Source code for pyro.ops.gaussian

# Copyright (c) 2017-2019 Uber Technologies, Inc.
# SPDX-License-Identifier: Apache-2.0

import math

import torch
from torch.distributions.utils import lazy_property
from torch.nn.functional import pad

from pyro.distributions.util import broadcast_shape


[docs]class Gaussian: """ Non-normalized Gaussian distribution. This represents an arbitrary semidefinite quadratic function, which can be interpreted as a rank-deficient scaled Gaussian distribution. The precision matrix may have zero eigenvalues, thus it may be impossible to work directly with the covariance matrix. :param torch.Tensor log_normalizer: a normalization constant, which is mainly used to keep track of normalization terms during contractions. :param torch.Tensor info_vec: information vector, which is a scaled version of the mean ``info_vec = precision @ mean``. We use this represention to make gaussian contraction fast and stable. :param torch.Tensor precision: precision matrix of this gaussian. """ def __init__(self, log_normalizer, info_vec, precision): # NB: using info_vec instead of mean to deal with rank-deficient problem assert info_vec.dim() >= 1 assert precision.dim() >= 2 assert precision.shape[-2:] == info_vec.shape[-1:] * 2 self.log_normalizer = log_normalizer self.info_vec = info_vec self.precision = precision
[docs] def dim(self): return self.info_vec.size(-1)
[docs] @lazy_property def batch_shape(self): return broadcast_shape(self.log_normalizer.shape, self.info_vec.shape[:-1], self.precision.shape[:-2])
[docs] def expand(self, batch_shape): n = self.dim() log_normalizer = self.log_normalizer.expand(batch_shape) info_vec = self.info_vec.expand(batch_shape + (n,)) precision = self.precision.expand(batch_shape + (n, n)) return Gaussian(log_normalizer, info_vec, precision)
[docs] def reshape(self, batch_shape): n = self.dim() log_normalizer = self.log_normalizer.reshape(batch_shape) info_vec = self.info_vec.reshape(batch_shape + (n,)) precision = self.precision.reshape(batch_shape + (n, n)) return Gaussian(log_normalizer, info_vec, precision)
[docs] def __getitem__(self, index): """ Index into the batch_shape of a Gaussian. """ assert isinstance(index, tuple) log_normalizer = self.log_normalizer[index] info_vec = self.info_vec[index + (slice(None),)] precision = self.precision[index + (slice(None), slice(None))] return Gaussian(log_normalizer, info_vec, precision)
[docs] @staticmethod def cat(parts, dim=0): """ Concatenate a list of Gaussians along a given batch dimension. """ if dim < 0: dim += len(parts[0].batch_shape) args = [torch.cat([getattr(g, attr) for g in parts], dim=dim) for attr in ["log_normalizer", "info_vec", "precision"]] return Gaussian(*args)
[docs] def event_pad(self, left=0, right=0): """ Pad along event dimension. """ lr = (left, right) log_normalizer = self.log_normalizer info_vec = pad(self.info_vec, lr) precision = pad(self.precision, lr + lr) return Gaussian(log_normalizer, info_vec, precision)
[docs] def event_permute(self, perm): """ Permute along event dimension. """ assert isinstance(perm, torch.Tensor) assert perm.shape == (self.dim(),) info_vec = self.info_vec[..., perm] precision = self.precision[..., perm][..., perm, :] return Gaussian(self.log_normalizer, info_vec, precision)
[docs] def __add__(self, other): """ Adds two Gaussians in log-density space. """ if isinstance(other, Gaussian): assert self.dim() == other.dim() return Gaussian(self.log_normalizer + other.log_normalizer, self.info_vec + other.info_vec, self.precision + other.precision) if isinstance(other, (int, float, torch.Tensor)): return Gaussian(self.log_normalizer + other, self.info_vec, self.precision) raise ValueError("Unsupported type: {}".format(type(other)))
