/brO4Fpfwx7IZggbAJyySA/fjfjKnpjIIPI6MkAZPAFU4bs/Z5JrhdrRDcyqCxAxngLkn8OT6VdOo4vmiZ1aUZx5ZbHkUbmztbvQNVttuniMmazIMr2WOVuLbGGltt2CNuGiPTGML1djrN3pqx2mrXy3NneIq2GpBVaJy2dgmZTtEhyMHhHxwQx2jaubfRfFtos1pc5ktXPlXEDATW8uOcH+E4+8rDBHDAiuDKaj4fnfTLq1hlS+Yh7cjbZ327O7yNxKwTt1aJvkc8qc7mHrxnCurde36r9fx25j5+dOphpJrbo/0a7du32d+U9P1HTk1O2ns7na8MyAKrLna4JIfOQcg4IxggjIOenA6hptzazxadrd43krgaZqmP39tOTtEczdG3ZABPyyAbX+bBazpN/c2EDS6EZNU0yE7JbOQ4vrJhyUAc5cAfwOd2OVZhha663u9F8TadKkTJeWsoaOWNhyM8MkiNhlPqGAIrmjz0X3j+X+T/B+aO2ThiF2l+D/AM15rVeTPN7O41vw3qE9jHGqO6tI9pEhMcoz891ZAnqCQZbfrnlTkgv6jb3C6ja/bdOukkguIgYXVdwB5+brz244xj8uMj0zzpG8IaxLIz26/aNNvQ2Jwi/Kfn/56wlgpJyHRhuBywqjp0+q6LqN2HiL3Kfvby2iGEuY84+2Wqno/wDz1izyenzEGTfEUo1VdfF+f9fg9H5c2FrSoe6/h/FPt/W61Xn6EiXDG58uVl3ZCb1ztfn5hyMr0wPY888fjh4zk8ar8BdL0C40DxBpk/w/+EXivR9XuLrTruxhh1UDTI444buZFjmaQW8jxyRFkKjcrHJA/Y3S7201G1+3WMxnt5juR85VgQPun+7/AFzUOv6Bo3irRL7w34is49Q0vUoXt7m3mG6OWKQYZGHcEcGvIaadme/GSauj8wNT8Cx3M+r3XgLwTremfBdtQ8LHWdDl0m9hmvZLddQGpzLpcka3NxGHk003e2J/tPkyYL4cnjbfwBNJ4dtvDJ+Hd7beDL7xB4tvbBdc8M6vrSWkck1sLFBo0LwMsk8LubSe72parEycOxA/YiikM/GSD4YfEVvAT3J8MapY+NdX+Gfw/tb2+k0W8urudNJ1S4HiGymlj8qR7h7L7Or2xnjmuo1VYydu5O60f4d3dlpGmL4k0O/1v4RjxTc3F1o9l4P1DTLOMSaP5cE0OgvPe30lgt5y0ZiRRcEzLH5f7yv1hooA/Nn+xNa8Kf8ABOHxxps+naloU8GjeK3gsr1ZINRgSS/vpYS7SfP5jRMjBj14YEg5rkrv4TWPjLWpdG8A+CtY0b4V6lrXhKOfTLzTb2wWS9gGo/2tcSWk6RyiB4ZLSK6lZfLuHVsu3zFv081/QNG8VaJfeG/EVnHqGl6lC9vc28w3RyxSDDIw7gjg1r0AfmFp/wAPNX074n29lpfhjVrP4haf45EttrS2F1/Z8Hg6NzshXUCotTa/2eFtxZiXeLgb/L3c14J8MfCPizXPgj4fv/gH4f13Stel+H+oxeJLw21zYvqlxMbU2qW95MsaXdw8ay/Y5IXf7NH+7LJvVR+3FZGgaBo3hXRLHw34ds49P0vTIUt7a3hG2OGGMbURR2CgYFAH5c+Ivhzo+u2njB/h54X13wx8Mbibw1JFYT+GLu6tJdYtzfG/uLrw9cCC7ubMxvZpc+XGTJMolG7ypGrifHnhHxn4r+GWi+HNS+Gh082fhrWYtIVtB13XF+2G/uUtl061knQ6PPNEkE8Ut62+2jkWFQFicD9k6KAPkv8AZy0jxvBJ8R9T162u9OvdaudDnt5dQtpY2kmXwzpcMspEoUvtnR0kHZ0dSQwIHwjYfDDxxJplrH8O/CfiHQfiRb/DLxHp/ifU59PvIPtniWWXTWk2X0giiurq6MVwYZ4ZcFCm2QBQI/2iooA/Jnw18H9R1rT00HTNIuz4O1LxN4ZeewtfCWo+FtPTykuxfSmzvLy4uWV0aGO8dkihkwBukZpTWX8QvhDfado91o8fhu//ALG8O614sj0LSbnwzfa5of2e6e1kgjigsHS7s5GO/wCwXkC+XB++UEF0U/r1UEX+tm+/94fe+790fd9vX3zQB+Yl38PNZ134j+H/ABj8QvAd2baDx5oty0M+nz6ybGO58DCzZxKEkeaCHVPs8cs4+RZYhJI6lCy+ufsxeH/FUvizXbDxC0otvhDaS+BNHmlPmecouPtTXJJPzl7BdLViTkSJKpJ5r7mrmvCPg7wv4D0OHw14O02HSdMgaSRYIFwu+Vy8jknJZnYlmYkkk5JoA3pEuG8zy5Qu5MJlc7X5+Y889uOOnXmiRLhvM8uULuTCZXO1+fmPPPbjjp15qeigCCRLhvM8uULuTCZXO1+fmPPPbjjp15okS4bzPLlC7kwmVztfn5jzz2446deanooAgkS4bzPLlC7kwmVztfn5jzz2446deaJEuG8zy5Qu5MJlc7X5+Y889uOOnXmp6KAIJEuG8zy5Qu5MJlc7X5+Y889uOOnXmiRLhvM8uULuTCZXO1+fmPPPbjjp15qeigCCRLhvM8uULuTCZXO1+fmPPPbjjp15okS4bzPLlC7kwmVztfn5jzz2446deanooAgkS4bzPLlC7kwmVztfn5jzz2446deaJEuG8zy5Qu5MJlc7X5+Y889uOOnXmp6KAKBS4a4uRHKy7okCblyiv83zDnntkcdB61YkS4bzPLlC7kwmVztfn5jzz2446deaRP8Aj5l+/wDdTr9zv9339fwqxQB4/wDBj/kDeJP+xm17/wBL5a9gryD4M/8AIG8Sf9jNr3/pfLXr9ABRRRQB/9P9/KKKKACiiigAooooAK+ZPhh/ycp8bv8Arl4Y/wDSOavpuvmT4Yf8nKfG7/rl4Y/9I5qAPpuoLb/Vn7/33+/1+8ent/d9sVPUFt/qz9/77/f6/ePT2/u+2KAJ6KKKACiiigAooooAxvD1vbWeiWdhaTC4SzjEBkAxueH5HJGTg7gcgk81pLJaxSraIyJIylxGCAxUEAkL1xkjJrMVtL0W7W3yYX1Wd2XO4o02zLAHopYKTjjJBI5zTodPkj1661QgGOe2ghBz8wMTysRj0O8flz2oAS3u4NXhuYPJubTK7XLo8DYYYyj8cj1U5H5VntDDL4dljdj4ltpOgJiZpYyRxuG1GKjkHgnA5zzXU1nJpdnbrdHTo0sprsfPJEiht+CA5BBBYZ7g+/FNNp3QpRTVmea3Gm3tssWt6e91JbhSEuAjfb7ZQTmOaJgDcwqf4WHmL1XcfmFqLUNJ1Jre91vbZXUuFg1SykKwTdgvmD7pzx5U2RngbjXQyWut3TxaZrCswDF4b+ybyijqpAMkbE4JBI/jRieQOBWPqelQ6TZHVZr+O3uZnMd1MIc2dxyQDdQglRwArSAqQe4X5a9OljFLSpo+/wDX/DeV3c8avl0oa0tV2f6f8On52SR0cOhXn9o2l/f6lJdixEnlqY40JaRdpLsgGcDPACjJyRwKs69pMup2yS2Ugt9QtG8y2mPRX7q2OqOPlcdxyOQCOL02aXRruCy806MJGUC2mPn2MwJ5+yzfKUJH3UJHtHjmvRLiVhPDHGxBLfMMdV59ux68jj8KyrKcJxafpp/lp+ZthvZzpyi1bvrd3+eq+dmedabqY0lX8RQxvBpd3My6jbvj/QrnOHlXH8G/iTt/y0HG4n1EEEZFc3plg6alrRlhYWl3IhxJgiRvLCuQOflICrz1wag8Pb9LuLjwvMSyWarJaMxyWtXJCqSepiIKf7u0nk0YjlneUd1+X+a2/wCGY8LzU2oy2d/vX6Pfyfqjq6KKK4D0wooooAKKKKACiiigAooooAKKKKACoIv9bN9/7w+9937o+77evvmp6gi/1s33/vD733fuj7vt6++aAJ6KKKACiiigAooooAKKKKACiiigAooooAKKKKAK6f8AHzL9/wC6nX7nf7vv6/hViq6f8fMv3/up1+53+77+v4VYoA8g+DP/ACBvEn/Yza9/6Xy16/XkHwZ/5A3iT/sZte/9L5a9foAKKKKAP//U/fyiiigAooooAKKKKACvmT4Yf8nKfG7/AK5eGP8A0jmr6br5k+GH/Jynxu/65eGP/SOagD6bqC2/1Z+/99/v9fvHp7f3fbFT1Bbf6s/f++/3+v3j09v7vtigCeiiigAooooAKKKKAILm1truMRXUSyorK4DDOGQ7lYehBAIPY1zc095rInsoC+m6lps6yKGJMcqA5U5GN8ci5U91OeNyiurqhqYvTZSf2bNHBdceW0o3Rls8KwBBwx44OfSgC6jq4JRg2CQcHPI6iseG/uZvEFzp3yJBa28UmOS7tMzAH0CrsI75J7Y5pJFZ6W154lZZYmmjU3VvFiVTIgxv2qCTIB8uVPzADIJAxJc6np66eniexhW9LxKIzGy+ZJG5B2Ic4Zv7q55PA5NAGlZDVklkTUGhli6xvEGRvoyksPxDc+grJ0VNBvJrrVNEfm4JW4iG5F83qTJC2CknrkAkHnPFbUWo2M9pFfQzq9vPsCODwxchVH4k4+tVrkTTJcCxiaC4dhEZsKrAAZEg3A7gueARycjpzQBhodT1O2ghGnRwW8TfZrywuUBRozt+eGQAqyqOQNuGHynYw4fHpFzYPInhjUQqwEK9pcEzwoSAwUHPmRZBBA3FQCMLUscOualBc6XrKrbtFIjJc25IWWMHcrIN26ORSoLBtyjgfMCaqRaUunXNpquuagBqAl8j7QgEIuY3yI45lA2k5PB/vfdxu21rTrSjotjGrh4Td3v32f3lqHW7+08wanpV0DvOXhxcx9P4Nu2THp8lJbTTat4gg1CG0mt7azgmjaSdDEZGmZCFVGwxA2ZJIHbGcnFrS78Xt1fWXmXENxZ3BaSOZVBMTEiMoQMGJgpKkHOQQTkEVs+XcjGJgfnLHcmfkP8ACMEYI9efpVqtFXajZmbw8nZSldb7LoWaKoyS38SoRAsxabadrbdsRP3vmHJAxkd+celSm6iUsJcxBXEYLjaGZgMbSeuc4Hvx1rnOos0UUUAFFFFABRRRQAUUUUAFFFFABUEX+tm+/wDeH3vu/dH3fb1981PUEX+tm+/94fe+790fd9vX3zQBPRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBXT/j5l+/8AdTr9zv8Ad9/X8KsVXT/j5l+/91Ov3O/3ff1/CrFAHkHwZ/5A3iT/ALGbXv8A0vlr1+vIPgz/AMgbxJ/2M2vf+l8tev0AFFFFAH//1f3b8W+JbDwZ4W1bxbqiSSWejWs13KsW3eyQoXYLvZUBIHVmVR1ZlGSOB8J/GPRfFPi2DwHLp13pHiL7Fd3tzY3fk+ZbRWsltGrFoZJEkjn+1K0MsTOjBJFLCRHRfGf2irDx/e6rGlna+NLzR3NkIR4QubKCNU88fbFullkiuWkaLcEILQ42jCtuLexfBrRtZtfDcV34msb6C5gL2+nNrQsn1iHTSsZENzJY7oeZVYqqsT5Yj8w+YGwAex0UUUAFFFFABXzJ8MP+TlPjd/1y8Mf+kc1fTdfMnww/5OU+N3/XLwx/6RzUAfTdQW3+rP3/AL7/AH+v3j09v7vtip6gtv8AVn7/AN9/v9fvHp7f3fbFAE9FFFABRRRQAUUUUAVbu48hAsZjM8uViSR/LDuAW25wx6AngE4BOKRbSMtI82ZTI6vtc7lVkAxsB6YIz9eacZP9MWHzUH7tm8vH7w8gBgc/dHQ8dSOR3WSdllijijMoZ9khUj90NhbLAnPJAGBz8wPTJoAxta06KSWDVo5ZYbmyy58jJaaJfvxMg++Dn5e6tgjuDi3dy9tNc2Gl28L2lpLG08Nt/wAfSCXMjT7CME7ucDLMNzBiwC109vGyTRAFLmdEEdxOdobKjIBUdCxbdjgAfhTl06NXjvXVJtQigMHnsoVmU4LA46AsoOBwO1AGXfafe2+oQXOlIv2a4nV7yND5chb5QsoYEbsBQHQ/eXpyMHcC3cZA3LKrSEkt8pWMg4AwDkg4HOOPfryWmWWvykSPM2lyIzRTw4WWJzjcssDHnBJx8w6ZBUMN1GlXNte6vd61oUxuobgbJo0baglj2LmRXO5ZNn3SqgMg+bPyEgHTTPZX8D2F8m1bsSwmKTgyKMq2BnkEcjHY5rGuo9K0jSIdG16R7yyuWNv5lyPMXEhOxJX/ACUM3U4ydx5NP1Ua7A0U5+wXEI8m4t5Y8SRzOAflLcMpG4KQCGzkHgitSexZYrlSPttvJEkYtZdpTC5DcsDneCM7sjj3NAGTPdXVtcf2PbiaOO6R4orviVoJgvyllb+AjBVjkFgQ3UZ2tMmvjawxax5SX+G3rExKsEbG9QeQGGDg525xk9Ti6bfSv4hvdNeB1itx+7lw2xhhSY+gCldwIwTuB4+6wEFz4bv9QmaHUL0mC3n+0WdxHhbqIsGBjyVK7VJGDzuX5HBGSwB2dFZEOolr06YWSS5iJaUcxlYm3FHUEHeOApIOM56Y21divIJGjjY+VLKHKxv8rkRkBiB3AyOfcetADUsoomBtiYB5rSuqYxIzAg7sg9Sd3GDkZz1BZ9sa3CjUAIyQ5LrnygFYKuWOMFgwOPrycZq/SMqupRwGVhgg8gigBaKpeVcQzBoD5iTS7pfMY/Inl4/djH94LwfVjnoKsQzRXEMdxC26OVQyn1VhkGgCWiiigAooooAKKKKACoIv9bN9/wC8Pvfd+6Pu+3r75qeoIv8AWzff+8Pvfd+6Pu+3r75oAnooooAKKKKACiiigAooooAKKKKACiiigAooooArp/x8y/f+6nX7nf7vv6/hViq6f8fMv3/up1+53+77+v4VYoA8g+DP/IG8Sf8AYza9/wCl8tev15B8Gf8AkDeJP+xm17/0vlr1+gAooooA/9b9/KKKKACiiigAooooAK+ZPhh/ycp8bv8Arl4Y/wDSOavpuvmT4Yf8nKfG7/rl4Y/9I5qAPpuoLb/Vn7/33+/1+8ent/d9sVPUFt/qz9/77/f6/ePT2/u+2KAJ6KKKACiiigAooqC4mWCPcWQMx2oHbYGc8Kueep46E+xoAp3e83cMcKSCSRHXzV5SNeDkgnaSSAF4JHpjNXooUhDFQNzkF2wAXYALubAGTgAfhVJ7QtasxiDXDETbDK+3zVwQA+MhcjHC4x/DyRWgjFkDMMEjkA5wfrQBXsm3wBzJHKWZstEMKfmI9W5HQ89R26Bkl07M0VkFmkikRJQW2hA2GYng8hTkDvkdAcilDcfaZrnT7WcSiFnSaQHY8LONyKgCbWIDdzkcZyTWyqqowowKAMi+0W21O0NrfSSSNvZ45VOySJi25SjJjBTgA9wOc5Oc6aSHw/Yi41O6jW9uALdr5odqlhu8kzbcAAZxklV3HA27gK3Z3kkkW1hZ4n+WQuEyu1XGUyRjLDI9QOfSnXdtHdxiKVVdM5ZWGQR0IwfWgDJtTearaz2GtWhtLlFAZ4jujYknbJC5GQQRkAgMpx7ElrZ6vpWntH9qOpyI0r5lG2RwwLKuQdoO7AzgKBwFGBTdYtNYmcCx8i5s5YzDNazgopDZG9ZFBPQ4ZSCCOmD1p21/DpVve6bZPJqFzp3lO8Mkyl0WbpGJZMZIGSu85OQC3IIAJ5XubiOKK0BlvLe5geQSnb5aMR5jAArwU3hcZBbgjGasajDDewpoWsQvcx6gJVMkaEImz5k3EElGxyrdNy8EEqDLZz22p24dvOhMjrIElJjkGwhgAAQduRz1BGRyDV1Z3gwl8y7mMjB1UqgQN8oYknDbSM88kEgAcUAUIRZCZbMMt7qmnQAq84CylZBgNvCcBynzFBjI6cYqxpN+2qWMd1PbPaTAlZIZB80ci8MM8bhnow4YYI4NZt1b6fok9rqJi/dwpDY26RrgxiWQIcncAy/c4I+XaSMk4rQvZdSjurOawUT2xkMdwnG5VPAkUkjlGGGXuCccgAgEpSawhJt1aeGGONEizl/lJBO9zliVx1OSR1yavJJHIWCMGKHa2D0PoafVOaFlcXFvlSrF5EQLmb5CoUk45+7g5HQDOKALlVHieObz4BvMhUSB5GCqozyq4YZ+mM9zwKnidpIkkdDEzAEo2Nyk9jtJGR7Ej3qSgCOKWKeJZoHWSNwCrKQVIPQgjqKkqgP9FudhYmK4OI0WIkIwBZiWUYAOOrY54zkgVfoAKKKKACiiigAqCL/Wzff+8Pvfd+6Pu+3r75qeoIv9bN9/7w+9937o+77evvmgCeiiigAooooAKKKKACiiigAooooAKKKKACiiigCun/HzL9/7qdfud/u+/r+FWKrp/wAfMv3/ALqdfud/u+/r+FWKAPIPgz/yBvEn/Yza9/6Xy16/XkHwZ/5A3iT/ALGbXv8A0vlr1+gAooooA//X/fyiiigAooooAKKKKACvmT4Yf8nKfG7/AK5eGP8A0jmr6br5k+GH/Jynxu/65eGP/SOagD6bqC2/1Z+/99/v9fvHp7f3fbFT1Bbf6s/f++/3+v3j09v7vtigCeiiigAooooAKpJItzdOI5I5I7c7WXblllwD97OB8p6Yzz17VZlljgieaZgkcYLMx4AA5JP0plt5nkIZnWRyMlkUopz3CksR+ZoAnrFuZE06TyVZLeK9fy4/Ljwwnk3MzMSdp3dRkDn+8SBW1VIQmaaWd1VHAMcUiNubYQC3BXCncOnzA4BPoABJEkgEMzTSyCJdjKFU+ZuwNzALnI6/Lgcnjpi9VW1aXa0EqyEw4XzJNmZeB842ccnrwvOcDGKrwMtg62UrKkLELA0k7SSyMQWZT5nORjj5myPTFAD5m8i+imluVjilXyhGxxukJ3LtyeuA3GMn8KsyTrHJHHhmMjFflBIUhS2WI6Djqe5A71HdmcxeXbhg8h27025jyD8/z8HHpg89sZqrpMC2tuYCXkmQgTSyLtaaQKAZDgAHIA6cDpxjFAFiKO6lWKS6fy2MZEkUZym9schyA3y4IB4znJGcYy9U8MaRq0ai4iKzRxiJJ0OJlQfw7zksp7q2VPcGugooA4bVNPSe6Syu9IOp6fczBHkcqHtRtwPLXaG8vdg5VsjLHoBWmLDS9T0j7Npkolt5LUW6nzGcCIr8pIbcCw9WBJ6HvW9B1l+/98/f/D7v+z6VkXOiWSalFrcUBaeBWUJCI0LGRslix2k4yxILbTknBbFAFe9tRC0MmqDzrKykt2g8sPvE3MZMijhlUkMPQkkj5Qa1ZVFqrTQcWyLK0kcSbnZ2YMWXbzkHdkAEsT6jmtd2/iAXLzafdwNCwGIZ4ScEDHEiOOD15U9etYd3b6lpOm+XbXKaeJ2Cs+WuCss0h3MglwSxyNuWIzxsPRgDpdH+3f2XajUjuulQCRum4jjcRgYLdSMDGcVpVjx2yaPEHRy0CjM8k8zswCLjeM5GeMt90dWPPXWR0kRZI2DIwBBByCD0INAGfOsGnu9+DDbQuS9y7LtZiAFVi+QOAMHcDx3GK0qKqWZYRmCSR5pITsZ3TYWOAc8AKeCOV4zkdQRQBNNEJomiLMm4YypKsPoRSQSPLEryRNCxzlX2lhg452lhz14NTVQj/dX8sSwuFmUSmTcSm4YUrgn5TgA8cHk9c5AL9FFFABRRRQAVBF/rZvv/AHh977v3R9329ffNT1BF/rZvv/eH3vu/dH3fb1980AT0UUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAV0/4+Zfv/dTr9zv9339fwqxVdP8Aj5l+/wDdTr9zv9339fwqxQB5B8Gf+QN4k/7GbXv/AEvlr1+vIPgz/wAgbxJ/2M2vf+l8tev0AFFFFAH/0P38ooooAKKKKACivm2H4r/GDxbqWsy/C7wJpGp6FompXmlPc6v4gl026uLnT5WguPKtoNOvQqCVWCNLKhdQG2hWUn1f4aePLH4meCNM8a6fbSWUd+JVeCVkdopreV4Jk3xlkcLJGwV0JVwAynBFAHd18yfDD/k5T43f9cvDH/pHNXsvir4gaF4OuILXV7XVrh7hC6nTtG1LU0ABx872NtOqH2YgnrjFfPnwS8R6f4j+PPxt8QafFeRWrJ4cXZeWN1Y3OYrObd/o11FFPg5+U+Xhv4c4NAH1zUFt/qz9/wC+/wB/r949Pb+77YokuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjMEVxHHHNuEx8re53IxJG5uFwPm6cAZONvHIyAXqKgkuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjJJcRx+ZuDnyk3nCM2Rz93AO48fdGT045GQCeioJLiOPzNwc+Um84RmyOfu4B3Hj7oyenHIySXEcfmbg58pN5wjNkc/dwDuPH3Rk9OORkAZeSGOEYlELOyKrFd3LMBjHv09utWqzr28S3KgyNHtKscRM+8EMdq4HLHaeBkjjjkVakuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjIA+V/LieQlRsUnLHCjA7nsPU1HawiC3jj8tIiBlljGEDHlsdOM5qrqU6JZ3Ksivsj3MJEYxsrZBHCnccA/KAT045GbclxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZAGXNv5oEsQQXEYbynddwQsPQFTg9wCM0yKYXSSwncksR2MdjKA2AcoXXDDngjIzx2NTSXEcfmbg58pN5wjNkc/dwDuPH3Rk9OORmrfPCAZ3aRDYgz5AfYco64O0fPgZO0ZIODjOKAKmmP5t1M08IaZFVEugqkTQjp86gchskrgAE8Ajk6c0G5xcxjM8asEyzKpLdmA4IyO4OO1ZVrdW89lHaXoUXNvb29zNFbh2RcksvllRll3xtgDkgcjnm39uNvI0F2ru6pJMXjicxiNW+VcgHL7SPlGScE4AxQBbhuYppJIQcSw7d6+m4ZH1HuOOvoasVSufIYlpBIGtgs2UV88buBtHzHAIKjJ56cioW1A2zNFeRuSkckzSRxsYwit8ozjlyvO0ZPB9VyAW4Osv3/vn7/4fd/2fSrFZ0F5AVnkQyyIg8zOxmBUjjyyoO4ccBcn25GbUlxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZAK6+RYyrCDDBDOxCLwjNM2XYAdCSMtxzwTz2v1m6leNbW0zxOI5IlV9zoxTBbGBgfMxwRtXJyRxyM25LiOPzNwc+Um84RmyOfu4B3Hj7oyenHIyAUNIins4m02UTOtrtVJ5nV2mUjO4kAYIOQRgdOOKtx+dFcNG3mzJLucO3lhI8YAjGNrc8kZB75boKzr0CO+a9itDJLbwhi6tIpYZYBMKpEmAWIXkg44GQav3f2aaKWG4R2WNVlO1XzwSRtKjJYFei89OORkAu1VyVvcF5CJI+E2jy12Hk7guQzbhwW5A4HDGoob0mJ/tK7ZoEDSrFvlVScnarbAWOB0A3cjjkZbczOlwjKJDHCrPLtBIKkHGAEJc5X7qnI4yOQCAaNVZxia3kAkbDFcIflGQeXGeQMfUHHvT5LiOPzNwc+Um84RmyOfu4B3Hj7oyenHIzVvp0jQkrO3kmNyIVYkgtjsPmAwSyjJx25GQDRoqCS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyMklxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZAJ6KgkuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjJJcRx+ZuDnyk3nCM2Rz93AO48fdGT045GQCeoIv8AWzff+8Pvfd+6Pu+3r75okuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjMAuEjlutwmPlAOcoxXG3pHgfN05Ayc9uRQBeoqCS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyMklxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZAJ6KgkuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjJJcRx+ZuDnyk3nCM2Rz93AO48fdGT045GQCeioJLiOPzNwc+Um84RmyOfu4B3Hj7oyenHIySXEcfmbg58pN5wjNkc/dwDuPH3Rk9OORkAnoqCS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyMklxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZAJ6KgkuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjJJcRx+ZuDnyk3nCM2Rz93AO48fdGT045GQCeioJLiOPzNwc+Um84RmyOfu4B3Hj7oyenHIySXEcfmbg58pN5wjNkc/dwDuPH3Rk9OORkAnoqCS4jj8zcHPlJvOEZsjn7uAdx4+6MnpxyMklxHH5m4OfKTecIzZHP3cA7jx90ZPTjkZAET/j5l+/91Ov3O/3ff1/CrFUjOkc9wWEp8uNXPyMy4+b7mAdzccgZPTjkZmkuI4/M3Bz5SbzhGbI5+7gHcePujJ6ccjIB5P8ABn/kDeJP+xm17/0vlr1+vH/gx/yBvEn/AGM2vf8ApfLXsFABRRRQB//R/fyiiigAooooA8a8Vfs8/BHxtrdx4k8U+C9Nv9TvNv2m4aEI9ztG0efs2+dhQB+83cADoBXq2laVpehaba6LolnDp+n2MSQ29tbxrFDDFGNqJHGgCqqgYAAAA6VfooAK+ZPhh/ycp8bv+uXhj/0jmr6br5k+GH/Jynxu/wCuXhj/ANI5qAPpuoLb/Vn7/wB9/v8AX7x6e3932xU9QW3+rP3/AL7/AH+v3j09v7vtigCeiiigAooooAq3TSosbRbv9YgYKobKk4OckYHOSe1Wqr3alraTG/IGQIzhzjnAzgZPTniplbcobBGRnB6igCtfqrWU4dwihCSxXeBjnO09celWVZXUOhyrDIPsadVOxnWaJkE3nSQMYpGK7DvXrkcfXjg9RxQBcpGUMpVgCCMEHkGlooAzYES8soo/NXMTIGNu21Q8DjcowcgblKsp7ZU96JDHco2mapEsn2tZgUClo2iB24YkYyUYZU9fmxkAmnSXSWd0kdzKqR3TBIVCEHzMEtufJX5gPlBA5B5OcC7JGksbRSDcjggg9CD1FAFO4mnso5roq90gO7YiguiBeQoHLnIzjrzgdhV7KnIBzjg1mokumwqke6e2hjkZtxaSckEFFX+9xuHJzwvXJNWDDFMWeB/KkZoyzIBuIUhgrZB4I4PGcHjBwQAQWdnBbki2DwpEXVY/ux/OQSQowMZHB9z61YjW8QRrI6SbY8O2CpaQY5AyQAeeO3FQW80yOyTRu3mTuqnqFXaWBOcYXjHfk+9XYpYpk8yFw65IypyMg4I/A8GgCBYZpTG10VICDdGFyvmAg7gTzwRx09euMW6KinmS3iaaQMVXqEVnb04VQSfwFAFRFgm1GWZWikaFBEwC5kQt85DNnoQVO3A9cnIxNZxCC3S3WKOFIsoiRfcVFOEAGBj5ccAYHQZHNOjElvATcSGZl3MWC84yTgKvoOB1P1NRWAzbCUwrA0xMhVAR945yQyqdxH3sgHOaAEdkt7xC0kUS3OUCkYkklA3DDZ5witxgnAznANPmi33du5gVxGHIkJ+aNiABtGOdwJycjHvmnXcvkxCQzJAA6As4yMFgCOo5boD2JHB6U1Ig15JcPDGCqhEkBy5U8sCMDaM+hOfagC3VW65ES4kOZF/1ZxjBz83I+Xjn1q1WfNtl1C3iWfY0IaVoxnLKQUBOCBgE9CDz9KANCiiigAooooAKgi/1s33/ALw+9937o+77evvmp6gi/wBbN9/7w+9937o+77evvmgCeiiigAooooAKKKKACiiigAooooAKKKKACiiigCun/HzL9/7qdfud/u+/r+FWKrp/x8y/f+6nX7nf7vv6/hVigDyD4M/8gbxJ/wBjNr3/AKXy16/XkHwZ/wCQN4k/7GbXv/S+WvX6ACiiigD/0v38ooooAKKKKACiiigAr5k+GH/Jynxu/wCuXhj/ANI5q+m6+ZPhh/ycp8bv+uXhj/0jmoA+m6gtv9Wfv/ff7/X7x6e3932xU9QW3+rP3/vv9/r949Pb+77YoAnooooAKKKKACqlmpiRoNkirE21WkfeXGAdwJZmxk4+bB46YwTbqjPF5dwl7HGpbHlyM0hQLFyc4wQxB9ccE89iAXqqSOYblC7uyT4RUCblVgGYsWUZUEcZY4yABgnm3UU8K3ELwOWVZFKkoxRgD6MpBB9wc0AS0VUt53Ltbzr5ciE7cuGLoOjjGD9cjg+2CbdADXRXXawBHuM1nxTSWMZjv5N0cKbjcPtVTyQd2MAEDBJwAe2K0qa6LIpRwGVgQQRkEHsaAHVUks4md5ov3M0nl7pEADMsbbgpJByOSPoTjB5pzW5BkeCRkaV0diSWHy4BAB4UMoxx9etPj8/fL523Zu/d7c524H3s9856dsUAZ8VzPA8guIpHWS58tCoL4UqCGYYG1c8ZGR3yBnFhrexvZIroYke2dwjIx+Vs7XB2n1GCD3HNR2puhbPgNJMHP+uO0Hp0IB4x0460sllLLJnzPJEcqyxmLKnphw4JKtuyw5HGQR8wBAA2eVbAQ+ddHE1xtG9dzMZCdsa7QMAE9cHCjk9TSRW99JNFc3TKskUkowjSbDCxOzK7gpfAUkkHHzAYzUkFrNbSZjYSCV2eZnzuJIAXbjjgAA8D169ZYriZvJWa3aN5AxbBDKhXsSPXtx+VAAtqWaOS5kMjwu7oRlAN2QAQDg7VOOfr16W6qQ3sUwhKpIvngsoaJ1xjruyPlPoGxntSx3LTGExwybJQxLMAmzGMBlYhsntx2OccZAK+qS+XDGg3hpZEVShxhs5UMRztJG1sAnB9MkWrWBLaERqiIWJd/LXapdzudsf7TEk9+eaz9Pt7hpvt975kdxs8p0JXynKH/WIoZyobqBuzj7wzWxQAVUtWMplnEjsjsVVHTZs2fKcAgMQSCQTkEYK8YJW8Mhi8mISZmOzfFs3R5B+f5+OPoeexqyo2qFyTgY55NAC0UUUAFFFFABUEX+tm+/8AeH3vu/dH3fb1981PUEX+tm+/94fe+790fd9vX3zQBPRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBXT/j5l+/91Ov3O/3ff1/CrFV0/wCPmX7/AN1Ov3O/3ff1/CrFAHkHwZ/5A3iT/sZte/8AS+WvX68g+DP/ACBvEn/Yza9/6Xy16/QAUUUUAf/T/fyiiigAooooAKKKKACvmT4Yf8nKfG7/AK5eGP8A0jmr6br5k+GH/Jynxu/65eGP/SOagD6bqC2/1Z+/99/v9fvHp7f3fbFT1Bbf6s/f++/3+v3j09v7vtigCeiiigAooooAKa6JIjRyKGVgQQRkEHqCKdRQBQtybaUWT42tkwiKFlRIkCgKzZZd2Tx93I6LwTV+oLm3ju4Ht5S6o4wTG7Rt+DIQw/A1GtztkMVyBEWfZESwPmfLu475wDkexPSgB91A88EiwyeTMyMqShQxjLDhgDwcHBweDio7eeUbba7/AOPhI0Z3VSsTM2QdhOe46ZJAIz1q5UNxbw3ULW9wgkjfqD/nt2oAmoqkwvYnLRkTrJKMhiE8uPaAduAdx3DPPqeeAKkS7hbaHPlNI7xorjazMmc7QevCkjHUc9KALNFFFAFeDrL9/wC+fv8A4fd/2fSrFV4Osv3/AL5+/wDh93/Z9KsUAFFRTzw2sMlzcyLFFEpd3Y4VVUZJJPQAdahlmuCZIrWLMihCGf5YzuJBwRkkqBnGPQZ54ALLukYBdgoJAGTjk8AfiaoBP7QVWuIsQHeDDKgJYq42P3x93IHXkZwRU/2SNpGknJm/eCSMOFIiIQJ8nAI7nnJyx5xgC1QAUhJAJAzjsO9BYKMk4qgkIvWS5uo1eMbJIUkjw8T4OWOc4bnHGMfnQA60iZna/lEiSTqv7qQofKAH3RsyOuSTlsnvgDF6iigAooooAKKKKACoIv8AWzff+8Pvfd+6Pu+3r75qeoIv9bN9/wC8Pvfd+6Pu+3r75oAnooooAKKKKACiiigAooooAKKKKACiiigAooooArp/x8y/f+6nX7nf7vv6/hViq6f8fMv3/up1+53+77+v4VYoA8g+DP8AyBvEn/Yza9/6Xy16/XkHwZ/5A3iT/sZte/8AS+WvX6ACiiigD//U/fyiiigAooooAKKKKACvmT4Yf8nKfG7/AK5eGP8A0jmr6br5k+GH/Jynxu/65eGP/SOagD6bqC2/1Z+/99/v9fvHp7f3fbFT1Bbf6s/f++/3+v3j09v7vtigCeiiigAooooAKKKKACkKq2NwBwcjPY0tFAGYGk0yDEzNNbW8LM0rZeX5OxVRliV7jkkdCTWnmiqF1YmaOf7HKbO4m25mjVS2UPGQwIPHB4zjuOCAC/SEA4JGcdKrSzTw+a7Q+ZGu3YIzlznhsg4HHXgnI7Z6vF1AWkTeAYmVGzxhmAIHPXO4Yx3460ARx2NvD5QgBiWHdtRGKp83XKg4PtkcdqdFBLGIlM7yCMENuC5fPQkgDke2Ks0EgDJoAzbSG5VgXuJHCM4cOgG8nGMegHbHWpfsjNEsc1xK5CsrMCELbu/yAYI7EYx9aktnVxIyMzAueW6f8B/2as0AV47W3jl89Ix5uxYy55copJALHkgEk9e9WKjlmhgTzJ3WNMgZYgDLHAGT6k4HvUIukadrZEdmQNk7SFBAU43HjJDDGPQ9waALVVZbtEdoogZpUMe6NCNyrI20MQSOByT3wDgE8VEYbm7hKXbeUs0Ox44mYMrsPmKyjaeOgICnv9LwUDJAxnrQBSjtXZxLess0kcrvEQu0RggqAOTztJBPfJxgcVeoooAKKKKACiiigAooooAKgi/1s33/ALw+9937o+77evvmp6gi/wBbN9/7w+9937o+77evvmgCeiiigAooooAKKKKACiiigAooooAKKKKACiiigCun/HzL9/7qdfud/u+/r+FWKrp/x8y/f+6nX7nf7vv6/hVigDyD4M/8gbxJ/wBjNr3/AKXy16/XkHwZ/wCQN4k/7GbXv/S+WvX6ACiiigD/1f38ooooAKKKKACiiigAr5k+GH/Jynxu/wCuXhj/ANI5q+m6+ZPhh/ycp8bv+uXhj/0jmoA+m6gtv9Wfv/ff7/X7x6e3932xU9QW3+rP3/vv9/r949Pb+77YoAnooooAKKKKACiiigAooooAKKKKACmsiONrqGGQcEZ5ByD+B5p1FAFVrO3O8hShkkWVijMhZ024JKkEjCgEdCOCCCRSm3G2cFmbz85DEkD5QuB6DjPHfNWaKAM22gkaGaJ5JUJkJyTkqOOFYjlf/r1ZktI5fMEjORIVJAdgBt6YwRjpz696WDrL9/75+/8Ah93/AGfSrFAEItrdXkkEahpSrOcDLFQACfXGBipqKKACiiigAooooAKKKKACiiigAooooAKgi/1s33/vD733fuj7vt6++anqCL/Wzff+8Pvfd+6Pu+3r75oAnooooAKKKKACiiigAooooAKKKKACiiigAooooArp/wAfMv3/ALqdfud/u+/r+FWKrp/x8y/f+6nX7nf7vv6/hVigDyD4M/8AIG8Sf9jNr3/pfLXr9eQfBn/kDeJP+xm17/0vlr1+gAooooA//9b9/KKKKACiiigAooooAK+ZPhh/ycp8bv8Arl4Y/wDSOavpuvmT4Yf8nKfG7/rl4Y/9I5qAPpuoLb/Vn7/33+/1+8ent/d9sVPUFt/qz9/77/f6/ePT2/u+2KAJ6KKKACiiigAooooAKKKKACiiigAooooAKKKKAK8HWX7/AN8/f/D7v+z6VYqvB1l+/wDfP3/w+7/s+lWKACiiigAooooAKKKKACiiigAooooAKKKKACoIv9bN9/7w+9937o+77evvmp6gi/1s33/vD733fuj7vt6++aAJ6KKKACiiigAooooAKKKKACiiigAooooAKKKKAK6f8fMv3/up1+53+77+v4VYqun/AB8y/f8Aup1+53+77+v4VYoA8g+DP/IG8Sf9jNr3/pfLXr9eQfBn/kDeJP8AsZte/wDS+WvX6ACiiigD/9f9/KKKKACiiigDx7xD8efhf4asIr+91K4u/tF/e6bFBp+n3uo3cl1psjRXipbWkMs7LA6kSOEKKOd2CM+ieGfE2g+MtAsPFPhe9j1HStTiWa3uIidro3scEEHgqQCpBBAIIr5Z8W/Cey+EOvn4v+GvEGvTS3WpTWMelILGa1tpfGmq2kN3cxb7Q3AWK4aO6ZGmZMRYwFJr6T8BeC9O+HvhOw8JaZcT3kVn5rvcXJQz3E88jTTzymNUTfLK7u2xFXLHaoGBQB2FfMnww/5OU+N3/XLwx/6RzV7L4p8A6P4vuILnU73VrVrdCijTtWv9OQgnOXS0niVz7sCe2cV5P8KfBOi+E/if8TrPS5L2U3q6O0s15fXN7cti2kUfv7mSSXCj7o3YXtigD6OqC2/1Z+/99/v9fvHp7f3fbFElusnm5Zx5qbDtdhgc8rg8HnqOenoKrxW6yRzZeYebvQ7mZSAGblcdOvBHONvPAoAv0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FACQdZfv/AHz9/wDD7v8As+lWKoJbrIbnLzDzcocsy4AzymOnXqOffgVYkt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPUEX+tm+/8AeH3vu/dH3fb1980SW6yeblnHmpsO12GBzyuDweeo56egquLdZJbvLzDzQEPzFVA29Y8dDzyRznvwKAL9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQBPRUElusnm5Zx5qbDtdhgc8rg8HnqOenoKJLdZPNyzjzU2Ha7DA55XB4PPUc9PQUAT0VBJbrJ5uWceamw7XYYHPK4PB56jnp6CiS3WTzcs481Nh2uwwOeVweDz1HPT0FAE9FQSW6yeblnHmpsO12GBzyuDweeo56egokt1k83LOPNTYdrsMDnlcHg89Rz09BQAif8fMv3/up1+53+77+v4VYqiYFlnuVLTDzI0Q4YqoHzcoR0bnkjnp6Cp5LdZPNyzjzU2Ha7DA55XB4PPUc9PQUAeT/Bn/kDeJP+xm17/wBL5a9frx/4Mf8AIG8Sf9jNr3/pfLXsFABRRRQB/9D9/KKKKACiiigAooooAK+eV8VWfgf4s+M5de0zWXg1WDSpLaax0XUtRgkEUcqOPNs7aaMMrDlCwYAg4wQT9DUUAeS/8Lq8Hf8AQP8AEn/hK67/APINcvZftJ/DO5jmNlaeK71Ibi4gaSHwd4juEEsEzxSx+ZFpzLmORWQrnKlSrYYEV9BV5r8KdC1Xw74XvtP1mA21xLrviK8VSytmC91i8uYHypI+eGVGx1GcEA5AAOT/AOGi/h//ANAnxf8A+ER4n/8AlbR/w0X8P/8AoE+L/wDwiPE//wAra92ooA8J/wCGi/h//wBAnxf/AOER4n/+VtH/AA0X8P8A/oE+L/8AwiPE/wD8ra92ooA8J/4aL+H/AP0CfF//AIRHif8A+VtH/DRfw/8A+gT4v/8ACI8T/wDytr3aigDwn/hov4f/APQJ8X/+ER4n/wDlbR/w0X8P/wDoE+L/APwiPE//AMra92ooA8J/4aL+H/8A0CfF/wD4RHif/wCVtH/DRfw//wCgT4v/APCI8T//ACtr3aigDwn/AIaL+H//AECfF/8A4RHif/5W0f8ADRfw/wD+gT4v/wDCI8T/APytr3aigDwn/hov4f8A/QJ8X/8AhEeJ/wD5W0f8NF/D/wD6BPi//wAIjxP/APK2vdqKAPnmw/aY+GN210tlaeLLo28zRShPBviSTypQATG2zTjsYAg7WwwyCRyK0P8Ahov4f/8AQJ8X/wDhEeJ//lbXW/DvQtV0S78YSapAYF1PXbi8tyWVvMgeCBFf5ScZZGGDg8dK9JoA8J/4aL+H/wD0CfF//hEeJ/8A5W0f8NF/D/8A6BPi/wD8IjxP/wDK2vdqKAPCf+Gi/h//ANAnxf8A+ER4n/8AlbR/w0X8P/8AoE+L/wDwiPE//wAra92ooA8J/wCGi/h//wBAnxf/AOER4n/+VtH/AA0X8P8A/oE+L/8AwiPE/wD8ra92ooA8J/4aL+H/AP0CfF//AIRHif8A+VtH/DRfw/8A+gT4v/8ACI8T/wDytr3aigDwn/hov4f/APQJ8X/+ER4n/wDlbR/w0X8P/wDoE+L/APwiPE//AMra92ooA8J/4aL+H/8A0CfF/wD4RHif/wCVtH/DRfw//wCgT4v/APCI8T//ACtr3aigDwn/AIaL+H//AECfF/8A4RHif/5W1n2n7THwxubq+gtbTxZcS2kqxzxr4N8SO0EhjSQI6rpxMZKOr7WAJDBgNrAn6GrzbwToWq6V4t+IOpahAYbbWdYtrm0fcp82FNKsbdmABJGJYnXDYPGcYIJAOS/4aL+H/wD0CfF//hEeJ/8A5W0f8NF/D/8A6BPi/wD8IjxP/wDK2vdqKAPCf+Gi/h//ANAnxf8A+ER4n/8AlbR/w0X8P/8AoE+L/wDwiPE//wAra92ooA8J/wCGi/h//wBAnxf/AOER4n/+VtH/AA0X8P8A/oE+L/8AwiPE/wD8ra92ooA8J/4aL+H/AP0CfF//AIRHif8A+VtH/DRfw/8A+gT4v/8ACI8T/wDytr3aigDwn/hov4f/APQJ8X/+ER4n/wDlbR/w0X8P/wDoE+L/APwiPE//AMra92ooA8J/4aL+H/8A0CfF/wD4RHif/wCVtH/DRfw//wCgT4v/APCI8T//ACtr3aigDwn/AIaL+H//AECfF/8A4RHif/5W0f8ADRfw/wD+gT4v/wDCI8T/APytr3aigDwfTf2hvhzqepXlhaW3iY3drHDJLbyeFNfjkjjmMgjcxvYK4DlHAYjDFSAeDje/4XV4O/6B/iT/AMJXXf8A5BrR0fQtVtfiz4q8RzwFdO1DSNFtoJdykPLazag0y7Qdw2iaM5IAO7jODj0mgDx34I/aJvDGrajNZ3djHqOva1dQJe2s1lOYJr2Vo3aC4SOVA6/Mu5QSCD0Ir2KiigAooooA/9H9/KKKKACiiigAooooAKKKKACiuC8ceNNW8H/2b/Zfg7WPFv2+YxSf2T9i/wBEUY/eT/bLq1+Tn/lnvbg/L0z8S/DTw/B4H+Nnh/xdOvhHxbL8RvE/iy1XULHSWj1/TpYnv7o79RlmZpIreOA2MyeTF5bGNQW5LgH6L0UUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUV578W73XtM+FPjPUfC28a1a6LqMtj5ed/2pLaRoduOd28DGO9AHoVFfEfwI0fwf4b+K2j2XwvSCHRdY8BWOp6oLQgpc3b3CLZ3k+0kPcTxm43TNl5AvzM20Y+3KACiiigAooooAKKKKACiiigAooooAKKKKACignAzX50fs0+HoPh58RPCdvIvhHxTcfEPQNX1b+39D0lrTVQ8F3bSTtdXkk0klzBO90ASyQlJUVSnOEAP0XooooAKKKKACiiigAooooAKKKKACiiigAoorzr4seGYfF3w+1rRbnUL7ToJLeR5X0+4a1mkSNSxiMyYkRHxh/LZWIyAwzQB6L15FFfm54T0XwT4u0f4M+Gvi1BbXfhSD4Tf2jBDfEfZhdQR6cs90ofCie2gYFJB88QdipXJJ+wv2eb/wAR6r8AvhrqnjCSSbXbzw1o81/JMSZXupLOJpmkzzvLkls980Aew0UUUAFFFFAH/9L9/KKKKACiiigAooooAKKKKACuM034cfD3RvFV7450jwxplj4j1JSl1qcFlDHezqxBYSTqgkcEgE5Y5wM9BXZ0UAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAHHeEfh54B+H6XsfgTw1pvhxdSl8+6GnWcNoJ5efnk8lV3tyeWyea7GiigAooooAKKKKACiiigAooooAKKKKACiiigArjPDvw4+HvhDWNS8QeFPDGmaNqmsnN9d2VlDbz3R3Fv30kaK0nzEt8xPJJ6k12dFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABUc0MVxE8E6LJFIpVlYAqykYIIPBBHUVJRQBwWvfCv4Y+KtG0rw54m8I6Rq2k6H5f9n2d3YW89vZ+Unlp5EToUi2p8o2AYXjpxXeAAAADAFLRQAUUUUAFFFFAH/9P9/KKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAK+bbT9pLw3d/EC38GrFbfZ7vxFdeF4pReqbwX9paS3Du9n5eVgLwyQh/MJ37Tt2tlfpKvnq2+AsEHi201RtVVtD0/xNeeLILIW5E41K9tJrZw1x5pBhD3E0wURBt5QbtqEOAfQtFeT/AAZ8F3XgHwne+HbmCWBBrOs3MHnXDXcr291fzTwyPM7yO7OjhiXYvk/N82a9YoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAP//U/fyiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAP/2Q==)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8LDpQmLAGLY2"
      },
      "source": [
        "- **Fundamental Theorem of Markov Chain** ensures that the chain converges to its stationary distribution."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jwGAgwqQXG0G"
      },
      "source": [
        "**Bayesian CAPM where Idiosyncratic Return follows Laplace Distribution!**\n",
        "\n",
        "- We want to model the relationship as\n",
        "$$\n",
        "r_t^{\\text{portf}}=\\alpha+\\beta~r_t^{\\text{S&P 500}} + \\epsilon\n",
        "$$\n",
        "where \n",
        "  - $r_t^{\\text{portf}}$ is portfolio return\n",
        "  - $r_t^{\\text{S&P 500}}$ is market index return\n",
        "  - $\\epsilon \\sim Laplace(0,\\lambda)$\n",
        "\n",
        "- That is the idiosyncratic Return follows Laplace Distribution\n",
        "\n",
        "- $$\\alpha \\sim Cauchy(0,1)$$\n",
        "- $$\\beta \\sim Cauchy(0,1)$$\n",
        "- $$\\lambda \\sim \\text{Half-Cauchy}(0,1)$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "GoXiHaJ6h2ii",
        "outputId": "20993437-1073-4993-c0bf-32d72d9288c4"
      },
      "source": [
        "install.packages('mvtnorm')\n",
        "library('mvtnorm')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Installing package into ‘/usr/local/lib/R/site-library’\n",
            "(as ‘lib’ is unspecified)\n",
            "\n"
          ],
          "name": "stderr"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "QG-tigdUhAnH",
        "outputId": "12940131-a53a-4aaa-c6f8-28e59e842ade"
      },
      "source": [
        "\n",
        "data = read.csv(file='https://www.cmi.ac.in/~sourish/StatFin2016/Program/R_Workshop/stock_treasury.csv')\n",
        "n = nrow(data)\n",
        "cat('Sample size = ',n,'\\n')\n",
        "\n",
        "# Risk Free Rate is in percentage and annualised. \n",
        "# So the following conversion is required.\n",
        "Rf = data$UST_Yr_1/(250)\n",
        "\n",
        "## Compute log-return in percentage\n",
        "ln_rt_snp500 = diff(log(data$SnP500))*100-Rf[2:n]\n",
        "ln_rt_ibm = diff(log(data$IBM_AdjClose))*100-Rf[2:n]\n",
        "ln_rt_apple = diff(log(data$Apple_AdjClose))*100-Rf[2:n]\n",
