AR1-Model-Octave.ipynb

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   "source": [
    "# AR(1) Model\n",
    "\n"
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   "cell_type": "markdown",
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   "source": [
    "## The Gibbs sampler"
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   "execution_count": 2,
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       "</g>\n",
       "\t</g>\n",
       "<g color=\"white\" fill=\"none\" stroke=\"rgb(  0,   0, 255)\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"2.00\">\n",
       "</g>\n",
       "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"2.00\">\n",
       "</g>\n",
       "<g color=\"black\" fill=\"none\" stroke=\"black\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n",
       "</g>\n",
       "<g color=\"black\" fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"butt\" stroke-linejoin=\"miter\" stroke-width=\"0.50\">\n",
       "</g>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nu = 0.4; beta = 100; \n",
    "nu_beta = nu/beta;  P = 1.2; x_0 = 1; x_1 = -6;\n",
    "T = 1000;\n",
    "R = zeros(1, T);\n",
    "A = zeros(1, T); A(1) = -6; R(1) = 0.00001;\n",
    "\n",
    "for t=2:T,\n",
    "    Sig = 1/(1/P + x_0^2*(1/R(t-1)));\n",
    "    mu = Sig*x_0*x_1*(1/R(t-1));\n",
    "    A(t) = sqrt(Sig)*randn + mu;\n",
    "\n",
    "    b = 0.5*(x_1.^2 - 2*x_1*x_0*A(t) + x_0.^2*A(t).^2) + nu_beta;\n",
    "    R(t) = 1/(gamrnd(nu+0.5, 1/b));\n",
    "end;\n",
    "\n",
    "semilogy(A, R, '.');\n",
    "\n",
    "set(gca, 'xlim', [-8 6])\n",
    "set(gca, 'ylim', [10^(-4) 10^(5)])\n",
    "\n",
    "xlabel('A')\n",
    "ylabel('R')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## VB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "E_A = -0.37043\n",
      "E_invR =  0.054835\n"
     ]
    }
   ],
   "source": [
    "nu = 0.4; beta = 100; \n",
    "nu_beta = nu/beta;  P = 1.2; x_0 = 1; x_1 = -6;\n",
    "T = 300;\n",
    "E_A = -6; \n",
    "E_A2 = E_A^2;\n",
    "E_invR = 1/0.00001;\n",
    "\n",
    "for t=2:T,\n",
    "    Sig = 1/(1/P + x_0^2*E_invR);\n",
    "    mu = Sig*x_0*x_1*E_invR;\n",
    "\n",
    "    E_A = mu;\n",
    "    E_A2 = mu.^2 + Sig; \n",
    "    \n",
    "    a = nu+0.5;\n",
    "    b = 0.5*(x_1.^2 - 2*x_1*x_0*E_A + x_0.^2*E_A2) + nu_beta;\n",
    "\n",
    "    E_invR = a/b;\n",
    "end;\n",
    "\n",
    "E_A\n",
    "E_invR"
   ]
  }
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     "text": "MetaKernel Magics",
     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
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