distribution.py

#   Copyright (c) 2020 PaddlePaddle 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.

# TODO: define the distribution functions 
# __all__ = ['Categorical',
#            'MultivariateNormalDiag',
#            'Normal',
#            'sampling_id',
#            'Uniform']

from __future__ import print_function

from .fluid.layers import control_flow
from .fluid.layers import tensor
from .fluid.layers import ops
from .fluid.layers import nn
from .fluid.layers import elementwise_mul, elementwise_div, elementwise_add, elementwise_sub
from .fluid import core
from .fluid.framework import in_dygraph_mode
from .tensor import arange, gather_nd, concat, multinomial
import math
import numpy as np
import warnings

from .fluid.data_feeder import convert_dtype, check_variable_and_dtype, check_type, check_dtype

__all__ = ['Distribution', 'Uniform', 'Normal', 'Categorical']


class Distribution(object):
    """
    The abstract base class for probability distributions. Functions are 
    implemented in specific distributions.
    """

    def __init__(self):
        super(Distribution, self).__init__()

    def sample(self):
        """Sampling from the distribution."""
        raise NotImplementedError

    def entropy(self):
        """The entropy of the distribution."""
        raise NotImplementedError

    def kl_divergence(self, other):
        """The KL-divergence between self distributions and other."""
        raise NotImplementedError

    def log_prob(self, value):
        """Log probability density/mass function."""
        raise NotImplementedError

    def probs(self, value):
        """Probability density/mass function."""
        raise NotImplementedError

    def _validate_args(self, *args):
        """
        Argument validation for distribution args
        Args:
            value (float, list, numpy.ndarray, Tensor)
        Raises
            ValueError: if one argument is Tensor, all arguments should be Tensor
        """
        is_variable = False
        is_number = False
        for arg in args:
            if isinstance(arg, tensor.Variable):
                is_variable = True
            else:
                is_number = True

        if is_variable and is_number:
            raise ValueError(
                'if one argument is Tensor, all arguments should be Tensor')

        return is_variable

    def _to_tensor(self, *args):
        """
        Argument convert args to Tensor

        Args:
            value (float, list, numpy.ndarray, Tensor)
        Returns:
            Tensor of args.
        """
        numpy_args = []
        variable_args = []
        tmp = 0.

        for arg in args:
            if isinstance(arg, float):
                arg = [arg]
            if not isinstance(arg, (list, tuple, np.ndarray, tensor.Variable)):
                raise TypeError(
                    "Type of input args must be float, list, numpy.ndarray or Tensor, but received type {}".
                    format(type(arg)))

            arg_np = np.array(arg)
            arg_dtype = arg_np.dtype
            if str(arg_dtype) != 'float32':
                if str(arg_dtype) != 'float64':
                    # "assign" op doesn't support float64. if dtype is float64, float32 variable will be generated
                    #  and converted to float64 later using "cast".
                    warnings.warn(
                        "data type of argument only support float32 and float64, your argument will be convert to float32."
                    )
                arg_np = arg_np.astype('float32')
            # tmp is used to support broadcast, it summarizes shapes of all the args and get the mixed shape.
            tmp = tmp + arg_np
            numpy_args.append(arg_np)

        dtype = tmp.dtype
        for arg in numpy_args:
            arg_broadcasted, _ = np.broadcast_arrays(arg, tmp)
            arg_variable = tensor.create_tensor(dtype=dtype)
            tensor.assign(arg_broadcasted, arg_variable)
            variable_args.append(arg_variable)

        return tuple(variable_args)

    def _check_values_dtype_in_probs(self, param, value):
        """
        Log_prob and probs methods have input ``value``, if value's dtype is different from param,
        convert value's dtype to be consistent with param's dtype.

        Args:
            param (Tensor): low and high in Uniform class, loc and scale in Normal class.
            value (Tensor): The input tensor.

