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Runs one step of the slice sampler using a hit and run approach.
Inherits From: TransitionKernel
tfp.experimental.substrates.numpy.mcmc.SliceSampler(
target_log_prob_fn, step_size, max_doublings, seed=None, name=None
)
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 ndimensions through use of a hitandrun approach: choose a
random direction in ndimensional space and take a step, as determined by the
onedimensional 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 logprobabilities 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 warmup.
In this example we sample from a standard univariate normal distribution using slice sampling.
from tensorflow_probability.python.internal.backend.numpy.compat import v2 as tf
import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.numpy
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.numpy.compat import v2 as tf
import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.numpy
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 , 705767. https://projecteuclid.org/download/pdf_1/euclid.aos/1056562461
[2]: C.J.P. Belisle, H.E. Romeijn, R.L. Smith. Hitandrun algorithms for generating multivariate distributions. Math. Oper. Res., 18(1993), 225266. https://www.jstor.org/stable/3690278?seq=1#page_scan_tab_contents
Args  

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) logdensity 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').

Attributes  

is_calibrated

Returns True if Markov chain converges to specified distribution.

max_doublings


name


parameters

Returns dict of __init__ arguments and their values.

seed


step_size


target_log_prob_fn

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

copy
copy(
**override_parameter_kwargs
)
Nondestructively 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
one_step(
current_state, previous_kernel_results, seed=None
)
Runs one iteration of Slice Sampler.
Args  

current_state

Tensor or Python list of Tensor s 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 Tensor s
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 Tensor s 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 .
