# Copyright 2020- The Blackjax Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
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# http://www.apache.org/licenses/LICENSE-2.0
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"""Public API for Barker's proposal with a Gaussian base kernel."""
from typing import Callable, NamedTuple
import jax
import jax.numpy as jnp
from jax.flatten_util import ravel_pytree
from jax.scipy import stats
from jax.tree_util import tree_leaves, tree_map
from blackjax.base import SamplingAlgorithm
from blackjax.mcmc.proposal import static_binomial_sampling
from blackjax.types import ArrayLikeTree, ArrayTree, PRNGKey
__all__ = ["BarkerState", "BarkerInfo", "init", "build_kernel", "as_top_level_api"]
[docs]
class BarkerState(NamedTuple):
"""State of the Barker's proposal algorithm.
The Barker algorithm takes one position of the chain and returns another
position. In order to make computations more efficient, we also store
the current log-probability density as well as the current gradient of the
log-probability density.
"""
[docs]
logdensity_grad: ArrayTree
[docs]
class BarkerInfo(NamedTuple):
"""Additional information on the Barker's proposal kernel transition.
This additional information can be used for debugging or computing
diagnostics.
proposal
The proposal that was sampled.
acceptance_rate
The acceptance rate of the transition.
is_accepted
Whether the proposed position was accepted or the original position
was returned.
"""
[docs]
def init(position: ArrayLikeTree, logdensity_fn: Callable) -> BarkerState:
grad_fn = jax.value_and_grad(logdensity_fn)
logdensity, logdensity_grad = grad_fn(position)
return BarkerState(position, logdensity, logdensity_grad)
[docs]
def build_kernel():
"""Build a Barker's proposal kernel.
Returns
-------
A kernel that takes a rng_key and a Pytree that contains the current state
of the chain and that returns a new state of the chain along with
information about the transition.
"""
def _compute_acceptance_probability(
state: BarkerState,
proposal: BarkerState,
) -> float:
"""Compute the acceptance probability of the Barker's proposal kernel."""
def ratio_proposal_nd(y, x, log_y, log_x):
num = -_log1pexp(-log_y * (x - y))
den = -_log1pexp(-log_x * (y - x))
return jnp.sum(num - den)
ratios_proposals = tree_map(
ratio_proposal_nd,
proposal.position,
state.position,
proposal.logdensity_grad,
state.logdensity_grad,
)
ratio_proposal = sum(tree_leaves(ratios_proposals))
return proposal.logdensity - state.logdensity + ratio_proposal
def kernel(
rng_key: PRNGKey, state: BarkerState, logdensity_fn: Callable, step_size: float
) -> tuple[BarkerState, BarkerInfo]:
"""Generate a new sample with the MALA kernel."""
grad_fn = jax.value_and_grad(logdensity_fn)
key_sample, key_rmh = jax.random.split(rng_key)
proposed_pos = _barker_sample(
key_sample, state.position, state.logdensity_grad, step_size
)
proposed_logdensity, proposed_logdensity_grad = grad_fn(proposed_pos)
proposed_state = BarkerState(
proposed_pos, proposed_logdensity, proposed_logdensity_grad
)
log_p_accept = _compute_acceptance_probability(state, proposed_state)
accepted_state, info = static_binomial_sampling(
key_rmh, log_p_accept, state, proposed_state
)
do_accept, p_accept, _ = info
return accepted_state, BarkerInfo(p_accept, do_accept, proposed_state)
return kernel
[docs]
def as_top_level_api(
logdensity_fn: Callable,
step_size: float,
) -> SamplingAlgorithm:
"""Implements the (basic) user interface for the Barker's proposal :cite:p:`Livingstone2022Barker` kernel with a
Gaussian base kernel.
The general Barker kernel builder (:meth:`blackjax.mcmc.barker.build_kernel`, alias `blackjax.barker.build_kernel`) can be
cumbersome to manipulate. Since most users only need to specify the kernel
parameters at initialization time, we provide a helper function that
specializes the general kernel.