        Returns:
            value (Tensor): Change value's dtype if value's dtype is different from param.
        """
        if in_dygraph_mode():
            if value.dtype != param.dtype and convert_dtype(
                    value.dtype) in ['float32', 'float64']:
                warnings.warn(
                    "dtype of input 'value' needs to be the same as parameters of distribution class. dtype of 'value' will be converted."
                )
                return core.ops.cast(value, 'in_dtype', value.dtype,
                                     'out_dtype', param.dtype)
            return value

        check_variable_and_dtype(value, 'value', ['float32', 'float64'],
                                 'log_prob')
        if value.dtype != param.dtype:
            warnings.warn(
                "dtype of input 'value' needs to be the same as parameters of distribution class. dtype of 'value' will be converted."
            )
            return tensor.cast(value, dtype=param.dtype)
        return value


class Uniform(Distribution):
    r"""Uniform distribution with `low` and `high` parameters.

    Mathematical Details

    The probability density function (pdf) is

    .. math::

        pdf(x; a, b) = \\frac{1}{Z}, \ a <=x <b

    .. math::

        Z = b - a

    In the above equation:

    * :math:`low = a`,
    * :math:`high = b`,
    * :math:`Z`: is the normalizing constant.

    The parameters `low` and `high` must be shaped in a way that supports
    [broadcasting](https://www.paddlepaddle.org.cn/documentation/docs/en/develop/beginners_guide/basic_concept/broadcasting_en.html) (e.g., `high - low` is a valid operation).

    Args:
        low(int|float|list|tuple|numpy.ndarray|Tensor): The lower boundary of uniform distribution.The data type is int, float, list, numpy.ndarray or Tensor
        high(int|float|list|tuple|numpy.ndarray|Tensor): The higher boundary of uniform distribution.The data type is int, float, list, numpy.ndarray or Tensor
        name(str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`.

    Examples:
        .. code-block:: python

          import paddle
          from paddle.distribution import Uniform

          # Without broadcasting, a single uniform distribution [3, 4]:
          u1 = Uniform(low=3.0, high=4.0)
          # 2 distributions [1, 3], [2, 4]
          u2 = Uniform(low=[1.0, 2.0], high=[3.0, 4.0])
          # 4 distributions
          u3 = Uniform(low=[[1.0, 2.0], [3.0, 4.0]],
                    high=[[1.5, 2.5], [3.5, 4.5]])

          # With broadcasting:
          u4 = Uniform(low=3.0, high=[5.0, 6.0, 7.0])

          # Complete example
          value_tensor = paddle.to_tensor([0.8], dtype="float32")

          uniform = Uniform([0.], [2.])

          sample = uniform.sample([2])
          # a random tensor created by uniform distribution with shape: [2, 1]
          entropy = uniform.entropy()
          # [0.6931472] with shape: [1]
          lp = uniform.log_prob(value_tensor)
          # [-0.6931472] with shape: [1]
          p = uniform.probs(value_tensor)
          # [0.5] with shape: [1]
    """

    def __init__(self, low, high, name=None):
        if not in_dygraph_mode():
            check_type(low, 'low',
                       (int, float, np.ndarray, tensor.Variable, list, tuple),
                       'Uniform')
            check_type(high, 'high',
                       (int, float, np.ndarray, tensor.Variable, list, tuple),
                       'Uniform')

        self.all_arg_is_float = False
        self.batch_size_unknown = False
        self.name = name if name is not None else 'Uniform'
        self.dtype = 'float32'

        if isinstance(low, int):
            low = float(low)
        if isinstance(high, int):
            high = float(high)

        if self._validate_args(low, high):
            self.batch_size_unknown = True
            self.low = low
            self.high = high
            self.dtype = convert_dtype(low.dtype)
        else:
            if isinstance(low, float) and isinstance(high, float):
                self.all_arg_is_float = True
            if isinstance(
                    low,
                    np.ndarray) and str(low.dtype) in ['float32', 'float64']:
                self.dtype = low.dtype
            elif isinstance(
                    high,
                    np.ndarray) and str(high.dtype) in ['float32', 'float64']:
                self.dtype = high.dtype
            self.low, self.high = self._to_tensor(low, high)
            if self.dtype != convert_dtype(self.low.dtype):
                self.low = tensor.cast(self.low, dtype=self.dtype)
                self.high = tensor.cast(self.high, dtype=self.dtype)

    def sample(self, shape, seed=0):
        """Generate samples of the specified shape.

