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mvn_tril.py
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mvn_tril.py
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# Copyright 2016 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 classes."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from tensorflow.contrib import linalg
from tensorflow.contrib.distributions.python.ops import mvn_linear_operator as mvn_linop
from tensorflow.python.framework import ops
from tensorflow.python.ops.distributions import util as distribution_util
__all__ = [
"MultivariateNormalTriL",
]
class MultivariateNormalTriL(
mvn_linop.MultivariateNormalLinearOperator):
"""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`
`scale` matrix; `covariance = scale @ scale.T` where `@` denotes
matrix-multiplication.
#### Mathematical Details
The probability density function (pdf) is,
```none
pdf(x; loc, scale) = exp(-0.5 ||y||**2) / Z,
y = inv(scale) @ (x - loc),
Z = (2 pi)**(0.5 k) |det(scale)|,
```
where:
* `loc` is a vector in `R^k`,
* `scale` is a matrix in `R^{k x k}`, `covariance = scale @ scale.T`,
* `Z` denotes the normalization constant, and,
* `||y||**2` denotes the squared Euclidean norm of `y`.
A (non-batch) `scale` matrix is:
```none
scale = scale_tril
```
where `scale_tril` is lower-triangular `k x k` matrix with non-zero diagonal,
i.e., `tf.diag_part(scale_tril) != 0`.
Additional leading dimensions (if any) will index batches.
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 as,
```none
X ~ MultivariateNormal(loc=0, scale=1) # Identity scale, zero shift.
Y = scale @ X + loc
```
Trainable (batch) lower-triangular matrices can be created with
`tf.contrib.distributions.matrix_diag_transform()` and/or
`tf.contrib.distributions.fill_triangular()`
#### Examples
```python
tfd = 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]]
scale = tf.cholesky(cov)
# ==> [[ 0.6, 0. , 0. ],
# [ 0.2, 0.5, 0. ],
# [ 0.1, -0.3, 0.4]])
mvn = tfd.MultivariateNormalTriL(
loc=mu,
scale_tril=scale)
mvn.mean().eval()
# ==> [1., 2, 3]
# Covariance agrees with cholesky(cov) parameterization.
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]
tril = ... # shape: [2, 3, 3], lower triangular, non-zero diagonal.
mvn = tfd.MultivariateNormalTriL(
loc=mu,
scale_tril=tril)
# 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]
# Instantiate a "learnable" MVN.
dims = 4
with tf.variable_scope("model"):
mvn = tfd.MultivariateNormalTriL(
loc=tf.get_variable(shape=[dims], dtype=tf.float32, name="mu"),
scale_tril=tfd.fill_triangular(
tf.get_variable(shape=[dims * (dims + 1) / 2],
dtype=tf.float32, name="chol_Sigma")))
```
"""
def __init__(self,
loc=None,
scale_tril=None,
validate_args=False,
allow_nan_stats=True,
name="MultivariateNormalTriL"):
"""Construct Multivariate Normal distribution on `R^k`.
The `batch_shape` is the broadcast shape between `loc` and `scale`
arguments.
The `event_shape` is given by last dimension of the matrix implied by
`scale`. The last dimension of `loc` (if provided) must broadcast with this.
Recall that `covariance = scale @ scale.T`. A (non-batch) `scale` matrix is:
```none
scale = scale_tril
```
where `scale_tril` is lower-triangular `k x k` matrix with non-zero
diagonal, i.e., `tf.diag_part(scale_tril) != 0`.
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.
scale_tril: Floating-point, lower-triangular `Tensor` with non-zero
diagonal elements. `scale_tril` 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 `scale_tril` are specified.
"""
parameters = locals()
def _convert_to_tensor(x, name):
return None if x is None else ops.convert_to_tensor(x, name=name)
if loc is None and scale_tril is None:
raise ValueError("Must specify one or both of `loc`, `scale_tril`.")
with ops.name_scope(name):
with ops.name_scope("init", values=[loc, scale_tril]):
loc = _convert_to_tensor(loc, name="loc")
scale_tril = _convert_to_tensor(scale_tril, name="scale_tril")
if scale_tril is None:
scale = linalg.LinearOperatorIdentity(
num_rows=distribution_util.dimension_size(loc, -1),
dtype=loc.dtype,
is_self_adjoint=True,
is_positive_definite=True,
assert_proper_shapes=validate_args)
else:
# No need to validate that scale_tril is non-singular.
# LinearOperatorLowerTriangular has an assert_non_singular
# method that is called by the Bijector.
scale = linalg.LinearOperatorLowerTriangular(
scale_tril,
is_non_singular=True,
is_self_adjoint=False,
is_positive_definite=False)
super(MultivariateNormalTriL, self).__init__(
loc=loc,
scale=scale,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
name=name)
self._parameters = parameters