Source code for torch.distributions.kl
import math
import warnings
from functools import total_ordering
from typing import Type, Dict, Callable, Tuple
import torch
from torch._six import inf
from .bernoulli import Bernoulli
from .beta import Beta
from .binomial import Binomial
from .categorical import Categorical
from .cauchy import Cauchy
from .continuous_bernoulli import ContinuousBernoulli
from .dirichlet import Dirichlet
from .distribution import Distribution
from .exponential import Exponential
from .exp_family import ExponentialFamily
from .gamma import Gamma
from .geometric import Geometric
from .gumbel import Gumbel
from .half_normal import HalfNormal
from .independent import Independent
from .laplace import Laplace
from .lowrank_multivariate_normal import (LowRankMultivariateNormal, _batch_lowrank_logdet,
_batch_lowrank_mahalanobis)
from .multivariate_normal import (MultivariateNormal, _batch_mahalanobis)
from .normal import Normal
from .one_hot_categorical import OneHotCategorical
from .pareto import Pareto
from .poisson import Poisson
from .transformed_distribution import TransformedDistribution
from .uniform import Uniform
from .utils import _sum_rightmost, euler_constant as _euler_gamma
_KL_REGISTRY = {} # Source of truth mapping a few general (type, type) pairs to functions.
_KL_MEMOIZE: Dict[Tuple[Type, Type], Callable] = {} # Memoized version mapping many specific (type, type) pairs to functions.
[docs]def register_kl(type_p, type_q):
"""
Decorator to register a pairwise function with :meth:`kl_divergence`.
Usage::
@register_kl(Normal, Normal)
def kl_normal_normal(p, q):
# insert implementation here
Lookup returns the most specific (type,type) match ordered by subclass. If
the match is ambiguous, a `RuntimeWarning` is raised. For example to
resolve the ambiguous situation::
@register_kl(BaseP, DerivedQ)
def kl_version1(p, q): ...
@register_kl(DerivedP, BaseQ)
def kl_version2(p, q): ...
you should register a third most-specific implementation, e.g.::
register_kl(DerivedP, DerivedQ)(kl_version1) # Break the tie.
Args:
type_p (type): A subclass of :class:`~torch.distributions.Distribution`.
type_q (type): A subclass of :class:`~torch.distributions.Distribution`.
"""
if not isinstance(type_p, type) and issubclass(type_p, Distribution):
raise TypeError('Expected type_p to be a Distribution subclass but got {}'.format(type_p))
if not isinstance(type_q, type) and issubclass(type_q, Distribution):
raise TypeError('Expected type_q to be a Distribution subclass but got {}'.format(type_q))
def decorator(fun):
_KL_REGISTRY[type_p, type_q] = fun
_KL_MEMOIZE.clear() # reset since lookup order may have changed
return fun
return decorator
@total_ordering
class _Match(object):
__slots__ = ['types']
def __init__(self, *types):
self.types = types
def __eq__(self, other):
return self.types == other.types
def __le__(self, other):
for x, y in zip(self.types, other.types):
if not issubclass(x, y):
return False
if x is not y:
break
return True
def _dispatch_kl(type_p, type_q):
"""
Find the most specific approximate match, assuming single inheritance.
"""
matches = [(super_p, super_q) for super_p, super_q in _KL_REGISTRY
if issubclass(type_p, super_p) and issubclass(type_q, super_q)]
if not matches:
return NotImplemented
# Check that the left- and right- lexicographic orders agree.
# mypy isn't smart enough to know that _Match implements __lt__
# see: https://github.com/python/typing/issues/760#issuecomment-710670503
left_p, left_q = min(_Match(*m) for m in matches).types # type: ignore[type-var]
right_q, right_p = min(_Match(*reversed(m)) for m in matches).types # type: ignore[type-var]
left_fun = _KL_REGISTRY[left_p, left_q]
right_fun = _KL_REGISTRY[right_p, right_q]
if left_fun is not right_fun:
warnings.warn('Ambiguous kl_divergence({}, {}). Please register_kl({}, {})'.format(
type_p.__name__, type_q.__name__, left_p.__name__, right_q.__name__),
RuntimeWarning)
return left_fun
def _infinite_like(tensor):
"""
Helper function for obtaining infinite KL Divergence throughout
"""
return torch.full_like(tensor, inf)
def _x_log_x(tensor):
"""
Utility function for calculating x log x
"""
return tensor * tensor.log()
def _batch_trace_XXT(bmat):
"""
Utility function for calculating the trace of XX^{T} with X having arbitrary trailing batch dimensions
"""
n = bmat.size(-1)
m = bmat.size(-2)
flat_trace = bmat.reshape(-1, m * n).pow(2).sum(-1)
return flat_trace.reshape(bmat.shape[:-2])
[docs]def kl_divergence(p, q):
r"""
Compute Kullback-Leibler divergence :math:`KL(p \| q)` between two distributions.