        Args:
          shape (list): 1D `int32`. Shape of the generated samples.
          seed (int): Python integer number.

        Returns:
          Tensor: A tensor with prepended dimensions shape.The data type is float32.

        """
        if not in_dygraph_mode():
            check_type(shape, 'shape', (list), 'sample')
            check_type(seed, 'seed', (int), 'sample')

        name = self.name + '_sample'
        batch_shape = list((self.low + self.high).shape)
        if self.batch_size_unknown:
            output_shape = shape + batch_shape
            zero_tmp = tensor.fill_constant_batch_size_like(
                self.low + self.high, batch_shape + shape, self.dtype, 0.)
            uniform_random_tmp = nn.uniform_random_batch_size_like(
                zero_tmp,
                zero_tmp.shape,
                dtype=self.dtype,
                min=0.,
                max=1.,
                seed=seed)
            zero_tmp_reshape = nn.reshape(zero_tmp, output_shape)
            uniform_random_tmp_reshape = nn.reshape(uniform_random_tmp,
                                                    output_shape)
            output = uniform_random_tmp_reshape * (
                zero_tmp_reshape + self.high - self.low)
            output = elementwise_add(output, self.low, name=name)
            return output
        else:
            output_shape = shape + batch_shape
            output = nn.uniform_random(
                output_shape, seed=seed, dtype=self.dtype) * (tensor.zeros(
                    output_shape, dtype=self.dtype) + (self.high - self.low))
            output = elementwise_add(output, self.low, name=name)
            if self.all_arg_is_float:
                return nn.reshape(output, shape, name=name)
            else:
                return output

    def log_prob(self, value):
        """Log probability density/mass function.

        Args:
          value (Tensor): The input tensor.

        Returns:
          Tensor: log probability.The data type is same with value.

        """
        value = self._check_values_dtype_in_probs(self.low, value)
        if in_dygraph_mode():
            # ensure value in [low, high]
            lb_bool = self.low < value
            ub_bool = value < self.high

            lb = core.ops.cast(lb_bool, 'in_dtype', lb_bool.dtype, 'out_dtype',
                               value.dtype)
            ub = core.ops.cast(ub_bool, 'in_dtype', ub_bool.dtype, 'out_dtype',
                               value.dtype)
            return nn.log(lb * ub) - nn.log(self.high - self.low)

        name = self.name + '_log_prob'
        lb_bool = self.low < value
        ub_bool = value < self.high
        lb = tensor.cast(lb_bool, dtype=value.dtype)
        ub = tensor.cast(ub_bool, dtype=value.dtype)
        return elementwise_sub(
            nn.log(lb * ub), nn.log(self.high - self.low), name=name)

    def probs(self, value):
        """Probability density/mass function.

        Args:
          value (Tensor): The input tensor.

        Returns:
          Tensor: probability.The data type is same with value.

        """
        value = self._check_values_dtype_in_probs(self.low, value)
        if in_dygraph_mode():
            lb_bool = self.low < value
            ub_bool = value < self.high

            lb = core.ops.cast(lb_bool, 'in_dtype', lb_bool.dtype, 'out_dtype',
                               value.dtype)
            ub = core.ops.cast(ub_bool, 'in_dtype', ub_bool.dtype, 'out_dtype',
                               value.dtype)
            return (lb * ub) / (self.high - self.low)

        name = self.name + '_probs'
        lb_bool = self.low < value
        ub_bool = value < self.high
        lb = tensor.cast(lb_bool, dtype=value.dtype)
        ub = tensor.cast(ub_bool, dtype=value.dtype)
        return elementwise_div((lb * ub), (self.high - self.low), name=name)

    def entropy(self):
        r"""Shannon entropy in nats.

        The entropy is

        .. math::

            entropy(low, high) = \\log (high - low)

        Returns:
          Tensor: Shannon entropy of uniform distribution.The data type is float32.

        """
        name = self.name + '_entropy'
        return nn.log(self.high - self.low, name=name)


class Normal(Distribution):
    r"""The Normal distribution with location `loc` and `scale` parameters.

