# Markov Chain Monte Carlo (MCMC)¶

class MCMC(sampler, *, num_warmup, num_samples, num_chains=1, thinning=1, postprocess_fn=None, chain_method='parallel', progress_bar=True, jit_model_args=False)[source]

Bases: object

Note

chain_method is an experimental arg, which might be removed in a future version.

Note

Setting progress_bar=False will improve the speed for many cases. But it might require more memory than the other option.

Note

If setting num_chains greater than 1 in a Jupyter Notebook, then you will need to have installed ipywidgets in the environment from which you launced Jupyter in order for the progress bars to render correctly. If you are using Jupyter Notebook or Jupyter Lab, please also install the corresponding extension package like widgetsnbextension or jupyterlab_widgets.

Parameters
• sampler (MCMCKernel) – an instance of MCMCKernel that determines the sampler for running MCMC. Currently, only HMC and NUTS are available.

• num_warmup (int) – Number of warmup steps.

• num_samples (int) – Number of samples to generate from the Markov chain.

• thinning (int) – Positive integer that controls the fraction of post-warmup samples that are retained. For example if thinning is 2 then every other sample is retained. Defaults to 1, i.e. no thinning.

• num_chains (int) – Number of MCMC chains to run. By default, chains will be run in parallel using jax.pmap(). If there are not enough devices available, chains will be run in sequence.

• postprocess_fn – Post-processing callable - used to convert a collection of unconstrained sample values returned from the sampler to constrained values that lie within the support of the sample sites. Additionally, this is used to return values at deterministic sites in the model.

• chain_method (str) – One of ‘parallel’ (default), ‘sequential’, ‘vectorized’. The method ‘parallel’ is used to execute the drawing process in parallel on XLA devices (CPUs/GPUs/TPUs), If there are not enough devices for ‘parallel’, we fall back to ‘sequential’ method to draw chains sequentially. ‘vectorized’ method is an experimental feature which vectorizes the drawing method, hence allowing us to collect samples in parallel on a single device.

• progress_bar (bool) – Whether to enable progress bar updates. Defaults to True.

• jit_model_args (bool) – If set to True, this will compile the potential energy computation as a function of model arguments. As such, calling MCMC.run again on a same sized but different dataset will not result in additional compilation cost. Note that currently, this does not take effect for the case num_chains > 1 and chain_method == 'parallel'.

property post_warmup_state

The state before the sampling phase. If this attribute is not None, run() will skip the warmup phase and start with the state specified in this attribute.

Note

This attribute can be used to sequentially draw MCMC samples. For example,

mcmc = MCMC(NUTS(model), num_warmup=100, num_samples=100)
mcmc.run(random.PRNGKey(0))
first_100_samples = mcmc.get_samples()
mcmc.post_warmup_state = mcmc.last_state
mcmc.run(mcmc.post_warmup_state.rng_key)  # or mcmc.run(random.PRNGKey(1))
second_100_samples = mcmc.get_samples()

property last_state

The final MCMC state at the end of the sampling phase.

warmup(rng_key, *args, extra_fields=(), collect_warmup=False, init_params=None, **kwargs)[source]

Run the MCMC warmup adaptation phase. After this call, self.warmup_state will be set and the run() method will skip the warmup adaptation phase. To run warmup again for the new data, it is required to run warmup() again.

Parameters
run(rng_key, *args, extra_fields=(), init_params=None, **kwargs)[source]

Run the MCMC samplers and collect samples.

Parameters
• rng_key (random.PRNGKey) – Random number generator key to be used for the sampling. For multi-chains, a batch of num_chains keys can be supplied. If rng_key does not have batch_size, it will be split in to a batch of num_chains keys.

• args – Arguments to be provided to the numpyro.infer.mcmc.MCMCKernel.init() method. These are typically the arguments needed by the model.

• extra_fields (tuple or list of str) – Extra fields (aside from “z”, “diverging”) to be collected during the MCMC run. Note that subfields can be accessed using dots, e.g. “adapt_state.step_size” can be used to collect step sizes at each step.

• init_params – Initial parameters to begin sampling. The type must be consistent with the input type to potential_fn.

• kwargs – Keyword arguments to be provided to the numpyro.infer.mcmc.MCMCKernel.init() method. These are typically the keyword arguments needed by the model.

Note

jax allows python code to continue even when the compiled code has not finished yet. This can cause troubles when trying to profile the code for speed. See https://jax.readthedocs.io/en/latest/async_dispatch.html and https://jax.readthedocs.io/en/latest/profiling.html for pointers on profiling jax programs.

get_samples(group_by_chain=False)[source]

Get samples from the MCMC run.

Parameters

group_by_chain (bool) – Whether to preserve the chain dimension. If True, all samples will have num_chains as the size of their leading dimension.

Returns

Samples having the same data type as init_params. The data type is a dict keyed on site names if a model containing Pyro primitives is used, but can be any jaxlib.pytree(), more generally (e.g. when defining a potential_fn for HMC that takes list args).

Example:

You can then pass those samples to Predictive:

posterior_samples = mcmc.get_samples()
predictive = Predictive(model, posterior_samples=posterior_samples)
samples = predictive(rng_key1, *model_args, **model_kwargs)

get_extra_fields(group_by_chain=False)[source]

Get extra fields from the MCMC run.

Parameters

group_by_chain (bool) – Whether to preserve the chain dimension. If True, all samples will have num_chains as the size of their leading dimension.

Returns

Extra fields keyed by field names which are specified in the extra_fields keyword of run().

print_summary(prob=0.9, exclude_deterministic=True)[source]

Print the statistics of posterior samples collected during running this MCMC instance.

Parameters
• prob (float) – the probability mass of samples within the credible interval.

• exclude_deterministic (bool) – whether or not print out the statistics at deterministic sites.

## MCMC Kernels¶

### MCMCKernel¶

class MCMCKernel[source]

Bases: abc.ABC

Defines the interface for the Markov transition kernel that is used for MCMC inference.

Example:

>>> from collections import namedtuple
>>> from jax import random
>>> import jax.numpy as jnp
>>> import numpyro
>>> import numpyro.distributions as dist
>>> from numpyro.infer import MCMC

>>> MHState = namedtuple("MHState", ["u", "rng_key"])

>>> class MetropolisHastings(numpyro.infer.mcmc.MCMCKernel):
...     sample_field = "u"
...
...     def __init__(self, potential_fn, step_size=0.1):
...         self.potential_fn = potential_fn
...         self.step_size = step_size
...
...     def init(self, rng_key, num_warmup, init_params, model_args, model_kwargs):
...         return MHState(init_params, rng_key)
...
...     def sample(self, state, model_args, model_kwargs):
...         u, rng_key = state
...         rng_key, key_proposal, key_accept = random.split(rng_key, 3)
...         u_proposal = dist.Normal(u, self.step_size).sample(key_proposal)
...         accept_prob = jnp.exp(self.potential_fn(u) - self.potential_fn(u_proposal))
...         u_new = jnp.where(dist.Uniform().sample(key_accept) < accept_prob, u_proposal, u)
...         return MHState(u_new, rng_key)

>>> def f(x):
...     return ((x - 2) ** 2).sum()

>>> kernel = MetropolisHastings(f)
>>> mcmc = MCMC(kernel, num_warmup=1000, num_samples=1000)
>>> mcmc.run(random.PRNGKey(0), init_params=jnp.array([1., 2.]))
>>> posterior_samples = mcmc.get_samples()
>>> mcmc.print_summary()

postprocess_fn(model_args, model_kwargs)[source]

Get a function that transforms unconstrained values at sample sites to values constrained to the site’s support, in addition to returning deterministic sites in the model.

