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# Source code for torch.distributions.multivariate_normal

# mypy: allow-untyped-defs
import math

import torch
from torch.distributions import constraints
from torch.distributions.distribution import Distribution
from torch.distributions.utils import _standard_normal, lazy_property

__all__ = ["MultivariateNormal"]

def _batch_mv(bmat, bvec):
r"""
Performs a batched matrix-vector product, with compatible but different batch shapes.

This function takes as input bmat, containing :math:n \times n matrices, and
bvec, containing length :math:n vectors.

Both bmat and bvec may have any number of leading dimensions, which correspond
to a batch shape. They are not necessarily assumed to have the same batch shape,
just ones which can be broadcasted.
"""

def _batch_mahalanobis(bL, bx):
r"""
Computes the squared Mahalanobis distance :math:\mathbf{x}^\top\mathbf{M}^{-1}\mathbf{x}
for a factored :math:\mathbf{M} = \mathbf{L}\mathbf{L}^\top.

Accepts batches for both bL and bx. They are not necessarily assumed to have the same batch
shape, but bL one should be able to broadcasted to bx one.
"""
n = bx.size(-1)
bx_batch_shape = bx.shape[:-1]

# Assume that bL.shape = (i, 1, n, n), bx.shape = (..., i, j, n),
# we are going to make bx have shape (..., 1, j,  i, 1, n) to apply batched tri.solve
bx_batch_dims = len(bx_batch_shape)
bL_batch_dims = bL.dim() - 2
outer_batch_dims = bx_batch_dims - bL_batch_dims
old_batch_dims = outer_batch_dims + bL_batch_dims
new_batch_dims = outer_batch_dims + 2 * bL_batch_dims
# Reshape bx with the shape (..., 1, i, j, 1, n)
bx_new_shape = bx.shape[:outer_batch_dims]
for sL, sx in zip(bL.shape[:-2], bx.shape[outer_batch_dims:-1]):
bx_new_shape += (sx // sL, sL)
bx_new_shape += (n,)
bx = bx.reshape(bx_new_shape)
# Permute bx to make it have shape (..., 1, j, i, 1, n)
permute_dims = (
list(range(outer_batch_dims))
+ list(range(outer_batch_dims, new_batch_dims, 2))
+ list(range(outer_batch_dims + 1, new_batch_dims, 2))
+ [new_batch_dims]
)
bx = bx.permute(permute_dims)

flat_L = bL.reshape(-1, n, n)  # shape = b x n x n
flat_x = bx.reshape(-1, flat_L.size(0), n)  # shape = c x b x n
flat_x_swap = flat_x.permute(1, 2, 0)  # shape = b x n x c
M_swap = (
torch.linalg.solve_triangular(flat_L, flat_x_swap, upper=False).pow(2).sum(-2)
)  # shape = b x c
M = M_swap.t()  # shape = c x b

# Now we revert the above reshape and permute operators.
permuted_M = M.reshape(bx.shape[:-1])  # shape = (..., 1, j, i, 1)
permute_inv_dims = list(range(outer_batch_dims))
for i in range(bL_batch_dims):
permute_inv_dims += [outer_batch_dims + i, old_batch_dims + i]
reshaped_M = permuted_M.permute(permute_inv_dims)  # shape = (..., 1, i, j, 1)
return reshaped_M.reshape(bx_batch_shape)

def _precision_to_scale_tril(P):
# Ref: https://nbviewer.jupyter.org/gist/fehiepsi/5ef8e09e61604f10607380467eb82006#Precision-to-scale_tril
Lf = torch.linalg.cholesky(torch.flip(P, (-2, -1)))
L_inv = torch.transpose(torch.flip(Lf, (-2, -1)), -2, -1)
Id = torch.eye(P.shape[-1], dtype=P.dtype, device=P.device)
L = torch.linalg.solve_triangular(L_inv, Id, upper=False)
return L

[docs]class MultivariateNormal(Distribution):
r"""
Creates a multivariate normal (also called Gaussian) distribution
parameterized by a mean vector and a covariance matrix.

The multivariate normal distribution can be parameterized either
in terms of a positive definite covariance matrix :math:\mathbf{\Sigma}
or a positive definite precision matrix :math:\mathbf{\Sigma}^{-1}
or a lower-triangular matrix :math:\mathbf{L} with positive-valued
diagonal entries, such that
:math:\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^\top. This triangular matrix
can be obtained via e.g. Cholesky decomposition of the covariance.