.. math::
KL(p \| q) = \int p(x) \log\frac {p(x)} {q(x)} \,dx
Args:
p (Distribution): A :class:`~torch.distributions.Distribution` object.
q (Distribution): A :class:`~torch.distributions.Distribution` object.
Returns:
Tensor: A batch of KL divergences of shape `batch_shape`.
Raises:
NotImplementedError: If the distribution types have not been registered via
:meth:`register_kl`.
"""
try:
fun = _KL_MEMOIZE[type(p), type(q)]
except KeyError:
fun = _dispatch_kl(type(p), type(q))
_KL_MEMOIZE[type(p), type(q)] = fun
if fun is NotImplemented:
raise NotImplementedError
return fun(p, q)
################################################################################
# KL Divergence Implementations
################################################################################
# Same distributions
@register_kl(Bernoulli, Bernoulli)
def _kl_bernoulli_bernoulli(p, q):
t1 = p.probs * (p.probs / q.probs).log()
t1[q.probs == 0] = inf
t1[p.probs == 0] = 0
t2 = (1 - p.probs) * ((1 - p.probs) / (1 - q.probs)).log()
t2[q.probs == 1] = inf
t2[p.probs == 1] = 0
return t1 + t2
@register_kl(Beta, Beta)
def _kl_beta_beta(p, q):
sum_params_p = p.concentration1 + p.concentration0
sum_params_q = q.concentration1 + q.concentration0
t1 = q.concentration1.lgamma() + q.concentration0.lgamma() + (sum_params_p).lgamma()
t2 = p.concentration1.lgamma() + p.concentration0.lgamma() + (sum_params_q).lgamma()
t3 = (p.concentration1 - q.concentration1) * torch.digamma(p.concentration1)
t4 = (p.concentration0 - q.concentration0) * torch.digamma(p.concentration0)
t5 = (sum_params_q - sum_params_p) * torch.digamma(sum_params_p)
return t1 - t2 + t3 + t4 + t5
@register_kl(Binomial, Binomial)
def _kl_binomial_binomial(p, q):
# from https://math.stackexchange.com/questions/2214993/
# kullback-leibler-divergence-for-binomial-distributions-p-and-q
if (p.total_count < q.total_count).any():
raise NotImplementedError('KL between Binomials where q.total_count > p.total_count is not implemented')
kl = p.total_count * (p.probs * (p.logits - q.logits) + (-p.probs).log1p() - (-q.probs).log1p())
inf_idxs = p.total_count > q.total_count
kl[inf_idxs] = _infinite_like(kl[inf_idxs])
return kl
@register_kl(Categorical, Categorical)
def _kl_categorical_categorical(p, q):
t = p.probs * (p.logits - q.logits)
t[(q.probs == 0).expand_as(t)] = inf
t[(p.probs == 0).expand_as(t)] = 0
return t.sum(-1)
@register_kl(ContinuousBernoulli, ContinuousBernoulli)
def _kl_continuous_bernoulli_continuous_bernoulli(p, q):
t1 = p.mean * (p.logits - q.logits)
t2 = p._cont_bern_log_norm() + torch.log1p(-p.probs)
t3 = - q._cont_bern_log_norm() - torch.log1p(-q.probs)
return t1 + t2 + t3
@register_kl(Dirichlet, Dirichlet)
def _kl_dirichlet_dirichlet(p, q):
# From http://bariskurt.com/kullback-leibler-divergence-between-two-dirichlet-and-beta-distributions/
sum_p_concentration = p.concentration.sum(-1)
sum_q_concentration = q.concentration.sum(-1)
t1 = sum_p_concentration.lgamma() - sum_q_concentration.lgamma()
t2 = (p.concentration.lgamma() - q.concentration.lgamma()).sum(-1)
t3 = p.concentration - q.concentration
t4 = p.concentration.digamma() - sum_p_concentration.digamma().unsqueeze(-1)
return t1 - t2 + (t3 * t4).sum(-1)
@register_kl(Exponential, Exponential)
def _kl_exponential_exponential(p, q):