Parameters
• model_args – Arguments to the model.

• model_kwargs – Keyword arguments to the model.

abstract init(rng_key, num_warmup, init_params, model_args, model_kwargs)[source]

Initialize the MCMCKernel and return an initial state to begin sampling from.

Parameters
• rng_key (random.PRNGKey) – Random number generator key to initialize the kernel.

• num_warmup (int) – Number of warmup steps. This can be useful when doing adaptation during warmup.

• init_params (tuple) – Initial parameters to begin sampling. The type must be consistent with the input type to potential_fn.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

The initial state representing the state of the kernel. This can be any class that is registered as a pytree.

abstract sample(state, model_args, model_kwargs)[source]

Given the current state, return the next state using the given transition kernel.

Parameters
• state

A pytree class representing the state for the kernel. For HMC, this is given by HMCState. In general, this could be any class that supports getattr.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

Next state.

property sample_field

The attribute of the state object passed to sample() that denotes the MCMC sample. This is used by postprocess_fn() and for reporting results in MCMC.print_summary().

property default_fields

The attributes of the state object to be collected by default during the MCMC run (when MCMC.run() is called).

get_diagnostics_str(state)[source]

Given the current state, returns the diagnostics string to be added to progress bar for diagnostics purpose.

### BarkerMH¶

This is a gradient-based MCMC algorithm of Metropolis-Hastings type that uses a skew-symmetric proposal distribution that depends on the gradient of the potential (the Barker proposal; see reference [1]). In particular the proposal distribution is skewed in the direction of the gradient at the current sample.

We expect this algorithm to be particularly effective for low to moderate dimensional models, where it may be competitive with HMC and NUTS.

Note

We recommend to use this kernel with progress_bar=False in MCMC to reduce JAX’s dispatch overhead.

References:

1. The Barker proposal: combining robustness and efficiency in gradient-based MCMC. Samuel Livingstone, Giacomo Zanella.

Parameters
• model – Python callable containing Pyro primitives. If model is provided, potential_fn will be inferred using the model.

• potential_fn – Python callable that computes the potential energy given input parameters. The input parameters to potential_fn can be any python collection type, provided that init_params argument to init() has the same type.

• step_size (float) – (Initial) step size to use in the Barker proposal.

• adapt_step_size (bool) – Whether to adapt the step size during warm-up. Defaults to adapt_step_size==True.

• adapt_mass_matrix (bool) – Whether to adapt the mass matrix during warm-up. Defaults to adapt_mass_matrix==True.

• dense_mass (bool) – Whether to use a dense (i.e. full-rank) or diagonal mass matrix. (defaults to dense_mass=False).

• target_accept_prob (float) – The target acceptance probability that is used to guide step size adapation. Defaults to target_accept_prob=0.4.

• init_strategy (callable) – a per-site initialization function. See Initialization Strategies section for available functions.

Example

>>> import jax
>>> import jax.numpy as jnp
>>> import numpyro
>>> import numpyro.distributions as dist
>>> from numpyro.infer import MCMC, BarkerMH

>>> def model():
...     x = numpyro.sample("x", dist.Normal().expand([10]))
...     numpyro.sample("obs", dist.Normal(x, 1.0), obs=jnp.ones(10))
>>>
>>> kernel = BarkerMH(model)
>>> mcmc = MCMC(kernel, num_warmup=1000, num_samples=1000, progress_bar=True)
>>> mcmc.run(jax.random.PRNGKey(0))
>>> mcmc.print_summary()

property model
property sample_field

The attribute of the state object passed to sample() that denotes the MCMC sample. This is used by postprocess_fn() and for reporting results in MCMC.print_summary().

get_diagnostics_str(state)[source]

Given the current state, returns the diagnostics string to be added to progress bar for diagnostics purpose.

init(rng_key, num_warmup, init_params, model_args, model_kwargs)[source]

Initialize the MCMCKernel and return an initial state to begin sampling from.

Parameters
• rng_key (random.PRNGKey) – Random number generator key to initialize the kernel.

• num_warmup (int) – Number of warmup steps. This can be useful when doing adaptation during warmup.

• init_params (tuple) – Initial parameters to begin sampling. The type must be consistent with the input type to potential_fn.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

The initial state representing the state of the kernel. This can be any class that is registered as a pytree.

postprocess_fn(args, kwargs)[source]

Get a function that transforms unconstrained values at sample sites to values constrained to the site’s support, in addition to returning deterministic sites in the model.

Parameters
• model_args – Arguments to the model.

• model_kwargs – Keyword arguments to the model.

sample(state, model_args, model_kwargs)[source]

Given the current state, return the next state using the given transition kernel.

Parameters
• state

A pytree class representing the state for the kernel. For HMC, this is given by HMCState. In general, this could be any class that supports getattr.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

Next state.

### HMC¶

class HMC(model=None, potential_fn=None, kinetic_fn=None, step_size=1.0, inverse_mass_matrix=None, adapt_step_size=True, adapt_mass_matrix=True, dense_mass=False, target_accept_prob=0.8, trajectory_length=6.283185307179586, init_strategy=<function init_to_uniform>, find_heuristic_step_size=False, forward_mode_differentiation=False, regularize_mass_matrix=True)[source]

Hamiltonian Monte Carlo inference, using fixed trajectory length, with provision for step size and mass matrix adaptation.

Note

Until the kernel is used in an MCMC run, postprocess_fn will return the identity function.

Note

The default init strategy init_to_uniform might not be a good strategy for some models. You might want to try other init strategies like init_to_median.

References:

1. MCMC Using Hamiltonian Dynamics, Radford M. Neal

Parameters
• model – Python callable containing Pyro primitives. If model is provided, potential_fn will be inferred using the model.

• potential_fn – Python callable that computes the potential energy given input parameters. The input parameters to potential_fn can be any python collection type, provided that init_params argument to init() has the same type.

• kinetic_fn – Python callable that returns the kinetic energy given inverse mass matrix and momentum. If not provided, the default is euclidean kinetic energy.

