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b/_images/f872d80acedc9f464f69f4829103b9c5bb0b73d3fe8a5c463eee4cc45be0f044.png new file mode 100644 index 0000000..b141566 Binary files /dev/null and b/_images/f872d80acedc9f464f69f4829103b9c5bb0b73d3fe8a5c463eee4cc45be0f044.png differ diff --git a/_sources/algorithms/mclmc.md b/_sources/algorithms/mclmc.md index 09d7e9b..77889ea 100644 --- a/_sources/algorithms/mclmc.md +++ b/_sources/algorithms/mclmc.md @@ -44,7 +44,7 @@ MCLMC in Blackjax comes with a tuning algorithm which attempts to find optimal v An example is given below, of tuning and running a chain for a 1000 dimensional Gaussian target (of which a 2 dimensional marginal is plotted): -```{code-cell} ipython3 +```{code-cell} :tags: [hide-cell] import matplotlib.pyplot as plt @@ -66,7 +66,7 @@ from numpyro.infer.util import initialize_model rng_key = jax.random.key(int(date.today().strftime("%Y%m%d"))) ``` -```{code-cell} ipython3 +```{code-cell} def run_mclmc(logdensity_fn, num_steps, initial_position, key, transform, desired_energy_variance= 5e-4): init_key, tune_key, run_key = jax.random.split(key, 3) @@ -115,7 +115,7 @@ def run_mclmc(logdensity_fn, num_steps, initial_position, key, transform, desire return samples, blackjax_state_after_tuning, blackjax_mclmc_sampler_params, run_key ``` -```{code-cell} ipython3 +```{code-cell} # run the algorithm on a high dimensional gaussian, and show two of the dimensions logdensity_fn = lambda x: -0.5 * jnp.sum(jnp.square(x)) @@ -134,13 +134,13 @@ samples, initial_state, params, chain_key = run_mclmc( samples.mean() ``` -```{code-cell} ipython3 +```{code-cell} plt.scatter(x=samples[:, 0], y=samples[:, 1], alpha=0.1) plt.axis("equal") plt.title("Scatter Plot of Samples") ``` -```{code-cell} ipython3 +```{code-cell} def visualize_results_gauss(samples, label, color): x1 = samples[:, 0] plt.hist(x1, bins= 30, density= True, histtype= 'step', lw= 4, color= color, label= label) @@ -165,12 +165,12 @@ ground_truth_gauss() A natural sanity check is to see if reducing $\epsilon$ changes the inferred distribution to an extent you care about. For example, we can inspect the 1D marginal with a stepsize $\epsilon$ as above, and compare it to a stepsize $\epsilon/2$ (and double the number of steps). We show this comparison below: -```{code-cell} ipython3 +```{code-cell} new_params = params._replace(step_size= params.step_size / 2) new_num_steps = num_steps * 2 ``` -```{code-cell} ipython3 +```{code-cell} sampling_alg = blackjax.mclmc( logdensity_fn, L=new_params.L, @@ -211,7 +211,7 @@ Our task is to find the posterior of the parameters $\{R_n\}_{n =1}^N$, $\sigma$ First, we get the data, define a model using NumPyro, and draw samples: -```{code-cell} ipython3 +```{code-cell} import matplotlib.dates as mdates from numpyro.examples.datasets import SP500, load_dataset from numpyro.distributions import StudentT @@ -243,7 +243,7 @@ def setup(): setup() ``` -```{code-cell} ipython3 +```{code-cell} def from_numpyro(model, rng_key, model_args): init_params, potential_fn_gen, *_ = initialize_model( rng_key, @@ -272,13 +272,13 @@ rng_key = jax.random.key(42) logp_sv, x_init = from_numpyro(stochastic_volatility, rng_key, model_args) ``` -```{code-cell} ipython3 +```{code-cell} num_steps = 20000 samples, initial_state, params, chain_key = run_mclmc(logdensity_fn= logp_sv, num_steps= num_steps, initial_position= x_init, key= sample_key, transform=lambda state, info: state.position) ``` -```{code-cell} ipython3 +```{code-cell} def visualize_results_sv(samples, color, label): R = np.exp(np.array(samples['s'])) # take an exponent to get R @@ -297,7 +297,7 @@ plt.legend() plt.show() ``` -```{code-cell} ipython3 +```{code-cell} new_params = params._replace(step_size = params.step_size/2) new_num_steps = num_steps * 2 @@ -318,10 +318,9 @@ _, new_samples = blackjax.util.run_inference_algorithm( transform=lambda state, info : state.position, progress_bar=True, ) - ``` -```{code-cell} ipython3 +```{code-cell} setup() visualize_results_sv(new_samples,'red', 'MCLMC', ) visualize_results_sv(samples,'teal', 'MCLMC (stepsize/2)', ) @@ -332,7 +331,7 @@ plt.show() Here, we have again inspected the effect of halving $\epsilon$. This looks OK, but suppose we are interested in the hierarchial