rate_ratio = q.rate / p.rate
t1 = -rate_ratio.log()
return t1 + rate_ratio - 1
@register_kl(ExponentialFamily, ExponentialFamily)
def _kl_expfamily_expfamily(p, q):
if not type(p) == type(q):
raise NotImplementedError("The cross KL-divergence between different exponential families cannot \
be computed using Bregman divergences")
p_nparams = [np.detach().requires_grad_() for np in p._natural_params]
q_nparams = q._natural_params
lg_normal = p._log_normalizer(*p_nparams)
gradients = torch.autograd.grad(lg_normal.sum(), p_nparams, create_graph=True)
result = q._log_normalizer(*q_nparams) - lg_normal
for pnp, qnp, g in zip(p_nparams, q_nparams, gradients):
term = (qnp - pnp) * g
result -= _sum_rightmost(term, len(q.event_shape))
return result
@register_kl(Gamma, Gamma)
def _kl_gamma_gamma(p, q):
t1 = q.concentration * (p.rate / q.rate).log()
t2 = torch.lgamma(q.concentration) - torch.lgamma(p.concentration)
t3 = (p.concentration - q.concentration) * torch.digamma(p.concentration)
t4 = (q.rate - p.rate) * (p.concentration / p.rate)
return t1 + t2 + t3 + t4
@register_kl(Gumbel, Gumbel)
def _kl_gumbel_gumbel(p, q):
ct1 = p.scale / q.scale
ct2 = q.loc / q.scale
ct3 = p.loc / q.scale
t1 = -ct1.log() - ct2 + ct3
t2 = ct1 * _euler_gamma
t3 = torch.exp(ct2 + (1 + ct1).lgamma() - ct3)
return t1 + t2 + t3 - (1 + _euler_gamma)
@register_kl(Geometric, Geometric)
def _kl_geometric_geometric(p, q):
return -p.entropy() - torch.log1p(-q.probs) / p.probs - q.logits
@register_kl(HalfNormal, HalfNormal)
def _kl_halfnormal_halfnormal(p, q):
return _kl_normal_normal(p.base_dist, q.base_dist)
@register_kl(Laplace, Laplace)
def _kl_laplace_laplace(p, q):
# From http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf
scale_ratio = p.scale / q.scale
loc_abs_diff = (p.loc - q.loc).abs()
t1 = -scale_ratio.log()
t2 = loc_abs_diff / q.scale
t3 = scale_ratio * torch.exp(-loc_abs_diff / p.scale)
return t1 + t2 + t3 - 1
@register_kl(LowRankMultivariateNormal, LowRankMultivariateNormal)
def _kl_lowrankmultivariatenormal_lowrankmultivariatenormal(p, q):
if p.event_shape != q.event_shape:
raise ValueError("KL-divergence between two Low Rank Multivariate Normals with\
different event shapes cannot be computed")
term1 = (_batch_lowrank_logdet(q._unbroadcasted_cov_factor, q._unbroadcasted_cov_diag,
q._capacitance_tril) -
_batch_lowrank_logdet(p._unbroadcasted_cov_factor, p._unbroadcasted_cov_diag,
p._capacitance_tril))
term3 = _batch_lowrank_mahalanobis(q._unbroadcasted_cov_factor, q._unbroadcasted_cov_diag,
q.loc - p.loc,
q._capacitance_tril)
# Expands term2 according to
# inv(qcov) @ pcov = [inv(qD) - inv(qD) @ qW @ inv(qC) @ qW.T @ inv(qD)] @ (pW @ pW.T + pD)
# = [inv(qD) - A.T @ A] @ (pD + pW @ pW.T)
qWt_qDinv = (q._unbroadcasted_cov_factor.mT /
q._unbroadcasted_cov_diag.unsqueeze(-2))
A = torch.linalg.solve_triangular(q._capacitance_tril, qWt_qDinv, upper=False)
term21 = (p._unbroadcasted_cov_diag / q._unbroadcasted_cov_diag).sum(-1)
term22 = _batch_trace_XXT(p._unbroadcasted_cov_factor *
q._unbroadcasted_cov_diag.rsqrt().unsqueeze(-1))
term23 = _batch_trace_XXT(A * p._unbroadcasted_cov_diag.sqrt().unsqueeze(-2))
term24 = _batch_trace_XXT(A.matmul(p._unbroadcasted_cov_factor))
term2 = term21 + term22 - term23 - term24