Example:

>>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_LAPACK)
>>> # xdoctest: +IGNORE_WANT("non-deterministic")
>>> m = MultivariateNormal(torch.zeros(2), torch.eye(2))
>>> m.sample()  # normally distributed with mean=[0,0] and covariance_matrix=I
tensor([-0.2102, -0.5429])

Args:
loc (Tensor): mean of the distribution
covariance_matrix (Tensor): positive-definite covariance matrix
precision_matrix (Tensor): positive-definite precision matrix
scale_tril (Tensor): lower-triangular factor of covariance, with positive-valued diagonal

Note:
Only one of :attr:covariance_matrix or :attr:precision_matrix or
:attr:scale_tril can be specified.

Using :attr:scale_tril will be more efficient: all computations internally
are based on :attr:scale_tril. If :attr:covariance_matrix or
:attr:precision_matrix is passed instead, it is only used to compute
the corresponding lower triangular matrices using a Cholesky decomposition.
"""
arg_constraints = {
"loc": constraints.real_vector,
"covariance_matrix": constraints.positive_definite,
"precision_matrix": constraints.positive_definite,
"scale_tril": constraints.lower_cholesky,
}
support = constraints.real_vector
has_rsample = True

def __init__(
self,
loc,
covariance_matrix=None,
precision_matrix=None,
scale_tril=None,
validate_args=None,
):
if loc.dim() < 1:
raise ValueError("loc must be at least one-dimensional.")
if (covariance_matrix is not None) + (scale_tril is not None) + (
precision_matrix is not None
) != 1:
raise ValueError(
"Exactly one of covariance_matrix or precision_matrix or scale_tril may be specified."
)

if scale_tril is not None:
if scale_tril.dim() < 2:
raise ValueError(
"scale_tril matrix must be at least two-dimensional, "
)
self.scale_tril = scale_tril.expand(batch_shape + (-1, -1))
elif covariance_matrix is not None:
if covariance_matrix.dim() < 2:
raise ValueError(
"covariance_matrix must be at least two-dimensional, "
)
covariance_matrix.shape[:-2], loc.shape[:-1]
)
self.covariance_matrix = covariance_matrix.expand(batch_shape + (-1, -1))
else:
if precision_matrix.dim() < 2:
raise ValueError(
"precision_matrix must be at least two-dimensional, "
)
precision_matrix.shape[:-2], loc.shape[:-1]
)
self.precision_matrix = precision_matrix.expand(batch_shape + (-1, -1))
self.loc = loc.expand(batch_shape + (-1,))

event_shape = self.loc.shape[-1:]
super().__init__(batch_shape, event_shape, validate_args=validate_args)

if scale_tril is not None:
elif covariance_matrix is not None:
else:  # precision_matrix is not None

[docs]    def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(MultivariateNormal, _instance)
batch_shape = torch.Size(batch_shape)
loc_shape = batch_shape + self.event_shape
cov_shape = batch_shape + self.event_shape + self.event_shape
new.loc = self.loc.expand(loc_shape)
if "covariance_matrix" in self.__dict__:
new.covariance_matrix = self.covariance_matrix.expand(cov_shape)
if "scale_tril" in self.__dict__:
new.scale_tril = self.scale_tril.expand(cov_shape)
if "precision_matrix" in self.__dict__:
new.precision_matrix = self.precision_matrix.expand(cov_shape)
super(MultivariateNormal, new).__init__(
batch_shape, self.event_shape, validate_args=False
)
new._validate_args = self._validate_args
return new

@lazy_property
def scale_tril(self):
self._batch_shape + self._event_shape + self._event_shape
)

@lazy_property
def covariance_matrix(self):
).expand(self._batch_shape + self._event_shape + self._event_shape)

@lazy_property
def precision_matrix(self):
self._batch_shape + self._event_shape + self._event_shape
)

@property
def mean(self):
return self.loc

@property
def mode(self):
return self.loc

@property
def variance(self):
return (
.sum(-1)
.expand(self._batch_shape + self._event_shape)
)

[docs]    def rsample(self, sample_shape=torch.Size()):
shape = self._extended_shape(sample_shape)
eps = _standard_normal(shape, dtype=self.loc.dtype, device=self.loc.device)

[docs]    def log_prob(self, value):
if self._validate_args:
self._validate_sample(value)
diff = value - self.loc
half_log_det = (
)
return -0.5 * (self._event_shape[0] * math.log(2 * math.pi) + M) - half_log_det

[docs]    def entropy(self):
half_log_det = (
)
H = 0.5 * self._event_shape[0] * (1.0 + math.log(2 * math.pi)) + half_log_det
if len(self._batch_shape) == 0:
return H
else:
return H.expand(self._batch_shape)


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