• step_size (float) – Determines the size of a single step taken by the verlet integrator while computing the trajectory using Hamiltonian dynamics. If not specified, it will be set to 1.

• inverse_mass_matrix (numpy.ndarray or dict) – Initial value for inverse mass matrix. This may be adapted during warmup if adapt_mass_matrix = True. If no value is specified, then it is initialized to the identity matrix. For a potential_fn with general JAX pytree parameters, the order of entries of the mass matrix is the order of the flattened version of pytree parameters obtained with jax.tree_flatten, which is a bit ambiguous (see more at https://jax.readthedocs.io/en/latest/pytrees.html). If model is not None, here we can specify a structured block mass matrix as a dictionary, where keys are tuple of site names and values are the corresponding block of the mass matrix. For more information about structured mass matrix, see dense_mass argument.

• adapt_step_size (bool) – A flag to decide if we want to adapt step_size during warm-up phase using Dual Averaging scheme.

• adapt_mass_matrix (bool) – A flag to decide if we want to adapt mass matrix during warm-up phase using Welford scheme.

• dense_mass (bool or list) –

This flag controls whether mass matrix is dense (i.e. full-rank) or diagonal (defaults to dense_mass=False). To specify a structured mass matrix, users can provide a list of tuples of site names. Each tuple represents a block in the joint mass matrix. For example, assuming that the model has latent variables “x”, “y”, “z” (where each variable can be multi-dimensional), possible specifications and corresponding mass matrix structures are as follows:

• dense_mass=[(“x”, “y”)]: use a dense mass matrix for the joint (x, y) and a diagonal mass matrix for z

• dense_mass=[] (equivalent to dense_mass=False): use a diagonal mass matrix for the joint (x, y, z)

• dense_mass=[(“x”, “y”, “z”)] (equivalent to full_mass=True): use a dense mass matrix for the joint (x, y, z)

• dense_mass=[(“x”,), (“y”,), (“z”)]: use dense mass matrices for each of x, y, and z (i.e. block-diagonal with 3 blocks)

• target_accept_prob (float) – Target acceptance probability for step size adaptation using Dual Averaging. Increasing this value will lead to a smaller step size, hence the sampling will be slower but more robust. Defaults to 0.8.

• trajectory_length (float) – Length of a MCMC trajectory for HMC. Default value is $$2\pi$$.

• init_strategy (callable) – a per-site initialization function. See Initialization Strategies section for available functions.

• find_heuristic_step_size (bool) – whether or not to use a heuristic function to adjust the step size at the beginning of each adaptation window. Defaults to False.

• forward_mode_differentiation (bool) – whether to use forward-mode differentiation or reverse-mode differentiation. By default, we use reverse mode but the forward mode can be useful in some cases to improve the performance. In addition, some control flow utility on JAX such as jax.lax.while_loop or jax.lax.fori_loop only supports forward-mode differentiation. See JAX’s The Autodiff Cookbook for more information.

• regularize_mass_matrix (bool) – whether or not to regularize the estimated mass matrix for numerical stability during warmup phase. Defaults to True. This flag does not take effect if adapt_mass_matrix == False.

property model
property sample_field

The attribute of the state object passed to sample() that denotes the MCMC sample. This is used by postprocess_fn() and for reporting results in MCMC.print_summary().

property default_fields

The attributes of the state object to be collected by default during the MCMC run (when MCMC.run() is called).

get_diagnostics_str(state)[source]

Given the current state, returns the diagnostics string to be added to progress bar for diagnostics purpose.

init(rng_key, num_warmup, init_params=None, model_args=(), model_kwargs={})[source]

Initialize the MCMCKernel and return an initial state to begin sampling from.

Parameters
• rng_key (random.PRNGKey) – Random number generator key to initialize the kernel.

• num_warmup (int) – Number of warmup steps. This can be useful when doing adaptation during warmup.

• init_params (tuple) – Initial parameters to begin sampling. The type must be consistent with the input type to potential_fn.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

The initial state representing the state of the kernel. This can be any class that is registered as a pytree.

postprocess_fn(args, kwargs)[source]

Get a function that transforms unconstrained values at sample sites to values constrained to the site’s support, in addition to returning deterministic sites in the model.

Parameters
• model_args – Arguments to the model.

• model_kwargs – Keyword arguments to the model.

sample(state, model_args, model_kwargs)[source]

Run HMC from the given HMCState and return the resulting HMCState.

Parameters
• state (HMCState) – Represents the current state.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

Next state after running HMC.

### NUTS¶

class NUTS(model=None, potential_fn=None, kinetic_fn=None, step_size=1.0, inverse_mass_matrix=None, adapt_step_size=True, adapt_mass_matrix=True, dense_mass=False, target_accept_prob=0.8, trajectory_length=None, max_tree_depth=10, init_strategy=<function init_to_uniform>, find_heuristic_step_size=False, forward_mode_differentiation=False, regularize_mass_matrix=True)[source]

Hamiltonian Monte Carlo inference, using the No U-Turn Sampler (NUTS) with adaptive path length and mass matrix adaptation.

Note

Until the kernel is used in an MCMC run, postprocess_fn will return the identity function.

Note

The default init strategy init_to_uniform might not be a good strategy for some models. You might want to try other init strategies like init_to_median.

References:

1. MCMC Using Hamiltonian Dynamics, Radford M. Neal

2. The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo, Matthew D. Hoffman, and Andrew Gelman.

3. A Conceptual Introduction to Hamiltonian Monte Carlo, Michael Betancourt

Parameters
• model – Python callable containing Pyro primitives. If model is provided, potential_fn will be inferred using the model.

• potential_fn – Python callable that computes the potential energy given input parameters. The input parameters to potential_fn can be any python collection type, provided that init_params argument to init_kernel has the same type.

• kinetic_fn – Python callable that returns the kinetic energy given inverse mass matrix and momentum. If not provided, the default is euclidean kinetic energy.

• step_size (float) – Determines the size of a single step taken by the verlet integrator while computing the trajectory using Hamiltonian dynamics. If not specified, it will be set to 1.

• inverse_mass_matrix (numpy.ndarray or dict) – Initial value for inverse mass matrix. This may be adapted during warmup if adapt_mass_matrix = True. If no value is specified, then it is initialized to the identity matrix. For a potential_fn with general JAX pytree parameters, the order of entries of the mass matrix is the order of the flattened version of pytree parameters obtained with jax.tree_flatten, which is a bit ambiguous (see more at https://jax.readthedocs.io/en/latest/pytrees.html). If model is not None, here we can specify a structured block mass matrix as a dictionary, where keys are tuple of site names and values are the corresponding block of the mass matrix. For more information about structured mass matrix, see dense_mass argument.

• adapt_step_size (bool) – A flag to decide if we want to adapt step_size during warm-up phase using Dual Averaging scheme.

• adapt_mass_matrix (bool) – A flag to decide if we want to adapt mass matrix during warm-up phase using Welford scheme.