return 0.5 * (term1 + term2 + term3 - p.event_shape[0])
@register_kl(MultivariateNormal, LowRankMultivariateNormal)
def _kl_multivariatenormal_lowrankmultivariatenormal(p, q):
if p.event_shape != q.event_shape:
raise ValueError("KL-divergence between two (Low Rank) Multivariate Normals with\
different event shapes cannot be computed")
term1 = (_batch_lowrank_logdet(q._unbroadcasted_cov_factor, q._unbroadcasted_cov_diag,
q._capacitance_tril) -
2 * p._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1))
term3 = _batch_lowrank_mahalanobis(q._unbroadcasted_cov_factor, q._unbroadcasted_cov_diag,
q.loc - p.loc,
q._capacitance_tril)
# Expands term2 according to
# inv(qcov) @ pcov = [inv(qD) - inv(qD) @ qW @ inv(qC) @ qW.T @ inv(qD)] @ p_tril @ p_tril.T
# = [inv(qD) - A.T @ A] @ p_tril @ p_tril.T
qWt_qDinv = (q._unbroadcasted_cov_factor.mT /
q._unbroadcasted_cov_diag.unsqueeze(-2))
A = torch.linalg.solve_triangular(q._capacitance_tril, qWt_qDinv, upper=False)
term21 = _batch_trace_XXT(p._unbroadcasted_scale_tril *
q._unbroadcasted_cov_diag.rsqrt().unsqueeze(-1))
term22 = _batch_trace_XXT(A.matmul(p._unbroadcasted_scale_tril))
term2 = term21 - term22
return 0.5 * (term1 + term2 + term3 - p.event_shape[0])
@register_kl(LowRankMultivariateNormal, MultivariateNormal)
def _kl_lowrankmultivariatenormal_multivariatenormal(p, q):
if p.event_shape != q.event_shape:
raise ValueError("KL-divergence between two (Low Rank) Multivariate Normals with\
different event shapes cannot be computed")
term1 = (2 * q._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1) -
_batch_lowrank_logdet(p._unbroadcasted_cov_factor, p._unbroadcasted_cov_diag,
p._capacitance_tril))
term3 = _batch_mahalanobis(q._unbroadcasted_scale_tril, (q.loc - p.loc))
# Expands term2 according to
# inv(qcov) @ pcov = inv(q_tril @ q_tril.T) @ (pW @ pW.T + pD)
combined_batch_shape = torch._C._infer_size(q._unbroadcasted_scale_tril.shape[:-2],
p._unbroadcasted_cov_factor.shape[:-2])
n = p.event_shape[0]
q_scale_tril = q._unbroadcasted_scale_tril.expand(combined_batch_shape + (n, n))
p_cov_factor = p._unbroadcasted_cov_factor.expand(combined_batch_shape +
(n, p.cov_factor.size(-1)))
p_cov_diag = (torch.diag_embed(p._unbroadcasted_cov_diag.sqrt())
.expand(combined_batch_shape + (n, n)))
term21 = _batch_trace_XXT(torch.linalg.solve_triangular(q_scale_tril, p_cov_factor, upper=False))
term22 = _batch_trace_XXT(torch.linalg.solve_triangular(q_scale_tril, p_cov_diag, upper=False))
term2 = term21 + term22
return 0.5 * (term1 + term2 + term3 - p.event_shape[0])
@register_kl(MultivariateNormal, MultivariateNormal)
def _kl_multivariatenormal_multivariatenormal(p, q):
# From https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Kullback%E2%80%93Leibler_divergence
if p.event_shape != q.event_shape:
raise ValueError("KL-divergence between two Multivariate Normals with\
different event shapes cannot be computed")
half_term1 = (q._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1) -
p._unbroadcasted_scale_tril.diagonal(dim1=-2, dim2=-1).log().sum(-1))
combined_batch_shape = torch._C._infer_size(q._unbroadcasted_scale_tril.shape[:-2],
p._unbroadcasted_scale_tril.shape[:-2])
n = p.event_shape[0]
q_scale_tril = q._unbroadcasted_scale_tril.expand(combined_batch_shape + (n, n))