    Mathematical details

    The probability density function (pdf) is

    .. math::

        pdf(x; \mu, \sigma) = \\frac{1}{Z}e^{\\frac {-0.5 (x - \mu)^2}  {\sigma^2} }

    .. math::

        Z = (2 \pi \sigma^2)^{0.5}

    In the above equation:

    * :math:`loc = \mu`: is the mean.
    * :math:`scale = \sigma`: is the std.
    * :math:`Z`: is the normalization constant.

    Args:
        loc(int|float|list|tuple|numpy.ndarray|Tensor): The mean of normal distribution.The data type is int, float, list, numpy.ndarray or Tensor.
        scale(int|float|list|tuple|numpy.ndarray|Tensor): The std of normal distribution.The data type is int, float, list, numpy.ndarray or Tensor.
        name(str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`.

    Examples:
        .. code-block:: python
          
          import paddle
          from paddle.distribution import Normal

          # Define a single scalar Normal distribution.
          dist = Normal(loc=0., scale=3.)
          # Define a batch of two scalar valued Normals.
          # The first has mean 1 and standard deviation 11, the second 2 and 22.
          dist = Normal(loc=[1., 2.], scale=[11., 22.])
          # Get 3 samples, returning a 3 x 2 tensor.
          dist.sample([3])

          # Define a batch of two scalar valued Normals.
          # Both have mean 1, but different standard deviations.
          dist = Normal(loc=1., scale=[11., 22.])

          # Complete example
          value_tensor = paddle.to_tensor([0.8], dtype="float32")

          normal_a = Normal([0.], [1.])
          normal_b = Normal([0.5], [2.])
          sample = normal_a.sample([2])
          # a random tensor created by normal distribution with shape: [2, 1]
          entropy = normal_a.entropy()
          # [1.4189385] with shape: [1]
          lp = normal_a.log_prob(value_tensor)
          # [-1.2389386] with shape: [1]
          p = normal_a.probs(value_tensor)
          # [0.28969154] with shape: [1]
          kl = normal_a.kl_divergence(normal_b)
          # [0.34939718] with shape: [1]
    """

    def __init__(self, loc, scale, name=None):
        if not in_dygraph_mode():
            check_type(loc, 'loc',
                       (int, float, np.ndarray, tensor.Variable, list, tuple),
                       'Normal')
            check_type(scale, 'scale',
                       (int, float, np.ndarray, tensor.Variable, list, tuple),
                       'Normal')

        self.batch_size_unknown = False
        self.all_arg_is_float = False
        self.name = name if name is not None else 'Normal'
        self.dtype = 'float32'

        if isinstance(loc, int):
            loc = float(loc)
        if isinstance(scale, int):
            scale = float(scale)

        if self._validate_args(loc, scale):
            self.batch_size_unknown = True
            self.loc = loc
            self.scale = scale
            self.dtype = convert_dtype(loc.dtype)
        else:
            if isinstance(loc, float) and isinstance(scale, float):
                self.all_arg_is_float = True
            if isinstance(
                    loc,
                    np.ndarray) and str(loc.dtype) in ['float32', 'float64']:
                self.dtype = loc.dtype
            elif isinstance(
                    scale,
                    np.ndarray) and str(scale.dtype) in ['float32', 'float64']:
                self.dtype = scale.dtype
            self.loc, self.scale = self._to_tensor(loc, scale)
            if self.dtype != convert_dtype(self.loc.dtype):
                self.loc = tensor.cast(self.loc, dtype=self.dtype)
                self.scale = tensor.cast(self.scale, dtype=self.dtype)

    def sample(self, shape, seed=0):
        """Generate samples of the specified shape.

        Args:
          shape (list): 1D `int32`. Shape of the generated samples.
          seed (int): Python integer number.

        Returns:
          Tensor: A tensor with prepended dimensions shape.The data type is float32.