• dense_mass (bool or list) –

This flag controls whether mass matrix is dense (i.e. full-rank) or diagonal (defaults to dense_mass=False). To specify a structured mass matrix, users can provide a list of tuples of site names. Each tuple represents a block in the joint mass matrix. For example, assuming that the model has latent variables “x”, “y”, “z” (where each variable can be multi-dimensional), possible specifications and corresponding mass matrix structures are as follows:

• dense_mass=[(“x”, “y”)]: use a dense mass matrix for the joint (x, y) and a diagonal mass matrix for z

• dense_mass=[] (equivalent to dense_mass=False): use a diagonal mass matrix for the joint (x, y, z)

• dense_mass=[(“x”, “y”, “z”)] (equivalent to full_mass=True): use a dense mass matrix for the joint (x, y, z)

• dense_mass=[(“x”,), (“y”,), (“z”)]: use dense mass matrices for each of x, y, and z (i.e. block-diagonal with 3 blocks)

• target_accept_prob (float) – Target acceptance probability for step size adaptation using Dual Averaging. Increasing this value will lead to a smaller step size, hence the sampling will be slower but more robust. Defaults to 0.8.

• trajectory_length (float) – Length of a MCMC trajectory for HMC. This arg has no effect in NUTS sampler.

• max_tree_depth (int) – Max depth of the binary tree created during the doubling scheme of NUTS sampler. Defaults to 10. This argument also accepts a tuple of integers (d1, d2), where d1 is the max tree depth during warmup phase and d2 is the max tree depth during post warmup phase.

• init_strategy (callable) – a per-site initialization function. See Initialization Strategies section for available functions.

• find_heuristic_step_size (bool) – whether or not to use a heuristic function to adjust the step size at the beginning of each adaptation window. Defaults to False.

• forward_mode_differentiation (bool) –

whether to use forward-mode differentiation or reverse-mode differentiation. By default, we use reverse mode but the forward mode can be useful in some cases to improve the performance. In addition, some control flow utility on JAX such as jax.lax.while_loop or jax.lax.fori_loop only supports forward-mode differentiation. See JAX’s The Autodiff Cookbook for more information.

### HMCGibbs¶

class HMCGibbs(inner_kernel, gibbs_fn, gibbs_sites)[source]

[EXPERIMENTAL INTERFACE]

HMC-within-Gibbs. This inference algorithm allows the user to combine general purpose gradient-based inference (HMC or NUTS) with custom Gibbs samplers.

Note that it is the user’s responsibility to provide a correct implementation of gibbs_fn that samples from the corresponding posterior conditional.

Parameters
• inner_kernel – One of HMC or NUTS.

• gibbs_fn – A Python callable that returns a dictionary of Gibbs samples conditioned on the HMC sites. Must include an argument rng_key that should be used for all sampling. Must also include arguments hmc_sites and gibbs_sites, each of which is a dictionary with keys that are site names and values that are sample values. Note that a given gibbs_fn may not need make use of all these sample values.

• gibbs_sites (list) – a list of site names for the latent variables that are covered by the Gibbs sampler.

Example

>>> from jax import random
>>> import jax.numpy as jnp
>>> import numpyro
>>> import numpyro.distributions as dist
>>> from numpyro.infer import MCMC, NUTS, HMCGibbs
...
>>> def model():
...     x = numpyro.sample("x", dist.Normal(0.0, 2.0))
...     y = numpyro.sample("y", dist.Normal(0.0, 2.0))
...     numpyro.sample("obs", dist.Normal(x + y, 1.0), obs=jnp.array([1.0]))
...
>>> def gibbs_fn(rng_key, gibbs_sites, hmc_sites):
...     y = hmc_sites['y']
...     new_x = dist.Normal(0.8 * (1-y), jnp.sqrt(0.8)).sample(rng_key)
...     return {'x': new_x}
...
>>> hmc_kernel = NUTS(model)
>>> kernel = HMCGibbs(hmc_kernel, gibbs_fn=gibbs_fn, gibbs_sites=['x'])
>>> mcmc = MCMC(kernel, num_warmup=100, num_samples=100, progress_bar=False)
>>> mcmc.run(random.PRNGKey(0))
>>> mcmc.print_summary()

sample_field = 'z'
property model
get_diagnostics_str(state)[source]

Given the current state, returns the diagnostics string to be added to progress bar for diagnostics purpose.

postprocess_fn(args, kwargs)[source]

Get a function that transforms unconstrained values at sample sites to values constrained to the site’s support, in addition to returning deterministic sites in the model.

Parameters
• model_args – Arguments to the model.

• model_kwargs – Keyword arguments to the model.

init(rng_key, num_warmup, init_params, model_args, model_kwargs)[source]

Initialize the MCMCKernel and return an initial state to begin sampling from.

Parameters
• rng_key (random.PRNGKey) – Random number generator key to initialize the kernel.

• num_warmup (int) – Number of warmup steps. This can be useful when doing adaptation during warmup.

• init_params (tuple) – Initial parameters to begin sampling. The type must be consistent with the input type to potential_fn.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

The initial state representing the state of the kernel. This can be any class that is registered as a pytree.

sample(state, model_args, model_kwargs)[source]

Given the current state, return the next state using the given transition kernel.

Parameters
• state

A pytree class representing the state for the kernel. For HMC, this is given by HMCState. In general, this could be any class that supports getattr.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

Next state.

### DiscreteHMCGibbs¶

class DiscreteHMCGibbs(inner_kernel, *, random_walk=False, modified=False)[source]

[EXPERIMENTAL INTERFACE]

A subclass of HMCGibbs which performs Metropolis updates for discrete latent sites.

Note

The site update order is randomly permuted at each step.

Note

This class supports enumeration of discrete latent variables. To marginalize out a discrete latent site, we can specify infer={‘enumerate’: ‘parallel’} keyword in its corresponding sample() statement.

Parameters
• inner_kernel – One of HMC or NUTS.

• random_walk (bool) – If False, Gibbs sampling will be used to draw a sample from the conditional p(gibbs_site | remaining sites). Otherwise, a sample will be drawn uniformly from the domain of gibbs_site. Defaults to False.

• modified (bool) – whether to use a modified proposal, as suggested in reference [1], which always proposes a new state for the current Gibbs site. Defaults to False. The modified scheme appears in the literature under the name “modified Gibbs sampler” or “Metropolised Gibbs sampler”.

