# Copyright 2018 The TensorFlow Probability 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
#
#     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.
# ============================================================================
"""The Half Normal distribution class."""

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

import math

# Dependency imports
import numpy as np
from tensorflow_probability.python.internal.backend.numpy.compat import v2 as tf

from tensorflow_probability.substrates.numpy.bijectors import softplus as softplus_bijector
from tensorflow_probability.substrates.numpy.distributions import distribution
from tensorflow_probability.substrates.numpy.distributions import kullback_leibler
from tensorflow_probability.substrates.numpy.internal import assert_util
from tensorflow_probability.substrates.numpy.internal import dtype_util
from tensorflow_probability.substrates.numpy.internal import parameter_properties
from tensorflow_probability.substrates.numpy.internal import prefer_static as ps
from tensorflow_probability.python.internal import reparameterization
from tensorflow_probability.substrates.numpy.internal import samplers
from tensorflow_probability.substrates.numpy.internal import tensor_util


__all__ = [
    'HalfNormal',
]


class HalfNormal(distribution.Distribution):
  """The Half Normal distribution with scale `scale`.

  #### Mathematical details

  The half normal is a transformation of a centered normal distribution.
  If some random variable `X` has normal distribution,
  ```none
  X ~ Normal(0.0, scale)
  Y = |X|
  ```
  Then `Y` will have half normal distribution. The probability density
  function (pdf) is:

  ```none
  pdf(x; scale, x > 0) = sqrt(2) / (scale * sqrt(pi)) *
    exp(- 1/2 * (x / scale) ** 2)
  )
  ```
  Where `scale = sigma` is the standard deviation of the underlying normal
  distribution.

  #### Examples

  Examples of initialization of one or a batch of distributions.

  ```python
  # Define a single scalar HalfNormal distribution.
  dist = tfp.distributions.HalfNormal(scale=3.0)

  # Evaluate the cdf at 1, returning a scalar.
  dist.cdf(1.)

  # Define a batch of two scalar valued HalfNormals.
  # The first has scale 11.0, the second 22.0
  dist = tfp.distributions.HalfNormal(scale=[11.0, 22.0])

  # Evaluate the pdf of the first distribution on 1.0, and the second on 1.5,
  # returning a length two tensor.
  dist.prob([1.0, 1.5])

  # Get 3 samples, returning a 3 x 2 tensor.
  dist.sample([3])
  ```

  """

  def __init__(self,
               scale,
               validate_args=False,
               allow_nan_stats=True,
               name='HalfNormal'):
    """Construct HalfNormals with scale `scale`.

    Args:
      scale: Floating point tensor; the scales of the distribution(s).
        Must contain only positive values.
      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.
    """
    parameters = dict(locals())
    with tf.name_scope(name) as name:
      dtype = dtype_util.common_dtype([scale], dtype_hint=tf.float32)
      self._scale = tensor_util.convert_nonref_to_tensor(
          scale, name='scale', dtype=dtype)
      super(HalfNormal, self).__init__(
          dtype=dtype,
          reparameterization_type=reparameterization.FULLY_REPARAMETERIZED,
          validate_args=validate_args,
          allow_nan_stats=allow_nan_stats,
          parameters=parameters,
          name=name)

  @classmethod
  def _parameter_properties(cls, dtype, num_classes=None):
    # pylint: disable=g-long-lambda
    return dict(
        scale=parameter_properties.ParameterProperties(
            default_constraining_bijector_fn=(
                lambda: softplus_bijector.Softplus(low=dtype_util.eps(dtype)))))
    # pylint: enable=g-long-lambda

  @property
  def scale(self):
    """Distribution parameter for the scale."""
    return self._scale

  def _batch_shape_tensor(self):
    return ps.shape(self.scale)

  def _batch_shape(self):
    return self.scale.shape

  def _event_shape_tensor(self):
    return tf.constant([], dtype=tf.int32)

  def _event_shape(self):
    return tf.TensorShape([])

  def _sample_n(self, n, seed=None):
    scale = tf.convert_to_tensor(self.scale)
    shape = ps.concat([[n], ps.shape(scale)], 0)
    sampled = samplers.normal(
        shape=shape, mean=0., stddev=1., dtype=self.dtype, seed=seed)
    return tf.abs(sampled * scale)

  def _log_prob(self, x):
    scale = tf.convert_to_tensor(self.scale)
    log_unnormalized = -0.5 * (x / scale)**2
    log_normalization = tf.math.log(scale) + tf.constant(0.5 * np.log(np.pi/2.),
                                                         dtype=self.dtype)
    return tf.where(x >= 0,
                    log_unnormalized - log_normalization,
                    tf.constant(-np.inf, dtype=self.dtype))

  def _cdf(self, x):
    truncated_x = tf.nn.relu(x)
    return tf.math.erf(truncated_x / self.scale / np.sqrt(2.0))

  def _survival_function(self, x):
    truncated_x = tf.nn.relu(x)
    return tf.math.erfc(truncated_x / self.scale / np.sqrt(2.0))

  def _log_survival_function(self, x):
    return tf.math.log(self._survival_function(x))

  def _entropy(self):
    return 0.5 * tf.math.log(np.pi * self.scale**2.0 / 2.0) + 0.5

  def _mean(self):
    return self.scale * math.sqrt(2.0) / math.sqrt(np.pi)

  def _quantile(self, p):
    return math.sqrt(2.0) * self.scale * tf.math.erfinv(p)

  def _mode(self):
    return tf.zeros(self.batch_shape_tensor())

  def _variance(self):
    return self.scale ** 2.0 * (1.0 - 2.0 / np.pi)

  def _default_event_space_bijector(self):
    return softplus_bijector.Softplus(validate_args=self.validate_args)

  def _parameter_control_dependencies(self, is_init):
    if not self.validate_args:
      return []
    assertions = []
    if is_init != tensor_util.is_ref(self._scale):
      assertions.append(assert_util.assert_positive(
          self._scale, message='Argument `scale` must be positive.'))
    return assertions

  def _sample_control_dependencies(self, x):
    """Checks the validity of a sample."""
    assertions = []
    if not self.validate_args:
      return assertions
    assertions.append(assert_util.assert_non_negative(
        x, message='Sample must be non-negative.'))
    return assertions


@kullback_leibler.RegisterKL(HalfNormal, HalfNormal)
def _kl_half_normal_half_normal(a, b, name=None):
  """Calculate the batched KL divergence KL(a || b) with a and b `HalfNormal`.

  Args:
    a: Instance of a `HalfNormal` distribution object.
    b: Instance of a `HalfNormal` distribution object.
    name: Python `str` name to use for created operations.
      Default value: `None` (i.e., `'kl_half_normal_half_normal'`).

  Returns:
    kl_div: Batchwise KL(a || b)
  """
  with tf.name_scope(name or 'kl_half_normal_half_normal'):
    # Consistent with
    # http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf, page 119
    a_scale = tf.convert_to_tensor(a.scale)
    b_scale = tf.convert_to_tensor(b.scale)
    return (tf.math.log(b_scale) - tf.math.log(a_scale) +
            (a_scale**2 - b_scale**2) / (2. * b_scale**2))

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#   `tensorflow_probability/substrates/numpy/distributions/half_normal.py`
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