We also add the general kernel and state generator as an attribute to this class so
users only need to pass `blackjax.barker` to SMC, adaptation, etc. algorithms.
Examples
--------
A new Barker kernel can be initialized and used with the following code:
.. code::
barker = blackjax.barker(logdensity_fn, step_size)
state = barker.init(position)
new_state, info = barker.step(rng_key, state)
Kernels are not jit-compiled by default so you will need to do it manually:
.. code::
step = jax.jit(barker.step)
new_state, info = step(rng_key, state)
Should you need to you can always use the base kernel directly:
.. code::
kernel = blackjax.barker.build_kernel(logdensity_fn)
state = blackjax.barker.init(position, logdensity_fn)
state, info = kernel(rng_key, state, logdensity_fn, step_size)
Parameters
----------
logdensity_fn
The log-density function we wish to draw samples from.
step_size
The value to use for the step size in the symplectic integrator.
Returns
-------
A ``SamplingAlgorithm``.
"""
kernel = build_kernel()
def init_fn(position: ArrayLikeTree, rng_key=None):
del rng_key
return init(position, logdensity_fn)
def step_fn(rng_key: PRNGKey, state):
return kernel(rng_key, state, logdensity_fn, step_size)
return SamplingAlgorithm(init_fn, step_fn)
def _barker_sample_nd(key, mean, a, scale):
"""
Sample from a multivariate Barker's proposal distribution. In 1D, this has the following probability density function:
.. math::
p(x; \\mu, a, \\sigma) = 2 \frac{N(x; \\mu, \\sigma^2)}{1 + \\exp(-a (x - \\mu)}
where :math:`N(x; \\mu, \\sigma^2)` is the normal distribution with mean :math:`\\mu` and standard deviation :math:`\\sigma`.
The multivariate Barker's proposal distribution is the product of one-dimensional Barker's proposal distributions.
Parameters
----------
key
A PRNG key.
mean
The mean of the normal distribution, an Array. This corresponds to :math:`\\mu` in the equation above.
a
The parameter :math:`a` in the equation above, an Array. This is a skewness parameter.
scale
The standard deviation of the normal distribution, a scalar. This corresponds to :math:`\\sigma` in the equation above.
It encodes the step size of the proposal.
Returns
-------
A sample from the Barker's multidimensional proposal distribution.
"""
key1, key2 = jax.random.split(key)
z = scale * jax.random.normal(key1, shape=mean.shape)
# Sample b=1 with probability p and 0 with probability 1 - p where
# p = 1 / (1 + exp(-a * (z - mean)))
log_p = -_log1pexp(-a * z)
b = jax.random.bernoulli(key2, p=jnp.exp(log_p), shape=mean.shape)
# return mean + z if b == 1 else mean - z
return mean + b * z - (1 - b) * z
def _barker_sample(key, mean, a, scale):
r"""
Sample from a multivariate Barker's proposal distribution for PyTrees.
Parameters
----------
key
A PRNG key.
mean
The mean of the normal distribution, a PyTree. This corresponds to :math:`\mu` in the equation above.
a
The parameter :math:`a` in the equation above, the same PyTree as `mean`. This is a skewness parameter.
scale
The standard deviation of the normal distribution, a scalar. This corresponds to :math:`\sigma` in the equation above.
It encodes the step size of the proposal.
"""
flat_mean, unravel_fn = ravel_pytree(mean)
flat_a, _ = ravel_pytree(a)
flat_sample = _barker_sample_nd(key, flat_mean, flat_a, scale)
return unravel_fn(flat_sample)
def _log1pexp(a):
return jnp.log1p(jnp.exp(a))
def _barker_logpdf(x, mean, a, scale):
logpdf = jnp.log(2) + stats.norm.logpdf(x, mean, scale) - _log1pexp(-a * (x - mean))
return logpdf
def _barker_pdf(x, mean, a, scale):
return jnp.exp(_barker_logpdf(x, mean, a, scale))