        "ln_rt_msft = diff(log(data$MSFT_AdjClose))*100-Rf[2:n]\n",
        "ln_rt_intel = diff(log(data$Intel_AdjClose))*100-Rf[2:n]\n",
        "\n",
        "## log-return of the portfolio\n",
        "ln_r = cbind(ln_rt_ibm,ln_rt_apple,ln_rt_msft,ln_rt_intel)\n",
        "\n",
        "### Portfolio\n",
        "w = c(0.2,0.3,0.25,0.25)\n",
        "ln_rt_portf = ln_r%*%w"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Sample size =  250 \n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ZQMbliBpXMzg"
      },
      "source": [
        "**Write log-likelihood function**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "9AGeLO1kXP_c"
      },
      "source": [
        "log_likelihood = function(param,y,x){\n",
        "  a = param[1]\n",
        "  b = param[2]\n",
        "  lambda = param[3]\n",
        "  pred = a + b*x \n",
        "  likelihoods = -log(2*lambda)-abs(y-pred)/lambda\n",
        "  sumll = sum(likelihoods)\n",
        "  return(sumll)   \n",
        "}\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "a9OCt_HBXUxv"
      },
      "source": [
        "**Write log-prior function**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "xJMTOnARXXdR"
      },
      "source": [
        "dHalfCauchy = function(theta,log=FALSE){\n",
        "  if(log==FALSE){\n",
        "    return(2/(pi*(1+theta^2)))\n",
        "  }else{\n",
        "    return(log(2)-log(pi)-log(1+theta^2))\n",
        "  }\n",
        "}\n",
        "\n",
        "log_prior = function(param,x){\n",
        "  a = param[1]\n",
        "  b = param[2]\n",
        "  lambda = param[3]\n",
        "  a_prior = dcauchy(a,0,1,log = T) \n",
        "  b_prior = dcauchy(b,0,1,log = T) \n",
        "  scale_prior = dHalfCauchy(theta=lambda,log = T)\n",
        "  \n",
        "  return(a_prior+b_prior+scale_prior)\n",
        "}\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FwzTZnXnXbcE"
      },
      "source": [
        "**Write log-posterior function**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ftdfuxaXXeHT"
      },
      "source": [
        "log_posterior = function(param,y,x){\n",
        "  like = log_likelihood(param=param,y=y,x=x)\n",
        "  prior = log_prior(param=param,x=x)\n",
        "  post  = like + prior\n",
        "  return ( post )\n",
        "}"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JJVsM4KYXo5E"
      },
      "source": [
        "**Bayesian CAPM Model Fitting with Metropolis-Hastings Algorithm**\n",
        "\n",
        "- Use conjugate posterior or posterior with flat prior as **proposal function**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "-lidDFZqXneI"
      },
      "source": [
        "proposalfunction = function(param,x){\n",
        "  X=cbind(rep(1,length(x)),x)\n",
        "  S=param[3]*solve(t(X)%*%X)\n",
        "  prop = c(rmvnorm(1\n",
        "                  ,mean = param[1:2]\n",
        "                  ,sigma = S)\n",
        "          ,rgamma(1,param[3]*5,5))\n",
        "  return(prop)\n",
        "}\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2Nm8d80ZXwHI"
      },
      "source": [
        "**Run Metropolis**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Z78f6KcwXzAm"
      },
      "source": [
        "run_metropolis = function(startvalue, N.sim, burnin){\n",
        "  iterations = N.sim + burnin\n",
        "  chain = array(dim = c(iterations+1,3))\n",
        "  chain[1,] = startvalue\n",
        "  for (i in 1:iterations){\n",
        "    proposal = proposalfunction(chain[i,],x=x)\n",
        "    \n",
        "    probab = exp(log_posterior(param=proposal\n",
        "                               ,y=y,x=x) \n",
        "                 - log_posterior(param=chain[i,]\n",
        "                                 ,y=y,x=x))\n",
        "    \n",
        "    if (runif(1) < probab){\n",
        "      chain[i+1,] = proposal\n",
        "    }else{\n",
        "      chain[i+1,] = chain[i,]\n",
        "    }\n",
        "  }\n",
        "  return(chain)\n",
        "}\n"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "le5t7BBfX5RB"
      },
      "source": [
        "options(warn=-1)\n",
        "y=ln_rt_portf\n",
        "x=ln_rt_snp500\n",
        "startvalue = c(1,0,1)\n",
        "N.sim=1000\n",
        "burnin=0\n",
        "\n",
        "set.seed(1)\n",
        "chain = run_metropolis(startvalue=startvalue\n",
        "                       ,N.sim=N.sim\n",
        "                       ,burnin=burnin)\n",
        "colnames(chain)=c(\"alpha\",\"beta\",\"lambda\")\n",
        "chain=chain[(burnin+1):nrow(chain),]\n",
        "options(warn=0)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cPM_-ujg-lO7"
      },
      "source": [
        "Purposefully we choose starting value far from MLE and keep burn-in at 0; so that we can study the convergence behaviour."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 437
        },
        "id": "5sVKBF9vX-ee",
        "outputId": "4692dd48-a2a3-4bf9-b7a8-3ff0cbd3ccbd"
      },