References:

1. Peskun’s theorem and a modified discrete-state Gibbs sampler, Liu, J. S. (1996)

Example

>>> from jax import random
>>> import jax.numpy as jnp
>>> import numpyro
>>> import numpyro.distributions as dist
>>> from numpyro.infer import DiscreteHMCGibbs, MCMC, NUTS
...
>>> def model(probs, locs):
...     c = numpyro.sample("c", dist.Categorical(probs))
...     numpyro.sample("x", dist.Normal(locs[c], 0.5))
...
>>> probs = jnp.array([0.15, 0.3, 0.3, 0.25])
>>> locs = jnp.array([-2, 0, 2, 4])
>>> kernel = DiscreteHMCGibbs(NUTS(model), modified=True)
>>> mcmc = MCMC(kernel, num_warmup=1000, num_samples=100000, progress_bar=False)
>>> mcmc.run(random.PRNGKey(0), probs, locs)
>>> mcmc.print_summary()
>>> samples = mcmc.get_samples()["x"]
>>> assert abs(jnp.mean(samples) - 1.3) < 0.1
>>> assert abs(jnp.var(samples) - 4.36) < 0.5

init(rng_key, num_warmup, init_params, model_args, model_kwargs)[source]

Initialize the MCMCKernel and return an initial state to begin sampling from.

Parameters
• rng_key (random.PRNGKey) – Random number generator key to initialize the kernel.

• num_warmup (int) – Number of warmup steps. This can be useful when doing adaptation during warmup.

• init_params (tuple) – Initial parameters to begin sampling. The type must be consistent with the input type to potential_fn.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

The initial state representing the state of the kernel. This can be any class that is registered as a pytree.

sample(state, model_args, model_kwargs)[source]

Given the current state, return the next state using the given transition kernel.

Parameters
• state

A pytree class representing the state for the kernel. For HMC, this is given by HMCState. In general, this could be any class that supports getattr.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

Next state.

### MixedHMC¶

class MixedHMC(inner_kernel, *, num_discrete_updates=None, random_walk=False, modified=False)[source]

Implementation of Mixed Hamiltonian Monte Carlo (reference [1]).

Note

The number of discrete sites to update at each MCMC iteration (n_D in reference [1]) is fixed at value 1.

References

1. Mixed Hamiltonian Monte Carlo for Mixed Discrete and Continuous Variables, Guangyao Zhou (2020)

2. Peskun’s theorem and a modified discrete-state Gibbs sampler, Liu, J. S. (1996)

Parameters
• inner_kernel – A HMC kernel.

• num_discrete_updates (int) – Number of times to update discrete variables. Defaults to the number of discrete latent variables.

• random_walk (bool) – If False, Gibbs sampling will be used to draw a sample from the conditional p(gibbs_site | remaining sites), where gibbs_site is one of the discrete sample sites in the model. Otherwise, a sample will be drawn uniformly from the domain of gibbs_site. Defaults to False.

• modified (bool) – whether to use a modified proposal, as suggested in reference [2], which always proposes a new state for the current Gibbs site (i.e. discrete site). Defaults to False. The modified scheme appears in the literature under the name “modified Gibbs sampler” or “Metropolised Gibbs sampler”.

Example

>>> from jax import random
>>> import jax.numpy as jnp
>>> import numpyro
>>> import numpyro.distributions as dist
>>> from numpyro.infer import HMC, MCMC, MixedHMC
...
>>> def model(probs, locs):
...     c = numpyro.sample("c", dist.Categorical(probs))
...     numpyro.sample("x", dist.Normal(locs[c], 0.5))
...
>>> probs = jnp.array([0.15, 0.3, 0.3, 0.25])
>>> locs = jnp.array([-2, 0, 2, 4])
>>> kernel = MixedHMC(HMC(model, trajectory_length=1.2), num_discrete_updates=20)
>>> mcmc = MCMC(kernel, num_warmup=1000, num_samples=100000, progress_bar=False)
>>> mcmc.run(random.PRNGKey(0), probs, locs)
>>> mcmc.print_summary()
>>> samples = mcmc.get_samples()
>>> assert "x" in samples and "c" in samples
>>> assert abs(jnp.mean(samples["x"]) - 1.3) < 0.1
>>> assert abs(jnp.var(samples["x"]) - 4.36) < 0.5

init(rng_key, num_warmup, init_params, model_args, model_kwargs)[source]

Initialize the MCMCKernel and return an initial state to begin sampling from.

Parameters
• rng_key (random.PRNGKey) – Random number generator key to initialize the kernel.

• num_warmup (int) – Number of warmup steps. This can be useful when doing adaptation during warmup.

• init_params (tuple) – Initial parameters to begin sampling. The type must be consistent with the input type to potential_fn.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

The initial state representing the state of the kernel. This can be any class that is registered as a pytree.

sample(state, model_args, model_kwargs)[source]

Given the current state, return the next state using the given transition kernel.

Parameters
• state

A pytree class representing the state for the kernel. For HMC, this is given by HMCState. In general, this could be any class that supports getattr.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

Next state.

### HMCECS¶

class HMCECS(inner_kernel, *, num_blocks=1, proxy=None)[source]

[EXPERIMENTAL INTERFACE]

HMC with Energy Conserving Subsampling.

A subclass of HMCGibbs for performing HMC-within-Gibbs for models with subsample statements using the plate primitive. This implements Algorithm 1 of reference [1] but uses a naive estimation (without control variates) of log likelihood, hence might incur a high variance.

The function can divide subsample indices into blocks and update only one block at each MCMC step to improve the acceptance rate of proposed subsamples as detailed in [3].

Note

New subsample indices are proposed randomly with replacement at each MCMC step.

References:

1. Hamiltonian Monte Carlo with energy conserving subsampling, Dang, K. D., Quiroz, M., Kohn, R., Minh-Ngoc, T., & Villani, M. (2019)

2. Speeding Up MCMC by Efficient Data Subsampling, Quiroz, M., Kohn, R., Villani, M., & Tran, M. N. (2018)

3. The Block Pseudo-Margional Sampler, Tran, M.-N., Kohn, R., Quiroz, M. Villani, M. (2017)

4. The Fundamental Incompatibility of Scalable Hamiltonian Monte Carlo and Naive Data Subsampling Betancourt, M. (2015)

Parameters

Example

>>> from jax import random
>>> import jax.numpy as jnp
>>> import numpyro
>>> import numpyro.distributions as dist
>>> from numpyro.infer import HMCECS, MCMC, NUTS
...
>>> def model(data):
...     x = numpyro.sample("x", dist.Normal(0, 1))
...     with numpyro.plate("N", data.shape[0], subsample_size=100):
...         batch = numpyro.subsample(data, event_dim=0)
...         numpyro.sample("obs", dist.Normal(x, 1), obs=batch)
...
>>> data = random.normal(random.PRNGKey(0), (10000,)) + 1
>>> kernel = HMCECS(NUTS(model), num_blocks=10)
>>> mcmc = MCMC(kernel, num_warmup=1000, num_samples=1000)
>>> mcmc.run(random.PRNGKey(0), data)
>>> samples = mcmc.get_samples()["x"]
>>> assert abs(jnp.mean(samples) - 1.) < 0.1

postprocess_fn(args, kwargs)[source]

Get a function that transforms unconstrained values at sample sites to values constrained to the site’s support, in addition to returning deterministic sites in the model.