parameters in particular, which tend to be harder to infer. We now inspect the marginal of a hierarchical parameter: -```{code-cell} ipython3 +```{code-cell} def visualize_results_sv_marginal(samples, color, label): # plt.subplot(1, 2, 1) # plt.hist(samples['nu'], bins = 20, histtype= 'step', lw= 4, density= True, color= color, label= label) @@ -354,9 +353,131 @@ If we care about this parameter in particular, we should reduce step size furthe +++ +## Adjusted MCLMC + +Blackjax also provides an adjusted version of the algorithm. This also has two hyperparameters, `step_size` and `L`. `L` is related to the `L` parameter of the unadjusted version, but not identical. The tuning algorithm is also similar, but uses a dual averaging scheme to tune the step size. We find in practice that a target MH acceptance rate of 0.9 is a good choice. + +```{code-cell} +from blackjax.mcmc.adjusted_mclmc import rescale +from blackjax.util import run_inference_algorithm + +def run_adjusted_mclmc( + logdensity_fn, + num_steps, + initial_position, + key, + transform=lambda state, _ : state.position, + diagonal_preconditioning=False, + random_trajectory_length=True, + L_proposal_factor=jnp.inf +): + + init_key, tune_key, run_key = jax.random.split(key, 3) + + initial_state = blackjax.mcmc.adjusted_mclmc.init( + position=initial_position, + logdensity_fn=logdensity_fn, + random_generator_arg=init_key, + ) + + if random_trajectory_length: + integration_steps_fn = lambda avg_num_integration_steps: lambda k: jnp.ceil( + jax.random.uniform(k) * rescale(avg_num_integration_steps)) + else: + integration_steps_fn = lambda avg_num_integration_steps: lambda _: jnp.ceil(avg_num_integration_steps) + + kernel = lambda rng_key, state, avg_num_integration_steps, step_size, sqrt_diag_cov: blackjax.mcmc.adjusted_mclmc.build_kernel( + integration_steps_fn=integration_steps_fn(avg_num_integration_steps), + sqrt_diag_cov=sqrt_diag_cov, + )( + rng_key=rng_key, + state=state, + step_size=step_size, + logdensity_fn=logdensity_fn, + L_proposal_factor=L_proposal_factor, + ) + + target_acc_rate = 0.9 # our recommendation + + ( + blackjax_state_after_tuning, + blackjax_mclmc_sampler_params, + ) = blackjax.adjusted_mclmc_find_L_and_step_size( + mclmc_kernel=kernel, + num_steps=num_steps, + state=initial_state, + rng_key=tune_key, + target=target_acc_rate, + frac_tune1=0.1, + frac_tune2=0.1, + frac_tune3=0.0, # our recommendation + diagonal_preconditioning=diagonal_preconditioning, + ) + + step_size = blackjax_mclmc_sampler_params.step_size + L = blackjax_mclmc_sampler_params.L + + alg = blackjax.adjusted_mclmc( + logdensity_fn=logdensity_fn, + step_size=step_size, + integration_steps_fn=lambda key: jnp.ceil( + jax.random.uniform(key) * rescale(L / step_size) + ), + sqrt_diag_cov=blackjax_mclmc_sampler_params.sqrt_diag_cov, + L_proposal_factor=L_proposal_factor, + ) + + _, out = run_inference_algorithm( + rng_key=run_key, + initial_state=blackjax_state_after_tuning, + inference_algorithm=alg, + num_steps=num_steps, + transform=transform, + progress_bar=False, + ) + + return out +``` + +```{code-cell} +# run the algorithm on a high dimensional gaussian, and show two of the dimensions + +sample_key, rng_key = jax.random.split(rng_key) +samples = run_adjusted_mclmc( + logdensity_fn=lambda x: -0.5 * jnp.sum(jnp.square(x)), + num_steps=1000, + initial_position=jnp.ones((1000,)), + key=sample_key, +) +plt.scatter(x=samples[:, 0], y=samples[:, 1], alpha=0.1) +plt.axis("equal") +plt.title("Scatter Plot of Samples") +``` + +```{code-cell} +# run the algorithm on a high dimensional gaussian, and show two of the dimensions + +sample_key, rng_key = jax.random.split(rng_key) +samples = run_adjusted_mclmc( + logdensity_fn=lambda x: -0.5 * jnp.sum(jnp.square(x)), + num_steps=1000, + initial_position=jnp.ones((1000,)), + key=sample_key, + random_trajectory_length=False, + L_proposal_factor=1.25, +) +plt.scatter(x=samples[:, 0], y=samples[:, 1], alpha=0.1) +plt.axis("equal") +plt.title("Scatter Plot of Samples") +``` + ```{bibliography} :filter: docname in docnames ``` +``` + +```{code-cell} + ``` diff --git a/algorithms/mclmc.html b/algorithms/mclmc.html index b053ee6..2916e34 100644 --- a/algorithms/mclmc.html +++ b/algorithms/mclmc.html @@ -386,6 +386,7 @@