        """
        if not in_dygraph_mode():
            check_type(shape, 'shape', (list), 'sample')
            check_type(seed, 'seed', (int), 'sample')

        batch_shape = list((self.loc + self.scale).shape)
        name = self.name + '_sample'

        if self.batch_size_unknown:
            output_shape = shape + batch_shape
            zero_tmp = tensor.fill_constant_batch_size_like(
                self.loc + self.scale, batch_shape + shape, self.dtype, 0.)
            zero_tmp_reshape = nn.reshape(zero_tmp, output_shape)
            zero_tmp_shape = nn.shape(zero_tmp_reshape)
            normal_random_tmp = nn.gaussian_random(
                zero_tmp_shape, mean=0., std=1., seed=seed, dtype=self.dtype)
            output = normal_random_tmp * (zero_tmp_reshape + self.scale)
            output = elementwise_add(output, self.loc, name=name)
            return output
        else:
            output_shape = shape + batch_shape
            output = nn.gaussian_random(output_shape, mean=0., std=1., seed=seed, dtype=self.dtype) * \
                     (tensor.zeros(output_shape, dtype=self.dtype) + self.scale)
            output = elementwise_add(output, self.loc, name=name)
            if self.all_arg_is_float:
                return nn.reshape(output, shape, name=name)
            else:
                return output

    def entropy(self):
        r"""Shannon entropy in nats.

        The entropy is

        .. math::

            entropy(\sigma) = 0.5 \\log (2 \pi e \sigma^2)

        In the above equation:

        * :math:`scale = \sigma`: is the std.

        Returns:
          Tensor: Shannon entropy of normal distribution.The data type is float32.

        """
        name = self.name + '_entropy'
        batch_shape = list((self.loc + self.scale).shape)
        zero_tmp = tensor.fill_constant_batch_size_like(
            self.loc + self.scale, batch_shape, self.dtype, 0.)
        return elementwise_add(
            0.5 + zero_tmp,
            0.5 * math.log(2 * math.pi) + nn.log((self.scale + zero_tmp)),
            name=name)

    def log_prob(self, value):
        """Log probability density/mass function.

        Args:
          value (Tensor): The input tensor.

        Returns:
          Tensor: log probability.The data type is same with value.

        """
        name = self.name + '_log_prob'
        value = self._check_values_dtype_in_probs(self.loc, value)

        var = self.scale * self.scale
        log_scale = nn.log(self.scale)
        return elementwise_sub(
            -1. * ((value - self.loc) * (value - self.loc)) / (2. * var),
            log_scale + math.log(math.sqrt(2. * math.pi)),
            name=name)

    def probs(self, value):
        """Probability density/mass function.

        Args:
          value (Tensor): The input tensor.

        Returns:
          Tensor: probability.The data type is same with value.

        """
        name = self.name + '_probs'
        value = self._check_values_dtype_in_probs(self.loc, value)

        var = self.scale * self.scale
        return elementwise_div(
            ops.exp(-1. * ((value - self.loc) * (value - self.loc)) /
                    (2. * var)), (math.sqrt(2 * math.pi) * self.scale),
            name=name)

    def kl_divergence(self, other):
        r"""The KL-divergence between two normal distributions.

        The probability density function (pdf) is

        .. math::

            KL\_divergence(\mu_0, \sigma_0; \mu_1, \sigma_1) = 0.5 (ratio^2 + (\\frac{diff}{\sigma_1})^2 - 1 - 2 \\ln {ratio})

        .. math::

            ratio = \\frac{\sigma_0}{\sigma_1}
        
        .. math::

            diff = \mu_1 - \mu_0

        In the above equation:

        * :math:`loc = \mu_0`: is the mean of current Normal distribution.
        * :math:`scale = \sigma_0`: is the std of current Normal distribution.
        * :math:`loc = \mu_1`: is the mean of other Normal distribution.
        * :math:`scale = \sigma_1`: is the std of other Normal distribution.
        * :math:`ratio`: is the ratio of scales.
        * :math:`diff`: is the difference between means.

        Args:
            other (Normal): instance of Normal.

        Returns:
            Tensor: kl-divergence between two normal distributions.The data type is float32.

