tfp.substrates.numpy.mcmc.RandomWalkMetropolis

Runs one step of the RWM algorithm with symmetric proposal.

Inherits From: TransitionKernel

Random Walk Metropolis is a gradient-free Markov chain Monte Carlo (MCMC) algorithm. The algorithm involves a proposal generating step proposal_state = current_state + perturb by a random perturbation, followed by Metropolis-Hastings accept/reject step. For more details see Section 2.1 of Roberts and Rosenthal (2004).

Current class implements RWM for normal and uniform proposals. Alternatively, the user can supply any custom proposal generating function.

The function one_step can update multiple chains in parallel. It assumes that all leftmost dimensions of current_state index independent chain states (and are therefore updated independently). The output of target_log_prob_fn(*current_state) should sum log-probabilities across all event dimensions. Slices along the rightmost dimensions may have different target distributions; for example, current_state[0, :] could have a different target distribution from current_state[1, :]. These semantics are governed by target_log_prob_fn(*current_state). (The number of independent chains is tf.size(target_log_prob_fn(*current_state)).)

Examples:

Sampling from the Standard Normal Distribution.
import numpy as np
from tensorflow_probability.python.internal.backend.numpy.compat import v2 as tf
import tensorflow_probability as tfp; tfp = tfp.substrates.numpy
tf.enable_v2_behavior()

tfd = tfp.distributions

dtype = np.float32

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

samples = tfp.mcmc.sample_chain(
  num_results=1000,
  current_state=dtype(1),
  kernel=tfp.mcmc.RandomWalkMetropolis(target.log_prob),
  num_burnin_steps=500,
  trace_fn=None,
  seed=42)

sample_mean = tf.math.reduce_mean(samples, axis=0)
sample_std = tf.sqrt(
    tf.math.reduce_mean(
        tf.math.squared_difference(samples, sample_mean),
        axis=0))

print('Estimated mean: {}'.format(sample_mean))
print('Estimated standard deviation: {}'.format(sample_std))
Sampling from a 2-D Normal Distribution.
import numpy as np
from tensorflow_probability.python.internal.backend.numpy.compat import v2 as tf
import tensorflow_probability as tfp; tfp = tfp.substrates.numpy
tf.enable_v2_behavior()

tfd = tfp.distributions

dtype = np.float32
true_mean = dtype([0, 0])
true_cov = dtype([[1, 0.5],
                  [0.5, 1]])
num_results = 500
num_chains = 100

# Target distribution is defined through the Cholesky decomposition `L`:
L = tf.linalg.cholesky(true_cov)
target = tfd.MultivariateNormalTriL(loc=true_mean, scale_tril=L)

# Initial state of the chain
init_state = np.ones([num_chains, 2], dtype=dtype)

# Run Random Walk Metropolis with normal proposal for `num_results`
# iterations for `num_chains` independent chains:
samples = tfp.mcmc.sample_chain(
    num_results=num_results,
    current_state=init_state,
    kernel=tfp.mcmc.RandomWalkMetropolis(target_log_prob_fn=target.log_prob),
    num_burnin_steps=200,
    num_steps_between_results=1,  # Thinning.
    trace_fn=None,
    seed=54)

sample_mean = tf.math.reduce_mean(samples, axis=0)
x = tf.squeeze(samples - sample_mean)
sample_cov = tf.matmul(tf.transpose(x, [1, 2, 0]),
                       tf.transpose(x, [1, 0, 2])) / num_results

mean_sample_mean = tf.math.reduce_mean(sample_mean)
mean_sample_cov = tf.math.reduce_mean(sample_cov, axis=0)
x = tf.reshape(sample_cov - mean_sample_cov, [num_chains, 2 * 2])
cov_sample_cov = tf.reshape(tf.matmul(x, x, transpose_a=True) / num_chains,
                            shape=[2 * 2, 2 * 2])

print('Estimated mean: {}'.format(mean_sample_mean))
print('Estimated avg covariance: {}'.format(mean_sample_cov))
print('Estimated covariance of covariance: {}'.format(cov_sample_cov))
Sampling from the Standard Normal Distribution using Cauchy proposal.
import numpy as np
from tensorflow_probability.python.internal.backend.numpy.compat import v2 as tf
import tensorflow_probability as tfp; tfp = tfp.substrates.numpy
tf.enable_v2_behavior()

tfd = tfp.distributions

dtype = np.float32
num_burnin_steps = 500
num_chain_results = 1000

def cauchy_new_state_fn(scale, dtype):
  cauchy = tfd.Cauchy(loc=dtype(0), scale=dtype(scale))
  def _fn(state_parts, seed):
    next_state_parts = []
    part_seeds = tfp.random.split_seed(
        seed, n=len(state_parts), salt='rwmcauchy')
    for sp, ps in zip(state_parts, part_seeds):
      next_state_parts.append(sp + cauchy.sample(
        sample_shape=sp.shape, seed=ps))
    return next_state_parts
  return _fn

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

samples = tfp.mcmc.sample_chain(
    num_results=num_chain_results,
    num_burnin_steps=num_burnin_steps,
    current_state=dtype(1),
    kernel=tfp.mcmc.RandomWalkMetropolis(
        target.log_prob,
        new_state_fn=cauchy_new_state_fn(scale=0.5, dtype=dtype)),
    trace_fn=None,
    seed=42)

sample_mean = tf.math.reduce_mean(samples, axis=0)
sample_std = tf.sqrt(
    tf.math.reduce_mean(
        tf.math.squared_difference(samples, sample_mean),
        axis=0))

print('Estimated mean: {}'.format(sample_mean))
print('Estimated standard deviation: {}'.format(sample_std))

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.
new_state_fn Python callable which takes a list of state parts and a seed; returns a same-type list of Tensors, each being a perturbation of the input state parts. The perturbation distribution is assumed to be a symmetric distribution centered at the input state part. Default value: None which is mapped to tfp.mcmc.random_walk_normal_fn().
seed Python integer to seed the random number generator. Deprecated, pass seed to tfp.mcmc.sample_chain.
name Python str name prefixed to Ops created by this function. Default value: None (i.e., 'rwm_kernel').

ValueError if there isn't one scale or a list with same length as current_state.

is_calibrated Returns True if Markov chain converges to specified distribution.

TransitionKernels which are "uncalibrated" are often calibrated by composing them with the tfp.mcmc.MetropolisHastings TransitionKernel.

name

new_state_fn

parameters Return dict of __init__ arguments and their values.
seed

target_log_prob_fn

Methods

bootstrap_results

View source

Creates initial previous_kernel_results using a supplied state.

copy

View source

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).

one_step

View source

Runs one iteration of Random Walk Metropolis with normal proposal.

Args
current_state Tensor or Python list of Tensors representing the current state(s) of the Markov chain(s). The first r dimensions index independent chains, r = tf.rank(target_log_prob_fn(*current_state)).
previous_kernel_results collections.namedtuple containing Tensors representing values from previous calls to this function (or from the bootstrap_results function.)
seed Optional, a seed for reproducible sampling.

Returns
next_state Tensor or Python list of Tensors representing the state(s) of the Markov chain(s) after taking exactly one step. Has same type and shape as current_state.
kernel_results collections.namedtuple of internal calculations used to advance the chain.

Raises
ValueError if there isn't one scale or a list with same length as current_state.