Contents

  • How to analyze the results of your MCLMC run
    • @@ -455,7 +456,7 @@

    How to run MCLMC in BlackJax -
    /opt/hostedtoolcache/Python/3.11.10/x64/lib/python3.11/site-packages/tqdm/auto.py:21: TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See https://ipywidgets.readthedocs.io/en/stable/user_install.html
+
    /opt/hostedtoolcache/Python/3.11.11/x64/lib/python3.11/site-packages/tqdm/auto.py:21: TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See https://ipywidgets.readthedocs.io/en/stable/user_install.html
   from .autonotebook import tqdm as notebook_tqdm
 
    
 
    
@@ -557,7 +558,7 @@ 
    How to run MCLMC in BlackJax
    Array(0.00264662, dtype=float32)
+    
    Array(0.00752303, dtype=float32)
    @@ -574,7 +575,7 @@

    How to run MCLMC in BlackJax
    Text(0.5, 1.0, 'Scatter Plot of Samples')

    -../_images/0ec1b4434d7309896a76cee60b2f67a770c5ddaeb10603670ec5fa8f8e14d4c1.png +../_images/f58a28d8bafef836b4f2d1ea0216cb04e354440f8ce63a6016047a6e39295bf5.png
    @@ -599,7 +600,7 @@

    How to run MCLMC in BlackJax -../_images/d1c7d7b84b3a6ac445ac29b9c2056d5ff847bed88f6d4718a6a5d831289668c4.png +../_images/f872d80acedc9f464f69f4829103b9c5bb0b73d3fe8a5c463eee4cc45be0f044.png

    @@ -662,7 +663,7 @@

    Validate the choice of \(\epsilon 100.00% [20000/20000 00:00<?] - ../_images/38d0c9943edeadd2477f421372f67a01eb1b0f4f12fc3a057b3f5a05794aac96.png + ../_images/94d18d3d4664c09e29ac2a9f661585be3ea67e4043777c3f339077f031456486.png

    So here the change has little effect in this case.

    @@ -671,8 +672,8 @@

    Validate the choice of \(\epsilon

    A more complex example#

    We now consider a more complex model, of stock volatility.