p_scale_tril = p._unbroadcasted_scale_tril.expand(combined_batch_shape + (n, n))
term2 = _batch_trace_XXT(torch.linalg.solve_triangular(q_scale_tril, p_scale_tril, upper=False))
term3 = _batch_mahalanobis(q._unbroadcasted_scale_tril, (q.loc - p.loc))
return half_term1 + 0.5 * (term2 + term3 - n)
@register_kl(Normal, Normal)
def _kl_normal_normal(p, q):
var_ratio = (p.scale / q.scale).pow(2)
t1 = ((p.loc - q.loc) / q.scale).pow(2)
return 0.5 * (var_ratio + t1 - 1 - var_ratio.log())
@register_kl(OneHotCategorical, OneHotCategorical)
def _kl_onehotcategorical_onehotcategorical(p, q):
return _kl_categorical_categorical(p._categorical, q._categorical)
@register_kl(Pareto, Pareto)
def _kl_pareto_pareto(p, q):
# From http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf
scale_ratio = p.scale / q.scale
alpha_ratio = q.alpha / p.alpha
t1 = q.alpha * scale_ratio.log()
t2 = -alpha_ratio.log()
result = t1 + t2 + alpha_ratio - 1
result[p.support.lower_bound < q.support.lower_bound] = inf
return result
@register_kl(Poisson, Poisson)
def _kl_poisson_poisson(p, q):
return p.rate * (p.rate.log() - q.rate.log()) - (p.rate - q.rate)
@register_kl(TransformedDistribution, TransformedDistribution)
def _kl_transformed_transformed(p, q):
if p.transforms != q.transforms:
raise NotImplementedError
if p.event_shape != q.event_shape:
raise NotImplementedError
return kl_divergence(p.base_dist, q.base_dist)
@register_kl(Uniform, Uniform)
def _kl_uniform_uniform(p, q):
result = ((q.high - q.low) / (p.high - p.low)).log()
result[(q.low > p.low) | (q.high < p.high)] = inf
return result
# Different distributions
@register_kl(Bernoulli, Poisson)
def _kl_bernoulli_poisson(p, q):
return -p.entropy() - (p.probs * q.rate.log() - q.rate)
@register_kl(Beta, ContinuousBernoulli)
def _kl_beta_continuous_bernoulli(p, q):
return -p.entropy() - p.mean * q.logits - torch.log1p(-q.probs) - q._cont_bern_log_norm()
@register_kl(Beta, Pareto)
def _kl_beta_infinity(p, q):
return _infinite_like(p.concentration1)
@register_kl(Beta, Exponential)
def _kl_beta_exponential(p, q):
return -p.entropy() - q.rate.log() + q.rate * (p.concentration1 / (p.concentration1 + p.concentration0))
@register_kl(Beta, Gamma)
def _kl_beta_gamma(p, q):
t1 = -p.entropy()
t2 = q.concentration.lgamma() - q.concentration * q.rate.log()
t3 = (q.concentration - 1) * (p.concentration1.digamma() - (p.concentration1 + p.concentration0).digamma())
t4 = q.rate * p.concentration1 / (p.concentration1 + p.concentration0)
return t1 + t2 - t3 + t4
# TODO: Add Beta-Laplace KL Divergence
@register_kl(Beta, Normal)
def _kl_beta_normal(p, q):
E_beta = p.concentration1 / (p.concentration1 + p.concentration0)
var_normal = q.scale.pow(2)
t1 = -p.entropy()
t2 = 0.5 * (var_normal * 2 * math.pi).log()
t3 = (E_beta * (1 - E_beta) / (p.concentration1 + p.concentration0 + 1) + E_beta.pow(2)) * 0.5
t4 = q.loc * E_beta
t5 = q.loc.pow(2) * 0.5
return t1 + t2 + (t3 - t4 + t5) / var_normal
@register_kl(Beta, Uniform)
def _kl_beta_uniform(p, q):
result = -p.entropy() + (q.high - q.low).log()
result[(q.low > p.support.lower_bound) | (q.high < p.support.upper_bound)] = inf
return result
# Note that the KL between a ContinuousBernoulli and Beta has no closed form
@register_kl(ContinuousBernoulli, Pareto)