def __sub__(self, other): if isinstance(other, (int, float, torch.Tensor)): return Gaussian(self.log_normalizer - other, self.info_vec, self.precision) raise ValueError("Unsupported type: {}".format(type(other)))
[docs] def log_density(self, value): """ Evaluate the log density of this Gaussian at a point value:: -0.5 * value.T @ precision @ value + value.T @ info_vec + log_normalizer This is mainly used for testing. """ if value.size(-1) == 0: batch_shape = broadcast_shape(value.shape[:-1], self.batch_shape) return self.log_normalizer.expand(batch_shape) result = (-0.5) * self.precision.matmul(value.unsqueeze(-1)).squeeze(-1) result = result + self.info_vec result = (value * result).sum(-1) return result + self.log_normalizer
[docs] def rsample(self, sample_shape=torch.Size()): """ Reparameterized sampler. """ P_chol = self.precision.cholesky() loc = self.info_vec.unsqueeze(-1).cholesky_solve(P_chol).squeeze(-1) shape = sample_shape + self.batch_shape + (self.dim(), 1) noise = torch.randn(shape, dtype=loc.dtype, device=loc.device) noise = noise.triangular_solve(P_chol, upper=False, transpose=True).solution.squeeze(-1) return loc + noise
[docs] def condition(self, value): """ Condition this Gaussian on a trailing subset of its state. This should satisfy:: g.condition(y).dim() == g.dim() - y.size(-1) Note that since this is a non-normalized Gaussian, we include the density of ``y`` in the result. Thus :meth:`condition` is similar to a ``functools.partial`` binding of arguments:: left = x[..., :n] right = x[..., n:] g.log_density(x) == g.condition(right).log_density(left) """ assert isinstance(value, torch.Tensor) assert value.size(-1) <= self.info_vec.size(-1) n = self.dim() - value.size(-1) info_a = self.info_vec[..., :n] info_b = self.info_vec[..., n:] P_aa = self.precision[..., :n, :n] P_ab = self.precision[..., :n, n:] P_bb = self.precision[..., n:, n:] b = value info_vec = info_a - P_ab.matmul(b.unsqueeze(-1)).squeeze(-1) precision = P_aa log_normalizer = (self.log_normalizer + -0.5 * P_bb.matmul(b.unsqueeze(-1)).squeeze(-1).mul(b).sum(-1) + b.mul(info_b).sum(-1)) return Gaussian(log_normalizer, info_vec, precision)
[docs] def marginalize(self, left=0, right=0): """ Marginalizing out variables on either side of the event dimension:: g.marginalize(left=n).event_logsumexp() = g.logsumexp() g.marginalize(right=n).event_logsumexp() = g.logsumexp() and for data ``x``: g.condition(x).event_logsumexp() = g.marginalize(left=g.dim() - x.size(-1)).log_density(x) """ if left == 0 and right == 0: return self if left > 0 and right > 0: raise NotImplementedError n = self.dim() n_b = left + right a = slice(left, n - right) # preserved b = slice(None, left) if left else slice(n - right, None) P_aa = self.precision[..., a, a] P_ba = self.precision[..., b, a] P_bb = self.precision[..., b, b] P_b = P_bb.cholesky() P_a = P_ba.triangular_solve(P_b, upper=False).solution P_at = P_a.transpose(-1, -2) precision = P_aa - P_at.matmul(P_a) info_a = self.info_vec[..., a] info_b = self.info_vec[..., b] b_tmp = info_b.unsqueeze(-1).triangular_solve(P_b, upper=False).solution info_vec = info_a - P_at.matmul(b_tmp).squeeze(-1) log_normalizer = (self.log_normalizer + 0.5 * n_b * math.log(2 * math.pi) - P_b.diagonal(dim1=-2, dim2=-1).log().sum(-1) + 0.5 * b_tmp.squeeze(-1).pow(2).sum(-1)) return Gaussian(log_normalizer, info_vec, precision)
[docs] def event_logsumexp(self): """ Integrates out all latent state (i.e. operating on event dimensions). """ n = self.dim() chol_P = self.precision.cholesky() chol_P_u = self.info_vec.unsqueeze(-1).triangular_solve(chol_P, upper=False).solution.squeeze(-1) u_P_u = chol_P_u.pow(2).sum(-1) return (self.log_normalizer + 0.5 * n * math.log(2 * math.pi) + 0.5 * u_P_u - chol_P.diagonal(dim1=-2, dim2=-1).log().sum(-1))