    The returns \(r_n\) are modeled by a Student’s-t distribution whose scale (volatility) \(R_n\) is time varying and unknown. The prior for \(\log R_n\) is a Gaussian random walk, with an exponential distribution of the random walk step-size \(\sigma\). An exponential prior is also taken for the Student’s-t degrees of freedom \(\nu\). The generative process of the data is:

    -
    -(1)#\[\begin{align} +
    +(1)#\[\begin{align} &r_n / R_n \sim \text{Student's-t}(\nu) \qquad &&\nu \sim \text{Exp}(\lambda = 1/10) \\ \nonumber &\log R_n \sim \mathcal{N}(\log R_{n-1}, \sigma) \qquad @@ -716,12 +717,10 @@

    A more complex example 
    Downloading - https://d2hg8soec8ck9v.cloudfront.net/datasets/SP500.csv.
+Download complete.
    -
    Download complete.
-
    -
    -../_images/f7fd34fb893d79efc3e37fb858a4589d8551017b468c16da4ac5f51a3ad224d1.png +../_images/007875a6641d5f45dcd208adcadda104db098d0a2c3a6adc8ab9a549a3ea8194.png

    @@ -808,7 +807,7 @@

    A more complex example 100.00% [20000/20000 00:00<?]

    -
    ../_images/0a70760fa6e9e436f76686fc34a0a16b4091cbcd9d13b793090929596eff7b8e.png + ../_images/c80fb9135c61b29129499d74efabdb728d01e90f1b18055728d11bf698744b5c.png
    @@ -870,7 +869,7 @@

    A more complex example 100.00% [40000/40000 00:00<?]

    - ../_images/50b50a28029252f98e42ed1b367172bcd81c5ebe172126bb0c87553f87976245.png + ../_images/adaeffd1cf285dd63535c2c781fb1286bb3bd072812b80b4aadc4cc675736d3a.png

    Here, we have again inspected the effect of halving \(\epsilon\). This looks OK, but suppose we are interested in the hierarchial parameters in particular, which tend to be harder to infer. We now inspect the marginal of a hierarchical parameter:

    @@ -895,10 +894,149 @@

    A more complex example
    -../_images/7a204e67bf4cd962187cf778534ae56bbaeb4dbc1bbf19681183439da815d942.png +../_images/71464f41a600b82a21fb8145d6da6372716a1d658eb09b075628598c1ca0c2c6.png

    If we care about this parameter in particular, we should reduce step size further, until the difference disappears.

    + +
    +

    Adjusted MCLMC#

    +

    Blackjax also provides an adjusted version of the algorithm. This also has two hyperparameters, step_size and L. L is related to the L parameter of the unadjusted version, but not identical. The tuning algorithm is also similar, but uses a dual averaging scheme to tune the step size. We find in practice that a target MH acceptance rate of 0.9 is a good choice.