def _kl_continuous_bernoulli_infinity(p, q):
return _infinite_like(p.probs)
@register_kl(ContinuousBernoulli, Exponential)
def _kl_continuous_bernoulli_exponential(p, q):
return -p.entropy() - torch.log(q.rate) + q.rate * p.mean
# Note that the KL between a ContinuousBernoulli and Gamma has no closed form
# TODO: Add ContinuousBernoulli-Laplace KL Divergence
@register_kl(ContinuousBernoulli, Normal)
def _kl_continuous_bernoulli_normal(p, q):
t1 = -p.entropy()
t2 = 0.5 * (math.log(2. * math.pi) + torch.square(q.loc / q.scale)) + torch.log(q.scale)
t3 = (p.variance + torch.square(p.mean) - 2. * q.loc * p.mean) / (2.0 * torch.square(q.scale))
return t1 + t2 + t3
@register_kl(ContinuousBernoulli, Uniform)
def _kl_continuous_bernoulli_uniform(p, q):
result = -p.entropy() + (q.high - q.low).log()
return torch.where(torch.max(torch.ge(q.low, p.support.lower_bound),
torch.le(q.high, p.support.upper_bound)),
torch.ones_like(result) * inf, result)
@register_kl(Exponential, Beta)
@register_kl(Exponential, ContinuousBernoulli)
@register_kl(Exponential, Pareto)
@register_kl(Exponential, Uniform)
def _kl_exponential_infinity(p, q):
return _infinite_like(p.rate)
@register_kl(Exponential, Gamma)
def _kl_exponential_gamma(p, q):
ratio = q.rate / p.rate
t1 = -q.concentration * torch.log(ratio)
return t1 + ratio + q.concentration.lgamma() + q.concentration * _euler_gamma - (1 + _euler_gamma)
@register_kl(Exponential, Gumbel)
def _kl_exponential_gumbel(p, q):
scale_rate_prod = p.rate * q.scale
loc_scale_ratio = q.loc / q.scale
t1 = scale_rate_prod.log() - 1
t2 = torch.exp(loc_scale_ratio) * scale_rate_prod / (scale_rate_prod + 1)
t3 = scale_rate_prod.reciprocal()
return t1 - loc_scale_ratio + t2 + t3
# TODO: Add Exponential-Laplace KL Divergence
@register_kl(Exponential, Normal)
def _kl_exponential_normal(p, q):
var_normal = q.scale.pow(2)
rate_sqr = p.rate.pow(2)
t1 = 0.5 * torch.log(rate_sqr * var_normal * 2 * math.pi)
t2 = rate_sqr.reciprocal()
t3 = q.loc / p.rate
t4 = q.loc.pow(2) * 0.5
return t1 - 1 + (t2 - t3 + t4) / var_normal
@register_kl(Gamma, Beta)
@register_kl(Gamma, ContinuousBernoulli)
@register_kl(Gamma, Pareto)
@register_kl(Gamma, Uniform)
def _kl_gamma_infinity(p, q):
return _infinite_like(p.concentration)
@register_kl(Gamma, Exponential)
def _kl_gamma_exponential(p, q):
return -p.entropy() - q.rate.log() + q.rate * p.concentration / p.rate
@register_kl(Gamma, Gumbel)
def _kl_gamma_gumbel(p, q):
beta_scale_prod = p.rate * q.scale
loc_scale_ratio = q.loc / q.scale
t1 = (p.concentration - 1) * p.concentration.digamma() - p.concentration.lgamma() - p.concentration
t2 = beta_scale_prod.log() + p.concentration / beta_scale_prod
t3 = torch.exp(loc_scale_ratio) * (1 + beta_scale_prod.reciprocal()).pow(-p.concentration) - loc_scale_ratio
return t1 + t2 + t3
# TODO: Add Gamma-Laplace KL Divergence
@register_kl(Gamma, Normal)
def _kl_gamma_normal(p, q):
var_normal = q.scale.pow(2)
beta_sqr = p.rate.pow(2)
t1 = 0.5 * torch.log(beta_sqr * var_normal * 2 * math.pi) - p.concentration - p.concentration.lgamma()
t2 = 0.5 * (p.concentration.pow(2) + p.concentration) / beta_sqr
t3 = q.loc * p.concentration / p.rate
t4 = 0.5 * q.loc.pow(2)
return t1 + (p.concentration - 1) * p.concentration.digamma() + (t2 - t3 + t4) / var_normal