Parameters
• model_args – Arguments to the model.

• model_kwargs – Keyword arguments to the model.

init(rng_key, num_warmup, init_params, model_args, model_kwargs)[source]

Initialize the MCMCKernel and return an initial state to begin sampling from.

Parameters
• rng_key (random.PRNGKey) – Random number generator key to initialize the kernel.

• num_warmup (int) – Number of warmup steps. This can be useful when doing adaptation during warmup.

• init_params (tuple) – Initial parameters to begin sampling. The type must be consistent with the input type to potential_fn.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

The initial state representing the state of the kernel. This can be any class that is registered as a pytree.

sample(state, model_args, model_kwargs)[source]

Given the current state, return the next state using the given transition kernel.

Parameters
• state

A pytree class representing the state for the kernel. For HMC, this is given by HMCState. In general, this could be any class that supports getattr.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

Next state.

static taylor_proxy(reference_params)[source]

This is just a convenient static method which calls taylor_proxy().

### SA¶

class SA(model=None, potential_fn=None, adapt_state_size=None, dense_mass=True, init_strategy=<function init_to_uniform>)[source]

This is a very fast (in term of n_eff / s) sampler but requires many warmup (burn-in) steps. In each MCMC step, we only need to evaluate potential function at one point.

Note that unlike in reference [1], we return a randomly selected (i.e. thinned) subset of approximate posterior samples of size num_chains x num_samples instead of num_chains x num_samples x adapt_state_size.

Note

We recommend to use this kernel with progress_bar=False in MCMC to reduce JAX’s dispatch overhead.

References:

Parameters
• model – Python callable containing Pyro primitives. If model is provided, potential_fn will be inferred using the model.

• potential_fn – Python callable that computes the potential energy given input parameters. The input parameters to potential_fn can be any python collection type, provided that init_params argument to init() has the same type.

• adapt_state_size (int) – The number of points to generate proposal distribution. Defaults to 2 times latent size.

• dense_mass (bool) – A flag to decide if mass matrix is dense or diagonal (default to dense_mass=True)

• init_strategy (callable) – a per-site initialization function. See Initialization Strategies section for available functions.

init(rng_key, num_warmup, init_params=None, model_args=(), model_kwargs={})[source]

Initialize the MCMCKernel and return an initial state to begin sampling from.

Parameters
• rng_key (random.PRNGKey) – Random number generator key to initialize the kernel.

• num_warmup (int) – Number of warmup steps. This can be useful when doing adaptation during warmup.

• init_params (tuple) – Initial parameters to begin sampling. The type must be consistent with the input type to potential_fn.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

The initial state representing the state of the kernel. This can be any class that is registered as a pytree.

property model
property sample_field

The attribute of the state object passed to sample() that denotes the MCMC sample. This is used by postprocess_fn() and for reporting results in MCMC.print_summary().

property default_fields

The attributes of the state object to be collected by default during the MCMC run (when MCMC.run() is called).

get_diagnostics_str(state)[source]

Given the current state, returns the diagnostics string to be added to progress bar for diagnostics purpose.

postprocess_fn(args, kwargs)[source]

Get a function that transforms unconstrained values at sample sites to values constrained to the site’s support, in addition to returning deterministic sites in the model.

Parameters
• model_args – Arguments to the model.

• model_kwargs – Keyword arguments to the model.

sample(state, model_args, model_kwargs)[source]

Run SA from the given SAState and return the resulting SAState.

Parameters
• state (SAState) – Represents the current state.

• model_args – Arguments provided to the model.

• model_kwargs – Keyword arguments provided to the model.

Returns

Next state after running SA.

hmc(potential_fn=None, potential_fn_gen=None, kinetic_fn=None, algo='NUTS')[source]

Hamiltonian Monte Carlo inference, using either fixed number of steps or the No U-Turn Sampler (NUTS) with adaptive path length.

References:

1. MCMC Using Hamiltonian Dynamics, Radford M. Neal

2. The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo, Matthew D. Hoffman, and Andrew Gelman.

3. A Conceptual Introduction to Hamiltonian Monte Carlo, Michael Betancourt

Parameters
• potential_fn – Python callable that computes the potential energy given input parameters. The input parameters to potential_fn can be any python collection type, provided that init_params argument to init_kernel has the same type.

• potential_fn_gen – Python callable that when provided with model arguments / keyword arguments returns potential_fn. This may be provided to do inference on the same model with changing data. If the data shape remains the same, we can compile sample_kernel once, and use the same for multiple inference runs.

• kinetic_fn – Python callable that returns the kinetic energy given inverse mass matrix and momentum. If not provided, the default is euclidean kinetic energy.

• algo (str) – Whether to run HMC with fixed number of steps or NUTS with adaptive path length. Default is NUTS.

Returns

a tuple of callables (init_kernel, sample_kernel), the first one to initialize the sampler, and the second one to generate samples given an existing one.

Warning

Instead of using this interface directly, we would highly recommend you to use the higher level MCMC API instead.

Example

>>> import jax
>>> from jax import random
>>> import jax.numpy as jnp
>>> import numpyro
>>> import numpyro.distributions as dist
>>> from numpyro.infer.hmc import hmc
>>> from numpyro.infer.util import initialize_model
>>> from numpyro.util import fori_collect

>>> true_coefs = jnp.array([1., 2., 3.])
>>> data = random.normal(random.PRNGKey(2), (2000, 3))
>>> labels = dist.Bernoulli(logits=(true_coefs * data).sum(-1)).sample(random.PRNGKey(3))
>>>
>>> def model(data, labels):
...     coefs = numpyro.sample('coefs', dist.Normal(jnp.zeros(3), jnp.ones(3)))
...     intercept = numpyro.sample('intercept', dist.Normal(0., 10.))
...     return numpyro.sample('y', dist.Bernoulli(logits=(coefs * data + intercept).sum(-1)), obs=labels)
>>>
>>> model_info = initialize_model(random.PRNGKey(0), model, model_args=(data, labels,))
>>> init_kernel, sample_kernel = hmc(model_info.potential_fn, algo='NUTS')
>>> hmc_state = init_kernel(model_info.param_info,
...                         trajectory_length=10,
...                         num_warmup=300)
>>> samples = fori_collect(0, 500, sample_kernel, hmc_state,
...                        transform=lambda state: model_info.postprocess_fn(state.z))
>>> print(jnp.mean(samples['coefs'], axis=0))
[0.9153987 2.0754058 2.9621222]

init_kernel(init_params, num_warmup, *, step_size=1.0, inverse_mass_matrix=None, adapt_step_size=True, adapt_mass_matrix=True, dense_mass=False, target_accept_prob=0.8, trajectory_length=6.283185307179586, max_tree_depth=10, find_heuristic_step_size=False, forward_mode_differentiation=False, regularize_mass_matrix=True, model_args=(), model_kwargs=None, rng_key=DeviceArray([0, 0], dtype=uint32))

Initializes the HMC sampler.