    +
    +
    +
    from blackjax.mcmc.adjusted_mclmc import rescale
+from blackjax.util import run_inference_algorithm
+
+def run_adjusted_mclmc(
+    logdensity_fn,
+    num_steps,
+    initial_position,
+    key,
+    transform=lambda state, _ : state.position,
+    diagonal_preconditioning=False,
+    random_trajectory_length=True,
+    L_proposal_factor=jnp.inf
+):
+
+    init_key, tune_key, run_key = jax.random.split(key, 3)
+
+    initial_state = blackjax.mcmc.adjusted_mclmc.init(
+        position=initial_position,
+        logdensity_fn=logdensity_fn,
+        random_generator_arg=init_key,
+    )
+
+    if random_trajectory_length:
+        integration_steps_fn = lambda avg_num_integration_steps: lambda k: jnp.ceil(
+            jax.random.uniform(k) * rescale(avg_num_integration_steps))
+    else:
+        integration_steps_fn = lambda avg_num_integration_steps: lambda _: jnp.ceil(avg_num_integration_steps)
+
+    kernel = lambda rng_key, state, avg_num_integration_steps, step_size, sqrt_diag_cov: blackjax.mcmc.adjusted_mclmc.build_kernel(
+        integration_steps_fn=integration_steps_fn(avg_num_integration_steps),
+        sqrt_diag_cov=sqrt_diag_cov,
+    )(
+        rng_key=rng_key,
+        state=state,
+        step_size=step_size,
+        logdensity_fn=logdensity_fn,
+        L_proposal_factor=L_proposal_factor,
+    )
+
+    target_acc_rate = 0.9 # our recommendation
+
+    (
+        blackjax_state_after_tuning,
+        blackjax_mclmc_sampler_params,
+    ) = blackjax.adjusted_mclmc_find_L_and_step_size(
+        mclmc_kernel=kernel,
+        num_steps=num_steps,
+        state=initial_state,
+        rng_key=tune_key,
+        target=target_acc_rate,
+        frac_tune1=0.1,
+        frac_tune2=0.1,
+        frac_tune3=0.0, # our recommendation
+        diagonal_preconditioning=diagonal_preconditioning,
+    )
+
+    step_size = blackjax_mclmc_sampler_params.step_size
+    L = blackjax_mclmc_sampler_params.L
+
+    alg = blackjax.adjusted_mclmc(
+        logdensity_fn=logdensity_fn,
+        step_size=step_size,
+        integration_steps_fn=lambda key: jnp.ceil(
+            jax.random.uniform(key) * rescale(L / step_size)
+        ),
+        sqrt_diag_cov=blackjax_mclmc_sampler_params.sqrt_diag_cov,
+        L_proposal_factor=L_proposal_factor,
+    )
+
+    _, out = run_inference_algorithm(
+        rng_key=run_key,
+        initial_state=blackjax_state_after_tuning,
+        inference_algorithm=alg,
+        num_steps=num_steps,
+        transform=transform,
+        progress_bar=False,
+    )
+
+    return out
+
    +
    +
    +
    +
    +
    +
    # run the algorithm on a high dimensional gaussian, and show two of the dimensions
+
+sample_key, rng_key = jax.random.split(rng_key)
+samples = run_adjusted_mclmc(
+    logdensity_fn=lambda x: -0.5 * jnp.sum(jnp.square(x)),
+    num_steps=1000,
+    initial_position=jnp.ones((1000,)),
+    key=sample_key,
+)
+plt.scatter(x=samples[:, 0], y=samples[:, 1], alpha=0.1)
+plt.axis("equal")
+plt.title("Scatter Plot of Samples")
+
    +
    +
    +
    +
    Text(0.5, 1.0, 'Scatter Plot of Samples')
+
    +
    +../_images/9ac388bb73a2903fcc26a9e56d67334a38b7217f1d9fb9fe422768ea6419656d.png +
    +
    +
    +
    +
    # run the algorithm on a high dimensional gaussian, and show two of the dimensions
+
+sample_key, rng_key = jax.random.split(rng_key)
+samples = run_adjusted_mclmc(
+    logdensity_fn=lambda x: -0.5 * jnp.sum(jnp.square(x)),
+    num_steps=1000,
+    initial_position=jnp.ones((1000,)),
+    key=sample_key,
+    random_trajectory_length=False,
+    L_proposal_factor=1.25,
+)
+plt.scatter(x=samples[:, 0], y=samples[:, 1], alpha=0.1)
+plt.axis("equal")
+plt.title("Scatter Plot of Samples")
+
    +
    +
    +
    +
    Text(0.5, 1.0, 'Scatter Plot of Samples')
+
    +
    +../_images/ce341e7b5e15603176fe4c01cc9731b1255965a6276c12e0db0cf60fa40c380e.png +
    +
    @@ -912,6 +1050,8 @@

