tfp.mcmc.ReplicaExchangeMC

Runs one step of the Replica Exchange Monte Carlo.

Inherits From: TransitionKernel

tfp.mcmc.ReplicaExchangeMC(
    target_log_prob_fn,
    inverse_temperatures,
    make_kernel_fn,
    swap_proposal_fn=default_swap_proposal_fn(1.0),
    state_includes_replicas=False,
    untempered_log_prob_fn=None,
    tempered_log_prob_fn=None,
    validate_args=False,
    name=None
)

Replica Exchange Monte Carlo is a Markov chain Monte Carlo (MCMC) algorithm that is also known as Parallel Tempering. This algorithm takes multiple samples (from tempered distributions) in parallel, then swaps these samples according to the Metropolis-Hastings criterion. See also the review paper [1].

The K replicas are parameterized in terms of inverse_temperature's, (beta[0], beta[1], ..., beta[K-1]). If the user provides target_log_prob_fn, then the kth replica samples from density p_k(x), with log(p_k(x)) = beta_k * target_log_prob(x). In this case, geometrically decaying beta often works well. That is, with R < 1, we recommend trying beta[k] = R^k so that 1.0 = beta[0] > beta[1] > ... > 0. See [2].

The user can also provide two functions, tempered_log_prob_fn and untempered_log_prob_fn. In this case, the kth replica samples from density p_k(x) with log(p_k(x)) = beta_k * tempered_log_prob_fn(x) + untempered_log_prob_fn(x). In this case, beta may be zero, and one often sets beta[-1] to zero. This means the last replica samples using untempered_log_prob_fn. In the Bayesian setup, untempered_log_prob_fn will often be the log prior, and tempered_log_prob_fn the likelihood.

In all cases,

beta[0] == 1 ==> First replica samples from the target density.
beta[k] < 1, for k = 1, ..., K-1 ==> Other replicas sample from "tempered" versions of target (peak is less high, valley less low). These distributions should allow easier exploration of separated modes.

By default, samples from adjacent replicas i, i + 1 are used as proposals for each other in a Metropolis step. This allows the lower beta samples, which explore less dense areas of p, to eventually swap state with the beta == 1 chain, allowing it to explore these new regions.

Samples from replica 0 are returned, and the others are discarded, unless state_includes_replicas.

Examples

Sampling from the Standard Normal Distribution.

import numpy as np
import tensorflow as tf
import tensorflow_probability as tfp
tfd = tfp.distributions

dtype = np.float32

target = tfd.Normal(loc=dtype(0), scale=dtype(1))

# Geometric decay is a good rule of thumb.
inverse_temperatures = 0.5**tf.range(4, dtype=dtype)

# If everything was Normal, step_size should be ~ sqrt(temperature).
step_size = 1.5 / tf.sqrt(inverse_temperatures)

def make_kernel_fn(target_log_prob_fn):
  return tfp.mcmc.HamiltonianMonteCarlo(
      target_log_prob_fn=target_log_prob_fn,
      step_size=step_size, num_leapfrog_steps=3)

remc = tfp.mcmc.ReplicaExchangeMC(
    target_log_prob_fn=target.log_prob,
    inverse_temperatures=inverse_temperatures,
    make_kernel_fn=make_kernel_fn)

def trace_swaps(unused_state, results):
  return (results.is_swap_proposed_adjacent,
          results.is_swap_accepted_adjacent)

samples, (is_swap_proposed_adjacent, is_swap_accepted_adjacent) = (
    tfp.mcmc.sample_chain(
        num_results=1000,
        current_state=1.0,
        kernel=remc,
        num_burnin_steps=500,
        trace_fn=trace_swaps)
)

# conditional_swap_prob[k] = P[ExchangeAccepted | ExchangeProposed],
# for the swap between replicas k and k+1.
conditional_swap_prob = (
    tf.reduce_sum(tf.cast(is_swap_accepted_adjacent, tf.float32), axis=0)
    /
    tf.reduce_sum(tf.cast(is_swap_proposed_adjacent, tf.float32), axis=0))

Sampling from a 2-D Mixture Normal Distribution.

import numpy as np
import tensorflow as tf
import tensorflow_probability as tfp
import matplotlib.pyplot as plt
tfd = tfp.distributions

dtype = np.float32

target = tfd.MixtureSameFamily(
    mixture_distribution=tfd.Categorical(probs=[0.5, 0.5]),
    components_distribution=tfd.MultivariateNormalDiag(
        loc=[[-1., -1], [1., 1.]],
        scale_diag=0.1*tf.ones([2, 2])))

inverse_temperatures = 0.2**tf.range(4, dtype=dtype)

# step_size must broadcast with all batch and event dimensions of target.
# Here, this means it must broadcast with:
#  [len(inverse_temperatures)] + target.event_shape
step_size = 0.075 / tf.reshape(tf.sqrt(inverse_temperatures), shape=(4, 1))

def make_kernel_fn(target_log_prob_fn):
  return tfp.mcmc.HamiltonianMonteCarlo(
      target_log_prob_fn=target_log_prob_fn,
      step_size=step_size, num_leapfrog_steps=3)

remc = tfp.mcmc.ReplicaExchangeMC(
    target_log_prob_fn=target.log_prob,
    inverse_temperatures=inverse_temperatures,
    make_kernel_fn=make_kernel_fn)

samples = tfp.mcmc.sample_chain(
    num_results=1000,
    # Start near the [1, 1] mode. Standard HMC would get stuck there.
    current_state=tf.ones(2, dtype=dtype),
    kernel=remc,
    trace_fn=None,
    num_burnin_steps=500)

plt.figure(figsize=(8, 8))
plt.xlim(-2, 2)
plt.ylim(-2, 2)
plt.plot(samples[:, 0], samples[:, 1], '.')
plt.show()

References

[1]: David J. Earl, Michael W. Deem Parallel Tempering: Theory, Applications, and New Perspectives https://arxiv.org/abs/physics/0508111 [2]: David A. Kofke On the acceptance probability of replica-exchange Monte Carlo trials. J. of Chem. Phys. Vol. 117 No. 5.

