# Copyright 2020- The Blackjax Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""Public API for marginal latent Gaussian sampling."""
from typing import Callable, NamedTuple, Optional
import jax
import jax.numpy as jnp
import jax.scipy.linalg as linalg
from blackjax.base import SamplingAlgorithm
from blackjax.mcmc.proposal import static_binomial_sampling
from blackjax.types import Array, PRNGKey
__all__ = [
"MarginalState",
"MarginalInfo",
"init",
"build_kernel",
"as_top_level_api",
]
# [TODO](https://github.com/blackjax-devs/blackjax/issues/237)
[docs]
class MarginalState(NamedTuple):
"""State of the RMH chain.
x
Current position of the chain.
log_p_x
Current value of the log-likelihood of the model
grad_x
Current value of the gradient of the log-likelihood of the model
U_x
Auxiliary attributes
U_grad_x
Gradient of the auxiliary attributes
"""
class CovarianceSVD(NamedTuple):
"""Singular Value Decomposition of the covariance matrix.
U
Unitary array of the covariance matrix.
Gamma
Singular values of the covariance matrix.
U_t
Transpose of the unitary array of the covariance matrix.
"""
U: Array
Gamma: Array
U_t: Array
def svd_from_covariance(covariance: Array) -> CovarianceSVD:
"""Compute the singular value decomposition of the covariance matrix.
Parameters
----------
covariance
The covariance matrix.
Returns
-------
A ``CovarianceSVD`` object.
"""
U, Gamma, U_t = jnp.linalg.svd(covariance, hermitian=True)
return CovarianceSVD(U, Gamma, U_t)
[docs]
class MarginalInfo(NamedTuple):
"""Additional information on the RMH chain.
This additional information can be used for debugging or computing
diagnostics.
acceptance_rate
The acceptance probability of the transition, linked to the energy
difference between the original and the proposed states.
is_accepted
Whether the proposed position was accepted or the original position
was returned.
proposal
The state proposed by the proposal.
"""
[docs]
proposal: MarginalState
def generate_mean_shifted_logprob(logdensity_fn, mean, covariance):
"""Generate a log-density function that is shifted by a constant
Parameters
----------
logdensity_fn
The original log-density function
mean
The mean of the prior Gaussian density
covariance
The covariance of the prior Gaussian density.
Returns
-------
A log-density function that is shifted by a constant
"""
shift = linalg.solve(covariance, mean, assume_a="pos")
def shifted_logdensity_fn(x):
return logdensity_fn(x) + jnp.dot(x, shift)
return shifted_logdensity_fn
[docs]
def init(position, logdensity_fn, U_t):
"""Initialize the marginal version of the auxiliary gradient-based sampler.
Parameters
----------
position
The initial position of the chain.
logdensity_fn
The logarithm of the likelihood function for the latent Gaussian model.
U_t
The unitary array of the covariance matrix.
"""
logdensity, logdensity_grad = jax.value_and_grad(logdensity_fn)(position)
return MarginalState(
position, logdensity, logdensity_grad, U_t @ position, U_t @ logdensity_grad
)
[docs]
def build_kernel(cov_svd: CovarianceSVD):
"""Build the marginal version of the auxiliary gradient-based sampler.
Parameters
----------
cov_svd
The singular value decomposition of the covariance matrix.
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.
"""
U, Gamma, U_t = cov_svd
def kernel(key: PRNGKey, state: MarginalState, logdensity_fn, delta):
y_key, u_key = jax.random.split(key, 2)
position, logdensity, logdensity_grad, U_x, U_grad_x = state
# Update Gamma(delta)
# TODO: Ideally, we could have a dichotomy, where we only update Gamma(delta) if delta changes,
# but this is hardly the most expensive part of the algorithm (the multiplication by U below is).
Gamma_1 = Gamma * delta / (delta + 2 * Gamma)
Gamma_3 = (delta + 2 * Gamma) / (delta + 4 * Gamma)
Gamma_2 = Gamma_1 / Gamma_3
# Propose a new y
temp = Gamma_1 * (U_x / (0.5 * delta) + U_grad_x)
temp = temp + jnp.sqrt(Gamma_2) * jax.random.normal(y_key, position.shape)
y = U @ temp
# Bookkeeping
log_p_y, grad_y = jax.value_and_grad(logdensity_fn)(y)
U_y = U_t @ y
U_grad_y = U_t @ grad_y
# Acceptance step
temp_x = Gamma_1 * (U_x / (0.5 * delta) + 0.5 * U_grad_x)
temp_y = Gamma_1 * (U_y / (0.5 * delta) + 0.5 * U_grad_y)
hxy = jnp.dot(U_x - temp_y, Gamma_3 * U_grad_y)
hyx = jnp.dot(U_y - temp_x, Gamma_3 * U_grad_x)
log_p_accept = log_p_y - logdensity + hxy - hyx
proposed_state = MarginalState(y, log_p_y, grad_y, U_y, U_grad_y)
accepted_state, info = static_binomial_sampling(
u_key, log_p_accept, state, proposed_state
)
do_accept, p_accept, _ = info
info = MarginalInfo(p_accept, do_accept, proposed_state)
return accepted_state, info
return kernel
[docs]
def as_top_level_api(
logdensity_fn: Callable,
covariance: Optional[Array] = None,
mean: Optional[Array] = None,
cov_svd: Optional[CovarianceSVD] = None,
step_size: float = 1.0,
) -> SamplingAlgorithm:
"""Implements the marginal sampler for latent Gaussian model of :cite:p:`titsias2018auxiliary`.
It uses a first order approximation to the log_likelihood of a model with Gaussian prior.
Interestingly, the only parameter that needs calibrating is the "step size" delta,
which can be done very efficiently.
Calibrating it to have an acceptance rate of roughly 50% is a good starting point.
Examples
--------
A new marginal latent Gaussian MCMC kernel for a model q(x) ∝ exp(f(x)) N(x; m, C)
can be initialized and used for a given "step size" delta with the following code:
.. code::
mgrad_gaussian = blackjax.mgrad_gaussian(f, C, mean=m, step_size=delta)
state = mgrad_gaussian.init(zeros) # Starting at the mean of the prior
new_state, info = mgrad_gaussian.step(rng_key, state)
We can JIT-compile the step function for better performance
.. code::
step = jax.jit(mgrad_gaussian.step)
new_state, info = step(rng_key, state)
Parameters
----------
logdensity_fn
The logarithm of the likelihood function for the latent Gaussian model.
covariance
The covariance of the prior Gaussian density.
mean: optional
Mean of the prior Gaussian density. Default is zero.
Returns
-------
A ``SamplingAlgorithm``.
"""
if cov_svd is None:
if covariance is None:
raise ValueError("Either covariance or cov_svd must be provided.")
cov_svd = svd_from_covariance(covariance)
U, Gamma, U_t = cov_svd
if mean is not None:
logdensity_fn = generate_mean_shifted_logprob(logdensity_fn, mean, covariance)
kernel = build_kernel(cov_svd)
def init_fn(position: Array, rng_key=None):
del rng_key
return init(position, logdensity_fn, U_t)
def step_fn(rng_key: PRNGKey, state):
return kernel(
rng_key,
state,
logdensity_fn,
step_size,
)
return SamplingAlgorithm(init_fn, step_fn)