    A more complex example 

    
+```{code-cell}
+
    @@ -989,6 +1129,7 @@

    A more complex exampleHow to analyze the results of your MCLMC run diff --git a/searchindex.js b/searchindex.js index 7e80acb..f724f3d 100644 --- a/searchindex.js +++ b/searchindex.js @@ -1 +1 @@ -Search.setIndex({"alltitles": {"A more complex example": [[5, "a-more-complex-example"]], "Adaptation": [[9, null]], "Algorithms": [[7, null]], "Arviz Plots": [[12, "arviz-plots"]], "Banana Density": [[0, "banana-density"]], "Bayesian Logistic Regression": [[14, null]], "Bayesian Logistic Regression With Latent Gaussian Sampler": [[10, null]], "Bayesian Regression With Latent Gaussian Sampler": [[9, null]], "Bibliography": [[17, "bibliography"]], "Change of Variable": [[12, "change-of-variable"]], "Change of Variable in HMC": [[12, null]], "Comparison": [[15, "comparison"]], "Conclusion": [[16, "conclusion"]], "Contour stochastic gradient Langevin dynamics": [[3, null]], "Cyclical SGLD": [[4, null], [4, "id2"]], "Cyclical schedule": [[4, "cyclical-schedule"]], "Data": [[13, "data"]], "Data Preparation": [[15, "data-preparation"]], "Diagnostics": [[8, "diagnostics"], [9, "diagnostics"]], "Ellipsis": [[0, "ellipsis"]], "Ellipsis + IAF": [[0, "ellipsis-iaf"]], "Example: Auxiliary Metropolis-Adjusted Langevin Algorithm (MALA)": [[9, "example-auxiliary-metropolis-adjusted-langevin-algorithm-mala"]], "Experimental Setup": [[1, "experimental-setup"], [1, "id1"]], "Exploring uncertainty": [[15, "exploring-uncertainty"]], "Fit Separate MLPs, One Per Task": [[13, "fit-separate-mlps-one-per-task"]], "Gaussian Mixture model": [[3, "gaussian-mixture-model"]], "Gaussian Regression with the Elliptical Slice Sampler": [[8, null]], "German credit dataset": [[17, "german-credit-dataset"]], "HMC Sampler": [[1, "hmc-sampler"]], "Hierarchical Bayesian Neural Networks": [[13, null]], "Hierarchical Model": [[13, "hierarchical-model"]], "How does Contour SGLD work?": [[3, "how-does-contour-sgld-work"]], "How to analyze the results of your MCLMC run": [[5, "how-to-analyze-the-results-of-your-mclmc-run"]], "How to run MCLMC in BlackJax": [[5, "how-to-run-mclmc-in-blackjax"]], "Hyperparameters": [[13, "hyperparameters"]], "IRMH tuning the covariance matrix.": [[2, "irmh-tuning-the-covariance-matrix"]], "IRMH tuning the diagonal of the covariance matrix": [[2, "irmh-tuning-the-diagonal-of-the-covariance-matrix"]], "IRMH without tuning": [[2, "irmh-without-tuning"]], "Imports": [[1, "imports"]], "Initial State and Sampler Parameters": [[0, "initial-state-and-sampler-parameters"]], "Latent Gaussian Models": [[9, "latent-gaussian-models"]], "Log-posterior densities via ProbDiffEq": [[16, "log-posterior-densities-via-probdiffeq"]], "MEADS": [[17, "meads"]], "MLP classifier": [[15, null]], "McLachlan": [[0, "mclachlan"]], "Microcanonical Langevin Monte Carlo": [[5, null]], "Models": [[7, null], [17, "models"]], "Motivation: Auxiliary Metropolis-Hastings samplers": [[9, "motivation-auxiliary-metropolis-hastings-samplers"]], "Multi-layer Perceptron": [[15, "multi-layer-perceptron"]], "NUTS Sampler": [[1, "nuts-sampler"]], "Parameter estimation in ODE models with a probabilistic ODE solver": [[16, null]], "Pathfinder": [[6, null]], "Pathfinder as a Variational Inference Method": [[6, "pathfinder-as-a-variational-inference-method"]], "Pathfinder as an Initialization Tool for MCMC Kernels": [[6, "pathfinder-as-an-initialization-tool-for-mcmc-kernels"]], "Pathfinder: Parallel Quasi-Newton Variational Inference": [[6, "pathfinder-parallel-quasi-newton-variational-inference"]], "Periodic