tfp.substrates.jax.mcmc.SliceSampler

Runs one step of the slice sampler using a hit and run approach.

Inherits From: TransitionKernel

Slice Sampling is a Markov Chain Monte Carlo (MCMC) algorithm based, as stated by [Neal (2003)][1], on the observation that "...one can sample from a distribution by sampling uniformly from the region under the plot of its density function. A Markov chain that converges to this uniform distribution can be constructed by alternately uniform sampling in the vertical direction with uniform sampling from the horizontal slice defined by the current vertical position, or more generally, with some update that leaves the uniform distribution over this slice invariant". Mathematical details and derivations can be found in [Neal (2003)][1]. The one dimensional slice sampler is extended to n-dimensions through use of a hit-and-run approach: choose a random direction in n-dimensional space and take a step, as determined by the one-dimensional slice sampling algorithm, along that direction [Belisle at al. 1993][2].

The one_step function 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)).)

Note that the sampler only supports states where all components have a common dtype.

Examples:

Simple chain with warm-up.

In this example we sample from a standard univariate normal distribution using slice sampling.

from tensorflow_probability.python.internal.backend.jax.compat import v2 as tf
import tensorflow_probability as tfp; tfp = tfp.substrates.jax
import numpy as np

tf.enable_v2_behavior()

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.SliceSampler(
        target.log_prob,
        step_size=1.0,
        max_doublings=5),
    num_burnin_steps=500,
    trace_fn=None,
    seed=1234)

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

print('Sample mean: ', sample_mean.numpy())
print('Sample Std: ', sample_std.numpy())

Sample from a Two Dimensional Normal.

In the following example we sample from a two dimensional Normal distribution using slice sampling.

from tensorflow_probability.python.internal.backend.jax.compat import v2 as tf
import tensorflow_probability as tfp; tfp = tfp.substrates.jax
import numpy as np

tf.enable_v2_behavior()

dtype = np.float32
true_mean = dtype([0, 0])
true_cov = dtype([[1, 0.5], [0.5, 1]])
num_results = 500
num_chains = 50

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

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

# Run Slice Samper for `num_results` iterations for `num_chains`
# independent chains:
@tf.function
def run_mcmc():
  states = tfp.mcmc.sample_chain(
      num_results=num_results,
      current_state=init_state,
      kernel=tfp.mcmc.SliceSampler(
          target_log_prob_fn=target.log_prob,
          step_size=1.0,
          max_doublings=5),
      num_burnin_steps=200,
      num_steps_between_results=1,
      trace_fn=None,
      seed=47)
  return states

states = run_mcmc()

sample_mean = tf.reduce_mean(states, axis=[0, 1])
z = (states - sample_mean)[..., tf.newaxis]
sample_cov = tf.reduce_mean(
    tf.matmul(z, tf.transpose(z, [0, 1, 3, 2])), [0, 1])

print('sample mean', sample_mean.numpy())
print('sample covariance matrix', sample_cov.numpy())

References

[1]: Radford M. Neal. Slice Sampling. The Annals of Statistics. 2003, Vol 31, No. 3 , 705-767. https://projecteuclid.org/download/pdf_1/euclid.aos/1056562461

[2]: C.J.P. Belisle, H.E. Romeijn, R.L. Smith. Hit-and-run algorithms for generating multivariate distributions. Math. Oper. Res., 18(1993), 225-266. https://www.jstor.org/stable/3690278?seq=1#page_scan_tab_contents

target_log_prob_fn Python callable which takes an argument like current_state (or *current_state if it is a list) and returns its (possibly unnormalized) log-density under the target distribution.
step_size Scalar or tf.Tensor with same dtype as and shape compatible with x_initial. The size of the initial interval.
max_doublings Scalar positive int32 tf.Tensor. The maximum number of doublings to consider.
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., 'slice_sampler_kernel').

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.

max_doublings

name

parameters Returns dict of __init__ arguments and their values.
seed

step_size

target_log_prob_fn

Methods

bootstrap_results

View source

Returns an object with the same type as returned by one_step(...)[1].

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

Returns
kernel_results A (possibly nested) tuple, namedtuple or list of Tensors representing internal calculations made within this function.

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 Slice Sampler.

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 step_size or a list with same length as current_state.
TypeError if not target_log_prob.dtype.is_floating.