tensorflow_probability.python.bijectors的一些使用

网上见到一个TensorFlow的代码，没见过这个形式的，是概率编程的代码：

# coding=utf-8
# Copyright 2020 The TF-Agents Authors.
#
# 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
#
#     https://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.

"""Tanh bijector."""

from __future__ import absolute_import
from __future__ import division
from __future__ import print_function

import tensorflow as tf  # pylint: disable=g-explicit-tensorflow-version-import
from tensorflow_probability.python.bijectors import bijector

__all__ = [
    "Tanh",
]

class Tanh(bijector.Bijector):
  """Bijector that computes `Y = tanh(X)`, therefore `Y in (-1, 1)`.

  This can be achieved by an affine transform of the Sigmoid bijector, i.e.,
  it is equivalent to
  ```
  tfb.Chain([tfb.Affine(shift=-1, scale=2.),
             tfb.Sigmoid(),
             tfb.Affine(scale=2.)])
  ```

  However, using the `Tanh` bijector directly is slightly faster and more
  numerically stable.
  """

  def __init__(self, validate_args=False, name="tanh"):
    parameters = dict(locals())
    super(Tanh, self).__init__(
        forward_min_event_ndims=0,
        validate_args=validate_args,
        parameters=parameters,
        name=name)

  def _forward(self, x):
    return tf.nn.tanh(x)

  def _inverse(self, y):
    # 0.99999997 is the maximum value such that atanh(x) is valid for both
    # tf.float32 and tf.float64
    y = tf.where(tf.less_equal(tf.abs(y), 1.),
                 tf.clip_by_value(y, -0.99999997, 0.99999997),
                 y)
    return tf.atanh(y)

  def _forward_log_det_jacobian(self, x):
    #  This formula is mathematically equivalent to
    #  `tf.log1p(-tf.square(tf.tanh(x)))`, however this code is more numerically
    #  stable.

    #  Derivation:
    #    log(1 - tanh(x)^2)
    #    = log(sech(x)^2)
    #    = 2 * log(sech(x))
    #    = 2 * log(2e^-x / (e^-2x + 1))
    #    = 2 * (log(2) - x - log(e^-2x + 1))
    #    = 2 * (log(2) - x - softplus(-2x))
    return 2.0 * (
        tf.math.log(tf.constant(2.0, dtype=x.dtype)) - x - tf.nn.softplus(
            -2.0 * x))

================================================

由于不是很理解这个代码的意思，于是找了下TensorFlow的官方文档：

https://tensorflow.google.cn/probability/api_docs/python/tfp/bijectors/Bijector

  class Exp(Bijector):

    def __init__(self, validate_args=False, name='exp'):
      super(Exp, self).__init__(
          validate_args=validate_args,
          forward_min_event_ndims=0,
          name=name)

    def _forward(self, x):
      return tf.exp(x)

    def _inverse(self, y):
      return tf.log(y)

    def _inverse_log_det_jacobian(self, y):
      return -self._forward_log_det_jacobian(self._inverse(y))

    def _forward_log_det_jacobian(self, x):
      # Notice that we needn't do any reducing, even when`event_ndims > 0`.
      # The base Bijector class will handle reducing for us; it knows how
      # to do so because we called `super` `__init__` with
      # `forward_min_event_ndims = 0`.
      return x
  ```

根据文档内容可以知道，这个bijectors是双向映射，也就是说知道g(X)=Y，知道X的概率分布及概率密度，现在要求Y的概率密度。上面代码中_forward函数和_inverse函数的含义比较好理解，也就是原函数与反函数，但是这个_forward_log_det_jacobian函数是什么意思就不是很好理解了，这里也就是说下个人的理解，不保证正确：

_forward_log_det_jacobian函数的输入变量为x，其函数需要返回的就是_forward函数的导数的log值，也就是log( _forward(x)导数 )，由于上面代码的_forward函数为tf.exp(x)，因此_forward(x)的导数也为tf.exp(x)，log( _forward(x)导数则为tf.log(tf.exp(x))=x，因此_forward_log_det_jacobian函数的输出已经为x。

--------------------------------------

例子：

class Identity(Bijector):

  def __init__(self, validate_args=False, name='identity'):
    super(Identity, self).__init__(
        is_constant_jacobian=True,
        validate_args=validate_args,
        forward_min_event_ndims=0,
        name=name)

  def _forward(self, x):
    return x

  def _inverse(self, y):
    return y

  def _inverse_log_det_jacobian(self, y):
    return -self._forward_log_det_jacobian(self._inverse(y))

  def _forward_log_det_jacobian(self, x):
    # The full log jacobian determinant would be tf.zero_like(x).
    # However, we circumvent materializing that, since the jacobian
    # calculation is input independent, and we specify it for one input.
    return tf.constant(0., x.dtype)

由于_forward(x)为返回值为x，因此_forward(x)导数为1，tf.log(tf.exp(x))=0.0 。

====================================

我们定义好Bijector的子类对象及其中的_forward函数、_inverse函数、_forward_log_det_jacobian函数，这样就可以通过双射函数一端的概率分布求得另一端的概率分布，比如g(X)=Y，我们知道X的概率分布也就能求得Y的概率密度。

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官方文档：

https://tensorflow.google.cn/probability/api_docs/python/tfp/bijectors/Bijector