Orbital MCMC": [[0, null]], "Posterior Sampling": [[12, "posterior-sampling"], [14, "posterior-sampling"]], "Posterior sampling": [[10, "posterior-sampling"]], "Predictive Distribution": [[14, "predictive-distribution"]], "Predictive distribution": [[10, "predictive-distribution"]], "Problem setting": [[16, "problem-setting"]], "Regime switching Hidden Markov model": [[11, null]], "Results": [[13, "results"], [13, "id1"]], "SGLD": [[4, "sgld"]], "Sample From the Posterior Distribution of the MLP\u2019s Weights": [[15, "sample-from-the-posterior-distribution-of-the-mlp-s-weights"]], "Sample with Contour SGLD": [[3, "sample-with-contour-sgld"]], "Sample with HMC": [[1, "sample-with-hmc"]], "Sample with NUTS": [[1, "sample-with-nuts"]], "Sampler Overview": [[9, "sampler-overview"]], "Sampling": [[8, "sampling"], [9, "sampling"]], "Sampling From a Bimodal Potential": [[1, "sampling-from-a-bimodal-potential"]], "Sampling from the Rastrigin Potential": [[1, "sampling-from-the-rastrigin-potential"]], "Sampling with BlackJAX": [[16, "sampling-with-blackjax"]], "Sampling with SGHMC": [[15, "sampling-with-sghmc"]], "Setup": [[13, "setup"]], "Some Caveats": [[6, "some-caveats"]], "Some Utility Functions": [[0, "some-utility-functions"]], "Sparse regression": [[17, null]], "Step function": [[4, "step-function"]], "TL;DR": [[16, "tl-dr"]], "Technical setup": [[16, "technical-setup"]], "Tempered SMC with HMC Kernel": [[1, 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"how-to-run-mclmc-in-blackjax"]], "Hyperparameters": [[13, "hyperparameters"]], "IRMH tuning the covariance matrix.": [[2, "irmh-tuning-the-covariance-matrix"]], "IRMH tuning the diagonal of the covariance matrix": [[2, "irmh-tuning-the-diagonal-of-the-covariance-matrix"]], "IRMH without tuning": [[2, "irmh-without-tuning"]], "Imports": [[1, "imports"]], "Initial State and Sampler Parameters": [[0, "initial-state-and-sampler-parameters"]], "Latent Gaussian Models": [[9, "latent-gaussian-models"]], "Log-posterior densities via ProbDiffEq": [[16, "log-posterior-densities-via-probdiffeq"]], "MEADS": [[17, "meads"]], "MLP classifier": [[15, null]], "McLachlan": [[0, "mclachlan"]], "Microcanonical Langevin Monte Carlo": [[5, null]], "Models": [[7, null], [17, "models"]], "Motivation: Auxiliary Metropolis-Hastings samplers": [[9, "motivation-auxiliary-metropolis-hastings-samplers"]], "Multi-layer Perceptron": [[15, "multi-layer-perceptron"]], "NUTS Sampler": [[1, "nuts-sampler"]], "Parameter estimation in ODE models with a probabilistic ODE solver": [[16, null]], "Pathfinder": [[6, null]], "Pathfinder as a Variational Inference Method": [[6, "pathfinder-as-a-variational-inference-method"]], "Pathfinder as an Initialization Tool for MCMC Kernels": [[6, "pathfinder-as-an-initialization-tool-for-mcmc-kernels"]], "Pathfinder: Parallel Quasi-Newton Variational Inference": [[6, "pathfinder-parallel-quasi-newton-variational-inference"]], "Periodic Orbital MCMC": [[0, null]], "Posterior Sampling": [[12, "posterior-sampling"], [14, "posterior-sampling"]], "Posterior sampling": [[10, "posterior-sampling"]], "Predictive Distribution": [[14, "predictive-distribution"]], "Predictive distribution": [[10, "predictive-distribution"]], "Problem setting": [[16, "problem-setting"]], "Regime switching Hidden Markov model": [[11, null]], "Results": [[13, "results"], [13, "id1"]], "SGLD": [[4, "sgld"]], "Sample From the Posterior Distribution of the MLP\u2019s Weights": [[15, 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