      "source": [
        "par(mfrow=c(3,2))\n",
        "plot(ts(chain[,\"alpha\"]),lwd=2\n",
        "     ,ylab = \"alpha\")\n",
        "plot(density(chain[,\"alpha\"]),lwd=2\n",
        "     ,main = \"\",col='red'\n",
        "     ,xlab = \"alpha\")\n",
        "plot(ts(chain[,\"beta\"]),lwd=2\n",
        "     ,ylab = \"beta\")\n",
        "plot(density(chain[,\"beta\"]),lwd=2\n",
        "     ,main = \"\",col='red'\n",
        "     ,xlab = \"beta\")\n",
        "plot(ts(chain[,\"lambda\"]),lwd=2\n",
        "     ,ylab = \"lambda\")\n",
        "plot(density(chain[,\"lambda\"]),lwd=2\n",
        "     ,main = \"\",col='red'\n",
        "     ,xlab = \"lambda\")\n",
        "\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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JQUY2PjGTNmuLu7JyUlDRs2TMuVaYGD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\n2VMAAAAASUVORK5CYII=",
            "text/plain": [
              "Plot with title “”"
            ]
          },
          "metadata": {
            "tags": [],
            "image/png": {
              "width": 420,
              "height": 420
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lriVK7_EzJVK"
      },
      "source": [
        "**What is the $\\mathbb{P}(\\alpha >0|y,x)$?**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "id": "YIocUi1yzKrW",
        "outputId": "564197ed-4468-499a-81a1-5bb3a30208a6"
      },
      "source": [
        "alpha_star = chain[,'alpha']\n",
        "no.of.alpha.gt.zero = length(alpha_star[alpha_star>0])\n",
        "total.no.of.alpha = length(alpha_star)\n",
        "\n",
        "prob_a_gt_zero = (no.of.alpha.gt.zero/total.no.of.alpha)\n",
        "\n",
        "round(prob_a_gt_zero,4)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "[1] 0.8811"
            ],
            "text/latex": "0.8811",
            "text/markdown": "0.8811",
            "text/html": [
              "0.8811"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uxIxuoYJz1wK"
      },
      "source": [
        "**If you invest \\$100 today in the portfolio, what is the probability that it will be more than \\$105 by the end of month? In one month you can consider 22 business-days. You can consider annualised risk-free rate to be 0.5\\%.**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "HPQAOaUu0R7l"
      },
      "source": [
        "rfr = 0.5/250\n",
        "days = 22 \n",
        "count = 0\n",
        "for(i in 1:N.sim){\n",
        "  a_str = chain[i,'alpha']\n",
        "  b_str = chain[i,'beta']\n",
        "  lambda_str = chain[i,'lambda']\n",
        "  for(t in 1:days){\n",
        "    \n",
        "\n",
        "  }\n",
        "\n",
        "}"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "079rMRvuuXpE"
      },
      "source": [
        "# **6. MCMC Convergence**\n",
        "\n",
        "## **Geweke Time-Series Disgnostics**\n",
        "\n",
        "- If we simulate large number of draws after burn-in, then mean (and variance) of a parameter's posterior distribution from first half of the chain will be equal to the mean (and variance) from the second half of the chain.\n",
        "\n",
        "- The value of this approach is that by allowing the algorithm to run for a very long time, it may reach areas of the posterior distribution that may not otherwise be reached.\n",
        "\n",
        "- The statistics to test convergence is\n",
        "$$\n",
        "z=\\frac{\\bar{\\theta}_a-\\bar{\\theta}_b}{\\sqrt{Var(\\theta_a)+Var(\\theta_b)}},\n",
        "$$\n",
        "where $a$ is the early interval and $b$ is the late interval.\n",
        "\n",
        "- If the chain has converged, the z-score should be within 2 standard deviation of zero.\n",
        "\n",
        "- That if $-2<z<2$ then we can say MCMC chain converged.\n",
        "\n",
        "- If $z<-2$ or $z>2$, then we can say MCMC chain has not converged."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "5tM46mG03103",
        "outputId": "500436e0-3e0c-4551-ce91-d36769be2e6c"
      },
      "source": [
        "## Geweke Statistics\n",
        "alpha_star = chain[1:200,'alpha']\n",
        "n_str = length(alpha_star)\n",
        "m_str = floor(n_str/2)\n",
        "#m_str\n",
        "alpha_star_1 = alpha_star[1:m_str]\n",
        "alpha_star_2 = alpha_star[(m_str+1):(n_str)]\n",
        "\n",
        "z = (mean(alpha_star_1)-mean(alpha_star_2))/sqrt(var(alpha_star_1)+var(alpha_star_2))\n",
        "cat('Geweke Statistics = ',z,'\\n')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Geweke Statistics =  1.048547 \n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7x1kQb2R5lJI"
      },
      "source": [
        "## **Gelman-Rubin Diagnostic**\n",
        "\n",
        "- Gelman (especially) argues that the best to identify non-converegence is to simulate multiple sequences for over-dispersed starting point\n",
        "\n",
        "- The intuition is that the behavior of all the chains should be basically same\n",
        "\n",