@register_kl(Gumbel, Beta)
@register_kl(Gumbel, ContinuousBernoulli)
@register_kl(Gumbel, Exponential)
@register_kl(Gumbel, Gamma)
@register_kl(Gumbel, Pareto)
@register_kl(Gumbel, Uniform)
def _kl_gumbel_infinity(p, q):
return _infinite_like(p.loc)
# TODO: Add Gumbel-Laplace KL Divergence
@register_kl(Gumbel, Normal)
def _kl_gumbel_normal(p, q):
param_ratio = p.scale / q.scale
t1 = (param_ratio / math.sqrt(2 * math.pi)).log()
t2 = (math.pi * param_ratio * 0.5).pow(2) / 3
t3 = ((p.loc + p.scale * _euler_gamma - q.loc) / q.scale).pow(2) * 0.5
return -t1 + t2 + t3 - (_euler_gamma + 1)
@register_kl(Laplace, Beta)
@register_kl(Laplace, ContinuousBernoulli)
@register_kl(Laplace, Exponential)
@register_kl(Laplace, Gamma)
@register_kl(Laplace, Pareto)
@register_kl(Laplace, Uniform)
def _kl_laplace_infinity(p, q):
return _infinite_like(p.loc)
@register_kl(Laplace, Normal)
def _kl_laplace_normal(p, q):
var_normal = q.scale.pow(2)
scale_sqr_var_ratio = p.scale.pow(2) / var_normal
t1 = 0.5 * torch.log(2 * scale_sqr_var_ratio / math.pi)
t2 = 0.5 * p.loc.pow(2)
t3 = p.loc * q.loc
t4 = 0.5 * q.loc.pow(2)
return -t1 + scale_sqr_var_ratio + (t2 - t3 + t4) / var_normal - 1
@register_kl(Normal, Beta)
@register_kl(Normal, ContinuousBernoulli)
@register_kl(Normal, Exponential)
@register_kl(Normal, Gamma)
@register_kl(Normal, Pareto)
@register_kl(Normal, Uniform)
def _kl_normal_infinity(p, q):
return _infinite_like(p.loc)
@register_kl(Normal, Gumbel)
def _kl_normal_gumbel(p, q):
mean_scale_ratio = p.loc / q.scale
var_scale_sqr_ratio = (p.scale / q.scale).pow(2)
loc_scale_ratio = q.loc / q.scale
t1 = var_scale_sqr_ratio.log() * 0.5
t2 = mean_scale_ratio - loc_scale_ratio
t3 = torch.exp(-mean_scale_ratio + 0.5 * var_scale_sqr_ratio + loc_scale_ratio)
return -t1 + t2 + t3 - (0.5 * (1 + math.log(2 * math.pi)))
@register_kl(Normal, Laplace)
def _kl_normal_laplace(p, q):
loc_diff = p.loc - q.loc
scale_ratio = p.scale / q.scale
loc_diff_scale_ratio = loc_diff / p.scale
t1 = torch.log(scale_ratio)
t2 = math.sqrt(2 / math.pi) * p.scale * torch.exp(-0.5 * loc_diff_scale_ratio.pow(2))
t3 = loc_diff * torch.erf(math.sqrt(0.5) * loc_diff_scale_ratio)
return -t1 + (t2 + t3) / q.scale - (0.5 * (1 + math.log(0.5 * math.pi)))
@register_kl(Pareto, Beta)
@register_kl(Pareto, ContinuousBernoulli)
@register_kl(Pareto, Uniform)
def _kl_pareto_infinity(p, q):
return _infinite_like(p.scale)
@register_kl(Pareto, Exponential)
def _kl_pareto_exponential(p, q):
scale_rate_prod = p.scale * q.rate
t1 = (p.alpha / scale_rate_prod).log()
t2 = p.alpha.reciprocal()
t3 = p.alpha * scale_rate_prod / (p.alpha - 1)
result = t1 - t2 + t3 - 1
result[p.alpha <= 1] = inf
return result
@register_kl(Pareto, Gamma)
def _kl_pareto_gamma(p, q):
common_term = p.scale.log() + p.alpha.reciprocal()
t1 = p.alpha.log() - common_term
t2 = q.concentration.lgamma() - q.concentration * q.rate.log()
t3 = (1 - q.concentration) * common_term
t4 = q.rate * p.alpha * p.scale / (p.alpha - 1)
result = t1 + t2 + t3 + t4 - 1
result[p.alpha <= 1] = inf
return result
# TODO: Add Pareto-Laplace KL Divergence
@register_kl(Pareto, Normal)
def _kl_pareto_normal(p, q):
var_normal = 2 * q.scale.pow(2)
common_term = p.scale / (p.alpha - 1)
t1 = (math.sqrt(2 * math.pi) * q.scale * p.alpha / p.scale).log()
