Implements Markov chain Monte Carlo via repeated TransitionKernel steps.

Used in the notebooks

Used in the tutorials

This function samples from an Markov chain at current_state and whose stationary distribution is governed by the supplied TransitionKernel instance (kernel).

The current_state can be represented as a single Tensor or a list of Tensors which collectively represent the current state.

This function can sample from multiple chains, in parallel. Whether or not there are multiple chains is dictated by how the kernel treats its inputs. Typically, the shape of the independent chains is shape of the result of the target_log_prob_fn used by the kernel when applied to the given current_state.

Since MCMC states are correlated, it is sometimes desirable to produce additional intermediate states, and then discard them, ending up with a set of states with decreased autocorrelation. See [Owen (2017)][1]. Such 'thinning' is made possible by setting num_steps_between_results > 0. The chain then takes num_steps_between_results extra steps between the steps that make it into the results. The extra steps are never materialized, and thus do not increase memory requirements.

In addition to returning the chain state, this function supports tracing of auxiliary variables used by the kernel. The traced values are selected by specifying trace_fn. By default, all kernel results are traced but in the future the default will be changed to no results being traced, so plan accordingly. See below for some examples of this feature.

num_results Integer number of Markov chain draws.
current_state Tensor or Python list of Tensors representing the current state(s) of the Markov chain(s).
previous_kernel_results A Tensor or a nested collection of Tensors representing internal calculations made within the previous call to this function (or as returned by bootstrap_results).
kernel An instance of tfp.mcmc.TransitionKernel which implements one step of the Markov chain.
num_burnin_steps Integer number of chain steps to take before starting to collect results. Default value: 0 (i.e., no burn-in).
num_steps_between_results Integer number of chain steps between collecting a result. Only one out of every num_steps_between_samples + 1 steps is included in the returned results. The number of returned chain states is still equal to num_results. Default value: 0 (i.e., no thinning).
trace_fn A callable that takes in the current chain state and the previous kernel results and return a Tensor or a nested collection of Tensors that is then traced along with the chain state.
return_final_kernel_results If True, then the final kernel results are returned alongside the chain state and the trace specified by the trace_fn.
parallel_iterations The number of iterations allowed to run in parallel. It must be a positive integer. See tf.while_loop for more details.
seed PRNG seed; see tfp.random.sanitize_seed for details.
name Python str name prefixed to Ops created by this function. Default value: None (i.e., 'mcmc_sample_chain').

checkpointable_states_and_trace if return_final_kernel_results is True. The return value is an instance of CheckpointableStatesAndTrace.
all_states if return_final_kernel_results is False and trace_fn is None. The return value is a Tensor or Python list of Tensors representing the state(s) of the Markov chain(s) at each result step. Has same shape as input current_state but with a prepended num_results-size dimension.
states_and_trace if return_final_kernel_results is False and trace_fn is not None. The return value is an instance of StatesAndTrace.


Sample from a diagonal-variance Gaussian.


for i=1..n:
  x[i] ~ MultivariateNormal(loc=0, scale=diag(true_stddev))  # likelihood
import tensorflow as tf
import tensorflow_probability as tfp
tfd = tfp.distributions

dims = 10
true_stddev = tf.sqrt(tf.linspace(1., 3., dims))
likelihood = tfd.MultivariateNormalDiag(loc=0., scale_diag=true_stddev)

states = tfp.mcmc.sample_chain(

sample_mean = tf.reduce_mean(states, axis=0)
# ==> approx all zeros

sample_stddev = tf.sqrt(tf.reduce_mean(
    tf.squared_difference(states, sample_mean),
# ==> approx equal true_stddev
Sampling from factor-analysis posteriors with known factors.


# prior
w ~ MultivariateNormal(loc=0, scale=eye(d))
for i=1..n:
  # likelihood
  x[i] ~ Normal(loc=w^T F[i], scale=1)

where F denotes factors.

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

# Specify model.
def make_prior(dims):
  return tfd.MultivariateNormalDiag(

def make_likelihood(weights, factors):
  return tfd.MultivariateNormalDiag(
      loc=tf.matmul(weights, factors, adjoint_b=True))

def joint_log_prob(num_weights, factors, x, w):
  return (make_prior(num_weights).log_prob(w) +
          make_likelihood(w, factors).log_prob(x))

def unnormalized_log_posterior(w):
  # Posterior is proportional to: `p(W, X=x | factors)`.
  return joint_log_prob(num_weights, factors, x, w)

# Setup data.
num_weights = 10 # == d
num_factors = 40 # == n
num_chains = 100

weights = make_prior(num_weights).sample(1)
factors = tf.random.normal([num_factors, num_weights])
x = make_likelihood(weights, factors).sample()

# Sample from Hamiltonian Monte Carlo Markov Chain.

# Get `num_results` samples from `num_chains` independent chains.
chains_states, kernels_results = tfp.mcmc.sample_chain(
    current_state=tf.zeros([num_chains, num_weights], name='init_weights'),

# Compute sample stats.
sample_mean = tf.reduce_mean(chains_states, axis=[0, 1])
# ==> approx equal to weights

sample_var = tf.reduce_mean(
    tf.squared_difference(chains_states, sample_mean),
    axis=[0, 1])
# ==> less than 1
Custom tracing functions.
import tensorflow as tf
import tensorflow_probability as tfp
tfd = tfp.distributions

likelihood = tfd.Normal(loc=0., scale=1.)

def sample_chain(trace_fn):
  return tfp.mcmc.sample_chain(

def trace_log_accept_ratio(states, previous_kernel_results):
  return previous_kernel_results.log_accept_ratio

def trace_everything(states, previous_kernel_results):
  return previous_kernel_results

_, log_accept_ratio = sample_chain(trace_fn=trace_log_accept_ratio)
_, kernel_results = sample_chain(trace_fn=trace_everything)

acceptance_prob = tf.math.exp(tf.minimum(log_accept_ratio, 0.))
# Equivalent to, but more efficient than:
acceptance_prob = tf.math.exp(tf.minimum(
    kernel_results.log_accept_ratio, 0.))


[1]: Art B. Owen. Statistically efficient thinning of a Markov chain sampler. Technical Report, 2017.