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Implements Markov chain Monte Carlo via repeated
tfp.experimental.substrates.numpy.mcmc.sample_chain( num_results, current_state, previous_kernel_results=None, kernel=None, num_burnin_steps=0, num_steps_between_results=0, trace_fn=(lambda current_state, kernel_results: kernel_results), return_final_kernel_results=False, parallel_iterations=10, name=None )
This function samples from an Markov chain at
current_state and whose
stationary distribution is governed by the supplied
This function can sample from multiple chains, in parallel. (Whether or not
there are multiple chains is dictated by the
current_state can be represented as a single
Tensor or a
Tensors which collectively represent the 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)]. Such "thinning"
is made possible by setting
num_steps_between_results > 0. The chain then
num_steps_between_results extra steps between the steps that make it
into the results. The extra steps are never materialized (in calls to
sess.run), 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
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.
Tensors representing the current state(s) of the Markov chain(s).
Tensoror a nested collection of
Tensors representing internal calculations made within the previous call to this function (or as returned by
kernel: An instance of
tfp.mcmc.TransitionKernelwhich 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 + 1steps 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
Tensoror a nested collection of
Tensors that is then traced along with the chain state.
True, then the final kernel results are returned alongside the chain state and the trace specified by the
parallel_iterations: The number of iterations allowed to run in parallel. It must be a positive integer. See
tf.while_loopfor more details.
strname prefixed to Ops created by this function. Default value:
True. The return value is an instance of
None. The return value is a
Tensoror Python list of
Tensors representing the state(s) of the Markov chain(s) at each result step. Has same shape as input
current_statebut with a prepended
None. The return value is an instance of
Sample from a diagonal-variance Gaussian.
for i=1..n: x[i] ~ MultivariateNormal(loc=0, scale=diag(true_stddev)) # likelihood
from tensorflow_probability.python.internal.backend import numpy as tf import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.numpy tfd = tfp.distributions dims = 10 true_stddev = np.sqrt(np.linspace(1., 3., dims)) likelihood = tfd.MultivariateNormalDiag(loc=0., scale_diag=true_stddev) states = tfp.mcmc.sample_chain( num_results=1000, num_burnin_steps=500, current_state=tf.zeros(dims), kernel=tfp.mcmc.HamiltonianMonteCarlo( target_log_prob_fn=likelihood.log_prob, step_size=0.5, num_leapfrog_steps=2), trace_fn=None) sample_mean = tf.reduce_mean(states, axis=0) # ==> approx all zeros sample_stddev = tf.sqrt(tf.reduce_mean( tf.squared_difference(states, sample_mean), axis=0)) # ==> 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)
F denotes factors.
from tensorflow_probability.python.internal.backend import numpy as tf import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.numpy tfd = tfp.distributions # Specify model. def make_prior(dims): return tfd.MultivariateNormalDiag( loc=tf.zeros(dims)) 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( num_results=1000, num_burnin_steps=500, current_state=tf.zeros([num_chains, num_weights], name='init_weights'), kernel=tfp.mcmc.HamiltonianMonteCarlo( target_log_prob_fn=unnormalized_log_posterior, step_size=0.1, num_leapfrog_steps=2)) # 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.
from tensorflow_probability.python.internal.backend import numpy as tf import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.numpy tfd = tfp.distributions likelihood = tfd.Normal(loc=0., scale=1.) def sample_chain(trace_fn): return tfp.mcmc.sample_chain( num_results=1000, num_burnin_steps=500, current_state=0., kernel=tfp.mcmc.HamiltonianMonteCarlo( target_log_prob_fn=likelihood.log_prob, step_size=0.5, num_leapfrog_steps=2), trace_fn=trace_fn) 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.))
: Art B. Owen. Statistically efficient thinning of a Markov chain sampler. Technical Report, 2017. http://statweb.stanford.edu/~owen/reports/bestthinning.pdf