Parameters
• init_params – Initial parameters to begin sampling. The type must be consistent with the input type to potential_fn.

• num_warmup (int) – Number of warmup steps; samples generated during warmup are discarded.

• step_size (float) – Determines the size of a single step taken by the verlet integrator while computing the trajectory using Hamiltonian dynamics. If not specified, it will be set to 1.

• inverse_mass_matrix (numpy.ndarray or dict) – Initial value for inverse mass matrix. This may be adapted during warmup if adapt_mass_matrix = True. If no value is specified, then it is initialized to the identity matrix. For a potential_fn with general JAX pytree parameters, the order of entries of the mass matrix is the order of the flattened version of pytree parameters obtained with jax.tree_flatten, which is a bit ambiguous (see more at https://jax.readthedocs.io/en/latest/pytrees.html). If model is not None, here we can specify a structured block mass matrix as a dictionary, where keys are tuple of site names and values are the corresponding block of the mass matrix. For more information about structured mass matrix, see dense_mass argument.

• adapt_step_size (bool) – A flag to decide if we want to adapt step_size during warm-up phase using Dual Averaging scheme.

• adapt_mass_matrix (bool) – A flag to decide if we want to adapt mass matrix during warm-up phase using Welford scheme.

• dense_mass (bool or list) –

This flag controls whether mass matrix is dense (i.e. full-rank) or diagonal (defaults to dense_mass=False). To specify a structured mass matrix, users can provide a list of tuples of site names. Each tuple represents a block in the joint mass matrix. For example, assuming that the model has latent variables “x”, “y”, “z” (where each variable can be multi-dimensional), possible specifications and corresponding mass matrix structures are as follows:

• dense_mass=[(“x”, “y”)]: use a dense mass matrix for the joint (x, y) and a diagonal mass matrix for z

• dense_mass=[] (equivalent to dense_mass=False): use a diagonal mass matrix for the joint (x, y, z)

• dense_mass=[(“x”, “y”, “z”)] (equivalent to full_mass=True): use a dense mass matrix for the joint (x, y, z)

• dense_mass=[(“x”,), (“y”,), (“z”)]: use dense mass matrices for each of x, y, and z (i.e. block-diagonal with 3 blocks)

• target_accept_prob (float) – Target acceptance probability for step size adaptation using Dual Averaging. Increasing this value will lead to a smaller step size, hence the sampling will be slower but more robust. Defaults to 0.8.

• trajectory_length (float) – Length of a MCMC trajectory for HMC. Default value is $$2\pi$$.

• max_tree_depth (int) – Max depth of the binary tree created during the doubling scheme of NUTS sampler. Defaults to 10. This argument also accepts a tuple of integers (d1, d2), where d1 is the max tree depth during warmup phase and d2 is the max tree depth during post warmup phase.

• find_heuristic_step_size (bool) – whether to a heuristic function to adjust the step size at the beginning of each adaptation window. Defaults to False.

• regularize_mass_matrix (bool) – whether or not to regularize the estimated mass matrix for numerical stability during warmup phase. Defaults to True. This flag does not take effect if adapt_mass_matrix == False.

• model_args (tuple) – Model arguments if potential_fn_gen is specified.

• model_kwargs (dict) – Model keyword arguments if potential_fn_gen is specified.

• rng_key (jax.random.PRNGKey) – random key to be used as the source of randomness.

sample_kernel(hmc_state, model_args=(), model_kwargs=None)

Given an existing HMCState, run HMC with fixed (possibly adapted) step size and return a new HMCState.

Parameters
• hmc_state – Current sample (and associated state).

• model_args (tuple) – Model arguments if potential_fn_gen is specified.

• model_kwargs (dict) – Model keyword arguments if potential_fn_gen is specified.

Returns

new proposed HMCState from simulating Hamiltonian dynamics given existing state.

taylor_proxy(reference_params)[source]

Control variate for unbiased log likelihood estimation using a Taylor expansion around a reference parameter. Suggest for subsampling in [1].

Parameters

reference_params (dict) – Model parameterization at MLE or MAP-estimate.

References:

[1] Towards scaling up Markov chainMonte Carlo: an adaptive subsampling approach

Bardenet., R., Doucet, A., Holmes, C. (2014)

BarkerMHState = <class 'numpyro.infer.barker.BarkerMHState'>

A namedtuple() consisting of the following fields:

• i - iteration. This is reset to 0 after warmup.

• z - Python collection representing values (unconstrained samples from the posterior) at latent sites.

• potential_energy - Potential energy computed at the given value of z.

• accept_prob - Acceptance probability of the proposal. Note that z does not correspond to the proposal if it is rejected.

• mean_accept_prob - Mean acceptance probability until current iteration during warmup adaptation or sampling (for diagnostics).

• adapt_state - A HMCAdaptState namedtuple which contains adaptation information during warmup:

• step_size - Step size to be used by the integrator in the next iteration.

• inverse_mass_matrix - The inverse mass matrix to be used for the next iteration.

• mass_matrix_sqrt - The square root of mass matrix to be used for the next iteration. In case of dense mass, this is the Cholesky factorization of the mass matrix.

• rng_key - random number generator seed used for generating proposals, etc.

HMCState = <class 'numpyro.infer.hmc.HMCState'>

A namedtuple() consisting of the following fields:

• i - iteration. This is reset to 0 after warmup.

• z - Python collection representing values (unconstrained samples from the posterior) at latent sites.

• potential_energy - Potential energy computed at the given value of z.

• energy - Sum of potential energy and kinetic energy of the current state.

• r - The current momentum variable. If this is None, a new momentum variable will be drawn at the beginning of each sampling step.

• trajectory_length - The amount of time to run HMC dynamics in each sampling step. This field is not used in NUTS.

• num_steps - Number of steps in the Hamiltonian trajectory (for diagnostics). In NUTS sampler, the tree depth of a trajectory can be computed from this field with tree_depth = np.log2(num_steps).astype(int) + 1.

• accept_prob - Acceptance probability of the proposal. Note that z does not correspond to the proposal if it is rejected.

• mean_accept_prob - Mean acceptance probability until current iteration during warmup adaptation or sampling (for diagnostics).

• diverging - A boolean value to indicate whether the current trajectory is diverging.

• adapt_state - A HMCAdaptState namedtuple which contains adaptation information during warmup:

• step_size - Step size to be used by the integrator in the next iteration.

• inverse_mass_matrix - The inverse mass matrix to be used for the next iteration.

• mass_matrix_sqrt - The square root of mass matrix to be used for the next iteration. In case of dense mass, this is the Cholesky factorization of the mass matrix.

• rng_key - random number generator seed used for the iteration.

