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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.
# ==============================================================================
"""Distribution of a vectorized Laplace, with uncorrelated components."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from tensorflow.contrib.distributions.python.ops import distribution_util
from tensorflow.contrib.distributions.python.ops import vector_laplace_linear_operator as vector_laplace_linop
from tensorflow.python.framework import ops
__all__ = [
"VectorLaplaceDiag",
]
class VectorLaplaceDiag(
vector_laplace_linop.VectorLaplaceLinearOperator):
"""The vectorization of the Laplace distribution on `R^k`.
The vector laplace distribution is defined over `R^k`, and parameterized by
a (batch of) length-`k` `loc` vector (the means) and a (batch of) `k x k`
`scale` matrix: `covariance = 2 * scale @ scale.T`, where `@` denotes
matrix-multiplication.
#### Mathematical Details
The probability density function (pdf) is,
```none
pdf(x; loc, scale) = exp(-||y||_1) / Z,
y = inv(scale) @ (x - loc),
Z = 2**k |det(scale)|,
```
where:
* `loc` is a vector in `R^k`,
* `scale` is a linear operator in `R^{k x k}`, `cov = scale @ scale.T`,
* `Z` denotes the normalization constant, and,
* `||y||_1` denotes the `l1` norm of `y`, `sum_i |y_i|.
A (non-batch) `scale` matrix is:
```none
scale = diag(scale_diag + scale_identity_multiplier * ones(k))
```
where:
* `scale_diag.shape = [k]`, and,
* `scale_identity_multiplier.shape = []`.
Additional leading dimensions (if any) will index batches.
If both `scale_diag` and `scale_identity_multiplier` are `None`, then
`scale` is the Identity matrix.
The VectorLaplace 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 = (X_1, ..., X_k), each X_i ~ Laplace(loc=0, scale=1)
Y = (Y_1, ...,Y_k) = scale @ X + loc
```
#### About `VectorLaplace` and `Vector` distributions in TensorFlow.
The `VectorLaplace` is a non-standard distribution that has useful properties.
The marginals `Y_1, ..., Y_k` are *not* Laplace random variables, due to
the fact that the sum of Laplace random variables is not Laplace.
Instead, `Y` is a vector whose components are linear combinations of Laplace
random variables. Thus, `Y` lives in the vector space generated by `vectors`
of Laplace distributions. This allows the user to decide the mean and
covariance (by setting `loc` and `scale`), while preserving some properties of
the Laplace distribution. In particular, the tails of `Y_i` will be (up to
polynomial factors) exponentially decaying.
To see this last statement, note that the pdf of `Y_i` is the convolution of
the pdf of `k` independent Laplace random variables. One can then show by
induction that distributions with exponential (up to polynomial factors) tails
are closed under convolution.
#### Examples
```python
tfd = tf.contrib.distributions
# Initialize a single 2-variate VectorLaplace.
vla = tfd.VectorLaplaceDiag(
loc=[1., -1],
scale_diag=[1, 2.])
vla.mean().eval()
# ==> [1., -1]
vla.stddev().eval()
# ==> [1., 2] * sqrt(2)
# Evaluate this on an observation in `R^2`, returning a scalar.
vla.prob([-1., 0]).eval() # shape: []
# Initialize a 3-batch, 2-variate scaled-identity VectorLaplace.
vla = tfd.VectorLaplaceDiag(
loc=[1., -1],
scale_identity_multiplier=[1, 2., 3])
vla.mean().eval() # shape: [3, 2]
# ==> [[1., -1]
# [1, -1],
# [1, -1]]
vla.stddev().eval() # shape: [3, 2]
# ==> sqrt(2) * [[1., 1],
# [2, 2],
# [3, 3]]
# Evaluate this on an observation in `R^2`, returning a length-3 vector.
vla.prob([-1., 0]).eval() # shape: [3]
# Initialize a 2-batch of 3-variate VectorLaplace's.
vla = tfd.VectorLaplaceDiag(
loc=[[1., 2, 3],
[11, 22, 33]] # shape: [2, 3]
scale_diag=[[1., 2, 3],
[0.5, 1, 1.5]]) # shape: [2, 3]
# Evaluate this on a two observations, each in `R^3`, returning a length-2
# vector.
x = [[-1., 0, 1],
[-11, 0, 11.]] # shape: [2, 3].
vla.prob(x).eval() # shape: [2]
```
"""
def __init__(self,
loc=None,
scale_diag=None,
scale_identity_multiplier=None,
validate_args=False,
allow_nan_stats=True,
name="VectorLaplaceDiag"):
"""Construct Vector Laplace 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 = 2 * scale @ scale.T`.
```none
scale = diag(scale_diag + scale_identity_multiplier * ones(k))
```
where:
* `scale_diag.shape = [k]`, and,
* `scale_identity_multiplier.shape = []`.
Additional leading dimensions (if any) will index batches.
If both `scale_diag` and `scale_identity_multiplier` are `None`, then
`scale` is the Identity matrix.
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_diag: Non-zero, floating-point `Tensor` representing a diagonal
matrix added to `scale`. May have shape `[B1, ..., Bb, k]`, `b >= 0`,
and characterizes `b`-batches of `k x k` diagonal matrices added to
`scale`. When both `scale_identity_multiplier` and `scale_diag` are
`None` then `scale` is the `Identity`.
scale_identity_multiplier: Non-zero, floating-point `Tensor` representing
a scaled-identity-matrix added to `scale`. May have shape
`[B1, ..., Bb]`, `b >= 0`, and characterizes `b`-batches of scaled
`k x k` identity matrices added to `scale`. When both
`scale_identity_multiplier` and `scale_diag` are `None` then `scale` is
the `Identity`.
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 at most `scale_identity_multiplier` is specified.
"""
parameters = distribution_util.parent_frame_arguments()
with ops.name_scope(name):
with ops.name_scope("init", values=[
loc, scale_diag, scale_identity_multiplier]):
# No need to validate_args while making diag_scale. The returned
# LinearOperatorDiag has an assert_non_singular method that is called by
# the Bijector.
scale = distribution_util.make_diag_scale(
loc=loc,
scale_diag=scale_diag,
scale_identity_multiplier=scale_identity_multiplier,
validate_args=False,
assert_positive=False)
super(VectorLaplaceDiag, self).__init__(
loc=loc,
scale=scale,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
name=name)
self._parameters = parameters
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