Args
`target_log_prob_fn`	Python callable which takes an argument like `current_state` (or `*current_state` if it's a list) and returns its (possibly unnormalized) log-density under the target distribution. Must be `None` if the pair `tempered/untempered_log_prob_fn` is provided
`inverse_temperatures`	`Tensor` of inverse temperatures to temper each replica. The leftmost dimension is the `num_replica` and the second dimension through the rightmost can provide different temperature to different batch members, doing a left-justified broadcast.
`make_kernel_fn`	Python callable which takes a `target_log_prob_fn` arg and returns a `tfp.mcmc.TransitionKernel` instance.
`swap_proposal_fn`	Python callable which take a number of replicas, and returns `swaps`, a shape `[num_replica] + batch_shape` `Tensor`, where axis 0 indexes a permutation of `{0,..., num_replica-1}`, designating replicas to swap.
`state_includes_replicas`	Boolean indicating whether the leftmost dimension of each state sample should index replicas. If `True`, the leftmost dimension of the `current_state` kwarg to `tfp.mcmc.sample_chain` will be interpreted as indexing replicas.
`untempered_log_prob_fn`	Python callable which takes an argument like `current_state` (or `*current_state` if it's a list) and returns its (possibly unnormalized) log-density under the target distribution. Must be `None` if `target_log_prob_fn` is provided.
`tempered_log_prob_fn`	Optional Python callable with same signature as `untempered_log_prob_fn`. Provide this arg if and only if `untempered_log_prob_fn` is provided.
`validate_args`	Python `bool`, default `False`. When `True` distribution parameters are checked for validity despite possibly degrading runtime performance. When `False` invalid inputs may silently render incorrect outputs.
`name`	Python `str` name prefixed to Ops created by this function. Default value: `None` (i.e., "remc_kernel").

Raises
`ValueError`	`inverse_temperatures` doesn't have statically known 1D shape.
`ValueError`	If wrong combination of log prob functions are provided.

Attributes
`experimental_shard_axis_names`	The shard axis names for members of the state.
`inverse_temperatures`
`is_calibrated`	Returns `True` if Markov chain converges to specified distribution. `TransitionKernel`s which are "uncalibrated" are often calibrated by composing them with the `tfp.mcmc.MetropolisHastings` `TransitionKernel`.
`make_kernel_fn`
`name`
`parameters`	Return `dict` of `__init__` arguments and their values.
`swap_proposal_fn`
`target_log_prob_fn`
`tempered_log_prob_fn`
`untempered_log_prob_fn`
`validate_args`

Methods

`bootstrap_results`

View source

bootstrap_results(
    init_state
)

Returns an object with the same type as returned by one_step.

Args
`init_state`	`Tensor` or Python `list` of `Tensor`s representing the initial state(s) of the Markov chain(s).

Returns
`kernel_results`	A (possibly nested) `tuple`, `namedtuple` or `list` of `Tensor`s representing internal calculations made within this function. This inculdes replica states.

`copy`

View source

copy(
    **override_parameter_kwargs
)

Non-destructively creates a deep copy of the kernel.

Args
`**override_parameter_kwargs`	Python String/value `dictionary` of initialization arguments to override with new values.

Returns
`new_kernel`	`TransitionKernel` object of same type as `self`, initialized with the union of self.parameters and override_parameter_kwargs, with any shared keys overridden by the value of override_parameter_kwargs, i.e., `dict(self.parameters, **override_parameters_kwargs)`.

`experimental_with_shard_axes`

View source

experimental_with_shard_axes(
    shard_axes
)

Returns a copy of the kernel with the provided shard axis names.

Args
`shard_axis_names`	a structure of strings indicating the shard axis names for each component of this kernel's state.

Returns
A copy of the current kernel with the shard axis information.

`num_replica`

View source

num_replica()

Integer (Tensor) number of replicas being tracked.

`one_step`

View source

one_step(
    current_state, previous_kernel_results, seed=None
)

Takes one step of the TransitionKernel.

Args
`current_state`	`Tensor` or Python `list` of `Tensor`s representing the current state(s) of the Markov chain(s).
`previous_kernel_results`	A (possibly nested) `tuple`, `namedtuple` or `list` of `Tensor`s representing internal calculations made within the previous call to this function (or as returned by `bootstrap_results`).
`seed`	PRNG seed; see `tfp.random.sanitize_seed` for details.

Returns
`next_state`	`Tensor` or Python `list` of `Tensor`s representing the next state(s) of the Markov chain(s).
`kernel_results`	A (possibly nested) `tuple`, `namedtuple` or `list` of `Tensor`s representing internal calculations made within this function. This inculdes replica states.

tfp.mcmc.ReplicaExchangeMC

Examples

Sampling from the Standard Normal Distribution.

Sampling from a 2-D Mixture Normal Distribution.

References

Args

Raises

Attributes

Methods

bootstrap_results

copy

experimental_with_shard_axes

num_replica

one_step

`bootstrap_results`

`copy`

`experimental_with_shard_axes`

`num_replica`

`one_step`