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mvn_full_covariance.py
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mvn_full_covariance.py
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# Copyright 2017 The TensorFlow Authors. All Rights Reserved.
#
# 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.
# ==============================================================================
"""Multivariate Normal distribution class initialized with a full covariance."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from tensorflow.contrib.distributions.python.ops import mvn_tril
from tensorflow.python.framework import ops
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import check_ops
from tensorflow.python.ops import control_flow_ops
from tensorflow.python.ops import linalg_ops
__all__ = [
"MultivariateNormalFullCovariance",
]
class MultivariateNormalFullCovariance(mvn_tril.MultivariateNormalTriL):
"""The multivariate normal distribution on `R^k`.
The Multivariate Normal distribution is defined over `R^k` and parameterized
by a (batch of) length-`k` `loc` vector (aka "mu") and a (batch of) `k x k`
`covariance_matrix` matrices that are the covariance.
This is different than the other multivariate normals, which are parameterized
by a matrix more akin to the standard deviation.
#### Mathematical Details
The probability density function (pdf) is, with `@` as matrix multiplication,
```none
pdf(x; loc, covariance_matrix) = exp(-0.5 ||y||**2) / Z,
y = (x - loc)^T @ inv(covariance_matrix) @ (x - loc)
Z = (2 pi)**(0.5 k) |det(covariance_matrix)|**(0.5).
```
where:
* `loc` is a vector in `R^k`,
* `covariance_matrix` is an `R^{k x k}` symmetric positive definite matrix,
* `Z` denotes the normalization constant, and,
* `||y||**2` denotes the squared Euclidean norm of `y`.
Additional leading dimensions (if any) in `loc` and `covariance_matrix` allow
for batch dimensions.
The MultivariateNormal distribution is a member of the [location-scale
family](https://en.wikipedia.org/wiki/Location-scale_family), i.e., it can be
constructed e.g. as,
```none
X ~ MultivariateNormal(loc=0, scale=1) # Identity scale, zero shift.
scale = Cholesky(covariance_matrix)
Y = scale @ X + loc
```
#### Examples
```python
ds = tf.contrib.distributions
# Initialize a single 3-variate Gaussian.
mu = [1., 2, 3]
cov = [[ 0.36, 0.12, 0.06],
[ 0.12, 0.29, -0.13],
[ 0.06, -0.13, 0.26]]
mvn = ds.MultivariateNormalFullCovariance(
loc=mu,
covariance_matrix=cov)
mvn.mean().eval()
# ==> [1., 2, 3]
# Covariance agrees with covariance_matrix.
mvn.covariance().eval()
# ==> [[ 0.36, 0.12, 0.06],
# [ 0.12, 0.29, -0.13],
# [ 0.06, -0.13, 0.26]]
# Compute the pdf of an observation in `R^3` ; return a scalar.
mvn.prob([-1., 0, 1]).eval() # shape: []
# Initialize a 2-batch of 3-variate Gaussians.
mu = [[1., 2, 3],
[11, 22, 33]] # shape: [2, 3]
covariance_matrix = ... # shape: [2, 3, 3], symmetric, positive definite.
mvn = ds.MultivariateNormalFullCovariance(
loc=mu,
covariance=covariance_matrix)
# Compute the pdf of two `R^3` observations; return a length-2 vector.
x = [[-0.9, 0, 0.1],
[-10, 0, 9]] # shape: [2, 3]
mvn.prob(x).eval() # shape: [2]
```
"""
def __init__(self,
loc=None,
covariance_matrix=None,
validate_args=False,
allow_nan_stats=True,
name="MultivariateNormalFullCovariance"):
"""Construct Multivariate Normal distribution on `R^k`.
The `batch_shape` is the broadcast shape between `loc` and
`covariance_matrix` arguments.
The `event_shape` is given by last dimension of the matrix implied by
`covariance_matrix`. The last dimension of `loc` (if provided) must
broadcast with this.
A non-batch `covariance_matrix` matrix is a `k x k` symmetric positive
definite matrix. In other words it is (real) symmetric with all eigenvalues
strictly positive.
Additional leading dimensions (if any) will index batches.
Args:
loc: Floating-point `Tensor`. If this is set to `None`, `loc` is
implicitly `0`. When specified, may have shape `[B1, ..., Bb, k]` where
`b >= 0` and `k` is the event size.
covariance_matrix: Floating-point, symmetric positive definite `Tensor` of
same `dtype` as `loc`. The strict upper triangle of `covariance_matrix`
is ignored, so if `covariance_matrix` is not symmetric no error will be
raised (unless `validate_args is True`). `covariance_matrix` has shape
`[B1, ..., Bb, k, k]` where `b >= 0` and `k` is the event size.
validate_args: Python `bool`, default `False`. When `True` distribution
parameters are checked for validity despite possibly degrading runtime
performance. When `False` invalid inputs may silently render incorrect
outputs.
allow_nan_stats: Python `bool`, default `True`. When `True`,
statistics (e.g., mean, mode, variance) use the value "`NaN`" to
indicate the result is undefined. When `False`, an exception is raised
if one or more of the statistic's batch members are undefined.
name: Python `str` name prefixed to Ops created by this class.
Raises:
ValueError: if neither `loc` nor `covariance_matrix` are specified.
"""
parameters = locals()
# Convert the covariance_matrix up to a scale_tril and call MVNTriL.
with ops.name_scope(name):
with ops.name_scope("init", values=[loc, covariance_matrix]):
if covariance_matrix is None:
scale_tril = None
else:
covariance_matrix = ops.convert_to_tensor(
covariance_matrix, name="covariance_matrix")
if validate_args:
assert_symmetric = check_ops.assert_equal(
covariance_matrix,
array_ops.matrix_transpose(covariance_matrix),
message="Matrix was not symmetric.")
covariance_matrix = control_flow_ops.with_dependencies(
[assert_symmetric], covariance_matrix)
# No need to validate that covariance_matrix is non-singular.
# LinearOperatorTriL has an assert_non_singular method that is called
# by the Bijector.
# However, cholesky() ignores the upper triangular part, so we do need
# to separately assert symmetric.
scale_tril = linalg_ops.cholesky(covariance_matrix)
super(MultivariateNormalFullCovariance, self).__init__(
loc=loc,
scale_tril=scale_tril,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats)
self._parameters = parameters