[docs]class AffineNormal: """ Represents a conditional diagonal normal distribution over a random variable ``Y`` whose mean is an affine function of a random variable ``X``. The likelihood of ``X`` is thus:: AffineNormal(matrix, loc, scale).condition(y).log_density(x) which is equivalent to:: Normal(x @ matrix + loc, scale).to_event(1).log_prob(y) :param torch.Tensor matrix: A transformation from ``X`` to ``Y``. Should have rightmost shape ``(x_dim, y_dim)``. :param torch.Tensor loc: A constant offset for ``Y``'s mean. Should have rightmost shape ``(y_dim,)``. :param torch.Tensor scale: Standard deviation for ``Y``. Should have rightmost shape ``(y_dim,)``. """ def __init__(self, matrix, loc, scale): assert loc.shape == scale.shape self.matrix = matrix self.loc = loc self.scale = scale @property def batch_shape(self): return self.matrix.shape[:-2]
[docs] def condition(self, value): """ Condition on a ``Y`` value. :param torch.Tensor value: A value of ``Y``. :return Gaussian: A gaussian likelihood over ``X``. """ assert value.size(-1) == self.loc.size(-1) prec_sqrt = self.matrix / self.scale.unsqueeze(-2) precision = prec_sqrt.matmul(prec_sqrt.transpose(-1, -2)) delta = (value - self.loc) / self.scale info_vec = prec_sqrt.matmul(delta.unsqueeze(-1)).squeeze(-1) log_normalizer = (-0.5 * self.loc.size(-1) * math.log(2 * math.pi) - 0.5 * delta.pow(2).sum(-1) - self.scale.log().sum(-1)) return Gaussian(log_normalizer, info_vec, precision)
[docs] def to_gaussian(self): mvn = torch.distributions.MultivariateNormal(self.loc, scale_tril=self.scale.diag_embed()) return matrix_and_mvn_to_gaussian(self.matrix, mvn)
[docs] def expand(self, batch_shape): matrix = self.matrix.expand(batch_shape + self.matrix.shape[-2:]) loc = self.loc.expand(batch_shape + self.loc.shape[-1:]) scale = self.scale.expand(batch_shape + self.scale.shape[-1:]) return AffineNormal(matrix, loc, scale)
[docs] def reshape(self, batch_shape): matrix = self.matrix.reshape(batch_shape + self.matrix.shape[-2:]) loc = self.loc.reshape(batch_shape + self.loc.shape[-1:]) scale = self.scale.reshape(batch_shape + self.scale.shape[-1:]) return AffineNormal(matrix, loc, scale)
[docs] def __getitem__(self, index): assert isinstance(index, tuple) matrix = self.matrix[index + (slice(None), slice(None))] loc = self.loc[index + (slice(None),)] scale = self.scale[index + (slice(None),)] return AffineNormal(matrix, loc, scale)
[docs] def event_permute(self, perm): return self.to_gaussian().event_permute(perm)
[docs] def __add__(self, other): return self.to_gaussian() + other
[docs] def marginalize(self, left=0, right=0): return self.to_gaussian().marginalize(left, right)
[docs]def mvn_to_gaussian(mvn): """ Convert a MultivariateNormal distribution to a Gaussian. :param ~torch.distributions.MultivariateNormal mvn: A multivariate normal distribution. :return: An equivalent Gaussian object. :rtype: ~pyro.ops.gaussian.Gaussian """ assert (isinstance(mvn, torch.distributions.MultivariateNormal) or (isinstance(mvn, torch.distributions.Independent) and isinstance(mvn.base_dist, torch.distributions.Normal))) if isinstance(mvn, torch.distributions.Independent): mvn = mvn.base_dist precision_diag = mvn.scale.pow(-2) precision = precision_diag.diag_embed() info_vec = mvn.loc * precision_diag scale_diag = mvn.scale else: precision = mvn.precision_matrix info_vec = precision.matmul(mvn.loc.unsqueeze(-1)).squeeze(-1) scale_diag = mvn.scale_tril.diagonal(dim1=-2, dim2=-1) n = mvn.loc.size(-1) log_normalizer = (-0.5 * n * math.log(2 * math.pi) + -0.5 * (info_vec * mvn.loc).sum(-1) - scale_diag.log().sum(-1)) return Gaussian(log_normalizer, info_vec, precision)