HMCGibbsState = <class 'numpyro.infer.hmc_gibbs.HMCGibbsState'>
• z - a dict of the current latent values (both HMC and Gibbs sites)

• hmc_state - current HMCState

• rng_key - random key for the current step

SAState = <class 'numpyro.infer.sa.SAState'>

A namedtuple() used in Sample Adaptive MCMC. This consists of the following fields:

• i - iteration. This is reset to 0 after warmup.

• z - Python collection representing values (unconstrained samples from the posterior) at latent sites.

• potential_energy - Potential energy computed at the given value of z.

• accept_prob - Acceptance probability of the proposal. Note that z does not correspond to the proposal if it is rejected.

• mean_accept_prob - Mean acceptance probability until current iteration during warmup or sampling (for diagnostics).

• diverging - A boolean value to indicate whether the new sample potential energy is diverging from the current one.

• adapt_state - A SAAdaptState namedtuple which contains adaptation information:

• zs - Step size to be used by the integrator in the next iteration.

• pes - Potential energies of zs.

• loc - Mean of those zs.

• inv_mass_matrix_sqrt - If using dense mass matrix, this is Cholesky of the covariance of zs. Otherwise, this is standard deviation of those zs.

• rng_key - random number generator seed used for the iteration.

## TensorFlow Kernels¶

Thin wrappers around TensorFlow Probability (TFP) distributions. For details on the TFP distribution interface, see its TransitionKernel docs.

## MCMC Utilities¶

initialize_model(rng_key, model, *, init_strategy=<function init_to_uniform>, dynamic_args=False, model_args=(), model_kwargs=None, forward_mode_differentiation=False, validate_grad=True)[source]

(EXPERIMENTAL INTERFACE) Helper function that calls get_potential_fn() and find_valid_initial_params() under the hood to return a tuple of (init_params_info, potential_fn, postprocess_fn, model_trace).

Parameters
• rng_key (jax.random.PRNGKey) – random number generator seed to sample from the prior. The returned init_params will have the batch shape rng_key.shape[:-1].

• model – Python callable containing Pyro primitives.

• init_strategy (callable) – a per-site initialization function. See Initialization Strategies section for available functions.

• dynamic_args (bool) – if True, the potential_fn and constraints_fn are themselves dependent on model arguments. When provided a *model_args, **model_kwargs, they return potential_fn and constraints_fn callables, respectively.

• model_args (tuple) – args provided to the model.

• model_kwargs (dict) – kwargs provided to the model.

• forward_mode_differentiation (bool) –

whether to use forward-mode differentiation or reverse-mode differentiation. By default, we use reverse mode but the forward mode can be useful in some cases to improve the performance. In addition, some control flow utility on JAX such as jax.lax.while_loop or jax.lax.fori_loop only supports forward-mode differentiation. See JAX’s The Autodiff Cookbook for more information.

• validate_grad (bool) – whether to validate gradient of the initial params. Defaults to True.

Returns

a namedtupe ModelInfo which contains the fields (param_info, potential_fn, postprocess_fn, model_trace), where param_info is a namedtuple ParamInfo containing values from the prior used to initiate MCMC, their corresponding potential energy, and their gradients; postprocess_fn is a callable that uses inverse transforms to convert unconstrained HMC samples to constrained values that lie within the site’s support, in addition to returning values at deterministic sites in the model.

fori_collect(lower, upper, body_fun, init_val, transform=<function identity>, progbar=True, return_last_val=False, collection_size=None, thinning=1, **progbar_opts)[source]

This looping construct works like fori_loop() but with the additional effect of collecting values from the loop body. In addition, this allows for post-processing of these samples via transform, and progress bar updates. Note that, progbar=False will be faster, especially when collecting a lot of samples. Refer to example usage in hmc().

Parameters
• lower (int) – the index to start the collective work. In other words, we will skip collecting the first lower values.

• upper (int) – number of times to run the loop body.

• body_fun – a callable that takes a collection of np.ndarray and returns a collection with the same shape and dtype.

• init_val – initial value to pass as argument to body_fun. Can be any Python collection type containing np.ndarray objects.

• transform – a callable to post-process the values returned by body_fn.

• progbar – whether to post progress bar updates.

• return_last_val (bool) – If True, the last value is also returned. This has the same type as init_val.

• thinning – Positive integer that controls the thinning ratio for retained values. Defaults to 1, i.e. no thinning.

• collection_size (int) – Size of the returned collection. If not specified, the size will be (upper - lower) // thinning. If the size is larger than (upper - lower) // thinning, only the top (upper - lower) // thinning entries will be non-zero.

• **progbar_opts – optional additional progress bar arguments. A diagnostics_fn can be supplied which when passed the current value from body_fun returns a string that is used to update the progress bar postfix. Also a progbar_desc keyword argument can be supplied which is used to label the progress bar.

Returns

collection with the same type as init_val with values collected along the leading axis of np.ndarray objects.

consensus(subposteriors, num_draws=None, diagonal=False, rng_key=None)[source]

Merges subposteriors following consensus Monte Carlo algorithm.

References:

1. Bayes and big data: The consensus Monte Carlo algorithm, Steven L. Scott, Alexander W. Blocker, Fernando V. Bonassi, Hugh A. Chipman, Edward I. George, Robert E. McCulloch

Parameters
• subposteriors (list) – a list in which each element is a collection of samples.

• num_draws (int) – number of draws from the merged posterior.

• diagonal (bool) – whether to compute weights using variance or covariance, defaults to False (using covariance).

• rng_key (jax.random.PRNGKey) – source of the randomness, defaults to jax.random.PRNGKey(0).

Returns

if num_draws is None, merges subposteriors without resampling; otherwise, returns a collection of num_draws samples with the same data structure as each subposterior.

parametric(subposteriors, diagonal=False)[source]

Merges subposteriors following (embarrassingly parallel) parametric Monte Carlo algorithm.

References:

1. Asymptotically Exact, Embarrassingly Parallel MCMC, Willie Neiswanger, Chong Wang, Eric Xing

Parameters
• subposteriors (list) – a list in which each element is a collection of samples.

• diagonal (bool) – whether to compute weights using variance or covariance, defaults to False (using covariance).

Returns

the estimated mean and variance/covariance parameters of the joined posterior

parametric_draws(subposteriors, num_draws, diagonal=False, rng_key=None)[source]

Merges subposteriors following (embarrassingly parallel) parametric Monte Carlo algorithm.

References:

1. Asymptotically Exact, Embarrassingly Parallel MCMC, Willie Neiswanger, Chong Wang, Eric Xing

Parameters
• subposteriors (list) – a list in which each element is a collection of samples.

• num_draws (int) – number of draws from the merged posterior.

• diagonal (bool) – whether to compute weights using variance or covariance, defaults to False (using covariance).

• rng_key (jax.random.PRNGKey) – source of the randomness, defaults to jax.random.PRNGKey(0).

Returns

a collection of num_draws samples with the same data structure as each subposterior.