t2 = p.alpha.reciprocal()
t3 = p.alpha * common_term.pow(2) / (p.alpha - 2)
t4 = (p.alpha * common_term - q.loc).pow(2)
result = t1 - t2 + (t3 + t4) / var_normal - 1
result[p.alpha <= 2] = inf
return result
@register_kl(Poisson, Bernoulli)
@register_kl(Poisson, Binomial)
def _kl_poisson_infinity(p, q):
return _infinite_like(p.rate)
@register_kl(Uniform, Beta)
def _kl_uniform_beta(p, q):
common_term = p.high - p.low
t1 = torch.log(common_term)
t2 = (q.concentration1 - 1) * (_x_log_x(p.high) - _x_log_x(p.low) - common_term) / common_term
t3 = (q.concentration0 - 1) * (_x_log_x((1 - p.high)) - _x_log_x((1 - p.low)) + common_term) / common_term
t4 = q.concentration1.lgamma() + q.concentration0.lgamma() - (q.concentration1 + q.concentration0).lgamma()
result = t3 + t4 - t1 - t2
result[(p.high > q.support.upper_bound) | (p.low < q.support.lower_bound)] = inf
return result
@register_kl(Uniform, ContinuousBernoulli)
def _kl_uniform_continuous_bernoulli(p, q):
result = -p.entropy() - p.mean * q.logits - torch.log1p(-q.probs) - q._cont_bern_log_norm()
return torch.where(torch.max(torch.ge(p.high, q.support.upper_bound),
torch.le(p.low, q.support.lower_bound)),
torch.ones_like(result) * inf, result)
@register_kl(Uniform, Exponential)
def _kl_uniform_exponetial(p, q):
result = q.rate * (p.high + p.low) / 2 - ((p.high - p.low) * q.rate).log()
result[p.low < q.support.lower_bound] = inf
return result
@register_kl(Uniform, Gamma)
def _kl_uniform_gamma(p, q):
common_term = p.high - p.low
t1 = common_term.log()
t2 = q.concentration.lgamma() - q.concentration * q.rate.log()
t3 = (1 - q.concentration) * (_x_log_x(p.high) - _x_log_x(p.low) - common_term) / common_term
t4 = q.rate * (p.high + p.low) / 2
result = -t1 + t2 + t3 + t4
result[p.low < q.support.lower_bound] = inf
return result
@register_kl(Uniform, Gumbel)
def _kl_uniform_gumbel(p, q):
common_term = q.scale / (p.high - p.low)
high_loc_diff = (p.high - q.loc) / q.scale
low_loc_diff = (p.low - q.loc) / q.scale
t1 = common_term.log() + 0.5 * (high_loc_diff + low_loc_diff)
t2 = common_term * (torch.exp(-high_loc_diff) - torch.exp(-low_loc_diff))
return t1 - t2
# TODO: Uniform-Laplace KL Divergence
@register_kl(Uniform, Normal)
def _kl_uniform_normal(p, q):
common_term = p.high - p.low
t1 = (math.sqrt(math.pi * 2) * q.scale / common_term).log()
t2 = (common_term).pow(2) / 12
t3 = ((p.high + p.low - 2 * q.loc) / 2).pow(2)
return t1 + 0.5 * (t2 + t3) / q.scale.pow(2)
@register_kl(Uniform, Pareto)
def _kl_uniform_pareto(p, q):
support_uniform = p.high - p.low
t1 = (q.alpha * q.scale.pow(q.alpha) * (support_uniform)).log()
t2 = (_x_log_x(p.high) - _x_log_x(p.low) - support_uniform) / support_uniform
result = t2 * (q.alpha + 1) - t1
result[p.low < q.support.lower_bound] = inf
return result
@register_kl(Independent, Independent)
def _kl_independent_independent(p, q):
if p.reinterpreted_batch_ndims != q.reinterpreted_batch_ndims:
raise NotImplementedError
result = kl_divergence(p.base_dist, q.base_dist)
return _sum_rightmost(result, p.reinterpreted_batch_ndims)
@register_kl(Cauchy, Cauchy)
def _kl_cauchy_cauchy(p, q):
# From https://arxiv.org/abs/1905.10965
t1 = ((p.scale + q.scale).pow(2) + (p.loc - q.loc).pow(2)).log()
t2 = (4 * p.scale * q.scale).log()
return t1 - t2