[docs]def matrix_and_mvn_to_gaussian(matrix, mvn): """ Convert a noisy affine function to a Gaussian. The noisy affine function is defined as:: y = x @ matrix + mvn.sample() :param ~torch.Tensor matrix: A matrix with rightmost shape ``(x_dim, y_dim)``. :param ~torch.distributions.MultivariateNormal mvn: A multivariate normal distribution. :return: A Gaussian with broadcasted batch shape and ``.dim() == x_dim + y_dim``. :rtype: ~pyro.ops.gaussian.Gaussian """ assert (isinstance(mvn, torch.distributions.MultivariateNormal) or (isinstance(mvn, torch.distributions.Independent) and isinstance(mvn.base_dist, torch.distributions.Normal))) assert isinstance(matrix, torch.Tensor) x_dim, y_dim = matrix.shape[-2:] assert mvn.event_shape == (y_dim,) batch_shape = broadcast_shape(matrix.shape[:-2], mvn.batch_shape) matrix = matrix.expand(batch_shape + (x_dim, y_dim)) mvn = mvn.expand(batch_shape) # Handle diagonal normal distributions as an efficient special case. if isinstance(mvn, torch.distributions.Independent): return AffineNormal(matrix, mvn.base_dist.loc, mvn.base_dist.scale) y_gaussian = mvn_to_gaussian(mvn) P_yy = y_gaussian.precision neg_P_xy = matrix.matmul(P_yy) P_xy = -neg_P_xy P_yx = P_xy.transpose(-1, -2) P_xx = neg_P_xy.matmul(matrix.transpose(-1, -2)) precision = torch.cat([torch.cat([P_xx, P_xy], -1), torch.cat([P_yx, P_yy], -1)], -2) info_y = y_gaussian.info_vec info_x = -matrix.matmul(info_y.unsqueeze(-1)).squeeze(-1) info_vec = torch.cat([info_x, info_y], -1) log_normalizer = y_gaussian.log_normalizer result = Gaussian(log_normalizer, info_vec, precision) assert result.batch_shape == batch_shape assert result.dim() == x_dim + y_dim return result
[docs]def gaussian_tensordot(x, y, dims=0): """ Computes the integral over two gaussians: `(x @ y)(a,c) = log(integral(exp(x(a,b) + y(b,c)), b))`, where `x` is a gaussian over variables (a,b), `y` is a gaussian over variables (b,c), (a,b,c) can each be sets of zero or more variables, and `dims` is the size of b. :param x: a Gaussian instance :param y: a Gaussian instance :param dims: number of variables to contract """ assert isinstance(x, Gaussian) assert isinstance(y, Gaussian) na = x.dim() - dims nb = dims nc = y.dim() - dims assert na >= 0 assert nb >= 0 assert nc >= 0 Paa, Pba, Pbb = x.precision[..., :na, :na], x.precision[..., na:, :na], x.precision[..., na:, na:] Qbb, Qbc, Qcc = y.precision[..., :nb, :nb], y.precision[..., :nb, nb:], y.precision[..., nb:, nb:] xa, xb = x.info_vec[..., :na], x.info_vec[..., na:] # x.precision @ x.mean yb, yc = y.info_vec[..., :nb], y.info_vec[..., nb:] # y.precision @ y.mean precision = pad(Paa, (0, nc, 0, nc)) + pad(Qcc, (na, 0, na, 0)) info_vec = pad(xa, (0, nc)) + pad(yc, (na, 0)) log_normalizer = x.log_normalizer + y.log_normalizer if nb > 0: B = pad(Pba, (0, nc)) + pad(Qbc, (na, 0)) b = xb + yb # Pbb + Qbb needs to be positive definite, so that we can malginalize out `b` (to have a finite integral) L = torch.cholesky(Pbb + Qbb) LinvB = torch.triangular_solve(B, L, upper=False)[0] LinvBt = LinvB.transpose(-2, -1) Linvb = torch.triangular_solve(b.unsqueeze(-1), L, upper=False)[0] precision = precision - torch.matmul(LinvBt, LinvB) # NB: precision might not be invertible for getting mean = precision^-1 @ info_vec if na + nc > 0: info_vec = info_vec - torch.matmul(LinvBt, Linvb).squeeze(-1) logdet = torch.diagonal(L, dim1=-2, dim2=-1).log().sum(-1) diff = 0.5 * nb * math.log(2 * math.pi) + 0.5 * Linvb.squeeze(-1).pow(2).sum(-1) - logdet log_normalizer = log_normalizer + diff return Gaussian(log_normalizer, info_vec, precision)