        "- or as Gelman and Rubin put it, the variance within the chain should be the same as the variance across the chains.\n",
        "\n",
        "\n",
        "- The Gelman-Rubin diagnostic uses an analysis of variance approach to assessing convergence. \n",
        "\n",
        "- It calculate Between-Chain variance (B) as\n",
        "$$\n",
        "B = \\frac{m}{m-1}\\sum_{j=1}^{m}(\\bar{\\theta}_{0j}-\\bar{\\theta}_{00})^2\n",
        "$$\n",
        "and within-chain variance (W) as,\n",
        "$$\n",
        "W=\\frac{1}{m}\\sum_{j=1}^{m}\\Big[\\frac{1}{n-1}\\sum_{i=1}^{n}(\\theta_{ij}-\\bar{\\theta}_{0j})^2\\Big]\n",
        "$$\n",
        "and assesses whether they are different enough to worry about the converegence.\n",
        "\n",
        "- The marginal posterior variance of $\\theta$ is\n",
        "$$\n",
        "Var(\\theta|Data) = \\frac{n-1}{n}W+\\frac{1}{n}B\n",
        "$$\n",
        "\n",
        "- **Gelman-Rubin Statistics** montor convergence as\n",
        "$$\n",
        "R=\\sqrt{\\frac{Var(\\theta|Data)}{W}}\n",
        "$$\n",
        "\n",
        "- If converges $R$ would be close to 1"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "7bj8F5vT9lnH"
      },
      "source": [
        "## Implement with 5 chains\n",
        "\n",
        "options(warn=-1)\n",
        "y=ln_rt_portf\n",
        "x=ln_rt_snp500\n",
        "\n",
        "N.sim=5500\n",
        "burnin=500\n",
        "\n",
        "set.seed(1)\n",
        "startvalue = c(1,0,1)\n",
        "chain1 = run_metropolis(startvalue=startvalue\n",
        "                       ,N.sim=N.sim\n",
        "                       ,burnin=burnin)\n",
        "colnames(chain1)=c(\"alpha\",\"beta\",\"lambda\")\n",
        "chain1=chain1[(burnin+1):nrow(chain1),]\n",
        "\n",
        "startvalue = c(0,0,1)\n",
        "chain2 = run_metropolis(startvalue=startvalue\n",
        "                       ,N.sim=N.sim\n",
        "                       ,burnin=burnin)\n",
        "colnames(chain2)=c(\"alpha\",\"beta\",\"lambda\")\n",
        "chain2=chain2[(burnin+1):nrow(chain2),]\n",
        "\n",
        "startvalue = c(0,1,0.5)\n",
        "chain3 = run_metropolis(startvalue=startvalue\n",
        "                       ,N.sim=N.sim\n",
        "                       ,burnin=burnin)\n",
        "colnames(chain3)=c(\"alpha\",\"beta\",\"lambda\")\n",
        "chain3=chain3[(burnin+1):nrow(chain3),]\n",
        "\n",
        "startvalue = c(1,0,1)\n",
        "chain4 = run_metropolis(startvalue=startvalue\n",
        "                       ,N.sim=N.sim\n",
        "                       ,burnin=burnin)\n",
        "colnames(chain4)=c(\"alpha\",\"beta\",\"lambda\")\n",
        "chain4=chain4[(burnin+1):nrow(chain4),]\n",
        "\n",
        "startvalue = c(1,0,1)\n",
        "chain5 = run_metropolis(startvalue=startvalue\n",
        "                       ,N.sim=N.sim\n",
        "                       ,burnin=burnin)\n",
        "colnames(chain5)=c(\"alpha\",\"beta\",\"lambda\")\n",
        "chain5=chain5[(burnin+1):nrow(chain5),]\n",
        "options(warn=0)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XL1vN-fvBtqX"
      },
      "source": [
        "### **Gelman Rubin Statistics**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "JXtEqPmK-cKK",
        "outputId": "3e0d3eed-4537-48ce-f8d3-3825b26a4213"
      },
      "source": [
        "## Gelman-Rubin Statistics for alpha\n",
        "\n",
        "n_str = length(chain1[(burnin+1):nrow(chain1),'alpha'])\n",
        "m_str = 5\n",
        "\n",
        "alpha_star_1 = chain1[(burnin+1):nrow(chain1),'alpha']\n",
        "alpha_star_2 = chain2[(burnin+1):nrow(chain2),'alpha']\n",
        "alpha_star_3 = chain3[(burnin+1):nrow(chain3),'alpha']\n",
        "alpha_star_4 = chain4[(burnin+1):nrow(chain4),'alpha']\n",
        "alpha_star_5 = chain5[(burnin+1):nrow(chain5),'alpha']\n",
        "\n",
        "a1 = mean(alpha_star_1)\n",
        "a2 = mean(alpha_star_2)\n",
        "a3 = mean(alpha_star_3)\n",
        "a4 = mean(alpha_star_4)\n",
        "a5 = mean(alpha_star_5)\n",
        "\n",
        "ai_bar = c(a1,a2,a3,a4,a5)\n",
        "\n",
        "a_bar = mean(ai_bar)\n",
        "\n",
        "B = m_str/(m_str-1)*sum((ai_bar-a_bar)^2)\n",
        "cat('B = ',B,'\\n')\n",
        "\n",
        "W =(1/m_str)*sum(var(alpha_star_1)+var(alpha_star_2)+var(alpha_star_3)+var(alpha_star_4)+var(alpha_star_5))\n",
        "cat('W = ',W,'\\n')\n",
        "\n",
        "marginal_var_alpha = ((n_str-1)/n_str)*W+(B/n_str)\n",
        "cat('marginal posterior variance = ',marginal_var_alpha,'\\n')\n",
        "\n",
        "R_hat = sqrt(marginal_var_alpha/W)\n",
        "cat('Gelman Rubin Statistic = ',round(R_hat,4),'\\n')\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "B =  2.270146e-05 \n",
            "W =  0.001166209 \n",
            "marginal posterior variance =  0.00116598 \n",
            "Gelman Rubin Statistic =  0.9999 \n"
          ],
          "name": "stdout"
        }
      ]
    }
  ]
}