# Copyright 2018 The JAX 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.


import numpy as np

from jax import lax
from jax._src.lax.lax import _const as _lax_const
from jax._src.numpy.util import promote_args_inexact
from jax._src.typing import Array, ArrayLike


def logpdf(x: ArrayLike, df: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
  r"""Student's T log probability distribution function.

  JAX implementation of :obj:`scipy.stats.t` ``logpdf``.

  The Student's T probability distribution function is given by

  .. math::

     f(x, \nu) = \frac{\Gamma((\nu + 1)/2)}{\sqrt{\pi\nu}\Gamma(\nu/2)}(1 + x^2/\nu)^{(\nu+1)/2}

  Where :math:`\Gamma` is the :func:`~jax.scipy.special.gamma` function, and :math:`\nu > 0`
  is the degrees of freedom (JAX follows the scipy convention of naming this ``df``).

  Args:
    x: arraylike, value at which to evaluate the PDF
    df: arraylike, distribution shape parameter
    loc: arraylike, distribution offset parameter
    scale: arraylike, distribution scale parameter

  Returns:
    array of logpdf values.

  See Also:
    :func:`jax.scipy.stats.t.pdf`
  """
  x, df, loc, scale = promote_args_inexact("t.logpdf", x, df, loc, scale)
  two = _lax_const(x, 2)
  scaled_x = lax.div(lax.sub(x, loc), scale)
  df_over_two = lax.div(df, two)
  df_plus_one_over_two = lax.add(df_over_two, _lax_const(x, 0.5))
  normalize_term_const = lax.mul(lax.mul(scale, scale), _lax_const(x, np.pi))
  normalize_term_tmp = lax.div(lax.log(lax.mul(normalize_term_const, df)), two)
  normalize_term = lax.sub(lax.add(lax.lgamma(df_over_two), normalize_term_tmp),
                           lax.lgamma(df_plus_one_over_two))
  quadratic = lax.div(lax.mul(scaled_x, scaled_x), df)
  return lax.neg(lax.add(normalize_term, lax.mul(df_plus_one_over_two, lax.log1p(quadratic))))


def pdf(x: ArrayLike, df: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
  r"""Student's T probability distribution function.

  JAX implementation of :obj:`scipy.stats.t` ``pdf``.

  The Student's T probability distribution function is given by

  .. math::

     f(x, \nu) = \frac{\Gamma((\nu + 1)/2)}{\sqrt{\pi\nu}\Gamma(\nu/2)}(1 + x^2/\nu)^{(\nu+1)/2}

  Where :math:`\Gamma` is the :func:`~jax.scipy.special.gamma` function, and :math:`\nu > 0`
  is the degrees of freedom (JAX follows the scipy convention of naming this ``df``).

  Args:
    x: arraylike, value at which to evaluate the PDF
    df: arraylike, distribution shape parameter
    loc: arraylike, distribution offset parameter
    scale: arraylike, distribution scale parameter

  Returns:
    array

  See Also:
    :func:`jax.scipy.stats.t.logpdf`
  """
  return lax.exp(logpdf(x, df, loc, scale))