        """
        if not in_dygraph_mode():
            check_type(other, 'other', Normal, 'kl_divergence')

        name = self.name + '_kl_divergence'
        var_ratio = self.scale / other.scale
        var_ratio = (var_ratio * var_ratio)
        t1 = (self.loc - other.loc) / other.scale
        t1 = (t1 * t1)
        return elementwise_add(
            0.5 * var_ratio, 0.5 * (t1 - 1. - nn.log(var_ratio)), name=name)


class Categorical(Distribution):
    r"""
    Categorical distribution is a discrete probability distribution that 
    describes the possible results of a random variable that can take on 
    one of K possible categories, with the probability of each category 
    separately specified.

    The probability mass function (pmf) is:

    .. math::

        pmf(k; p_i) = \prod_{i=1}^{k} p_i^{[x=i]}

    In the above equation:

    * :math:`[x=i]` : it evaluates to 1 if :math:`x==i` , 0 otherwise.

    Args:
        logits(list|tuple|numpy.ndarray|Tensor): The logits input of categorical distribution. The data type is float32 or float64.
        name(str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`.

    Examples:
        .. code-block:: python

            import paddle
            from paddle.distribution import Categorical

            paddle.seed(100) # on CPU device
            x = paddle.rand([6])
            print(x)
            # [0.5535528  0.20714243 0.01162981
            #  0.51577556 0.36369765 0.2609165 ]

            paddle.seed(200) # on CPU device
            y = paddle.rand([6])
            print(y)
            # [0.77663314 0.90824795 0.15685187
            #  0.04279523 0.34468332 0.7955718 ]

            cat = Categorical(x)
            cat2 = Categorical(y)

            paddle.seed(1000) # on CPU device
            cat.sample([2,3])
            # [[0, 0, 5],
            #  [3, 4, 5]]

            cat.entropy()
            # [1.77528]

            cat.kl_divergence(cat2)
            # [0.071952]

            value = paddle.to_tensor([2,1,3])
            cat.probs(value)
            # [0.00608027 0.108298 0.269656]

            cat.log_prob(value)
            # [-5.10271 -2.22287 -1.31061]

    """

    def __init__(self, logits, name=None):
        """
        Args:
            logits(list|tuple|numpy.ndarray|Tensor): The logits input of categorical distribution. The data type is float32 or float64.
            name(str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`.
        """
        if not in_dygraph_mode():
            check_type(logits, 'logits',
                       (np.ndarray, tensor.Variable, list, tuple),
                       'Categorical')

        self.name = name if name is not None else 'Categorical'
        self.dtype = 'float32'

        if self._validate_args(logits):
            self.logits = logits
            self.dtype = convert_dtype(logits.dtype)
        else:
            if isinstance(logits, np.ndarray) and str(
                    logits.dtype) in ['float32', 'float64']:
                self.dtype = logits.dtype
            self.logits = self._to_tensor(logits)[0]
            if self.dtype != convert_dtype(self.logits.dtype):
                self.logits = tensor.cast(self.logits, dtype=self.dtype)

    def sample(self, shape):
        """Generate samples of the specified shape.

        Args:
            shape (list): Shape of the generated samples.

        Returns:
            Tensor: A tensor with prepended dimensions shape.
        
        Examples:
            .. code-block:: python

                import paddle
                from paddle.distribution import Categorical

                paddle.seed(100) # on CPU device
                x = paddle.rand([6])
                print(x)
                # [0.5535528  0.20714243 0.01162981
                #  0.51577556 0.36369765 0.2609165 ]

                cat = Categorical(x)

                paddle.seed(1000) # on CPU device
                cat.sample([2,3])
                # [[0, 0, 5],
                #  [3, 4, 5]]

        """
        name = self.name + '_sample'
        if not in_dygraph_mode():
            check_type(shape, 'shape', (list), 'sample')

        num_samples = np.prod(np.array(shape))

        logits_shape = list(self.logits.shape)
        if len(logits_shape) > 1:
            sample_shape = shape + logits_shape[:-1]
            logits = nn.reshape(self.logits,
                                [np.prod(logits_shape[:-1]), logits_shape[-1]])
        else:
            sample_shape = shape
            logits = self.logits

        sample_index = multinomial(logits, num_samples, True)
        return nn.reshape(sample_index, sample_shape, name=name)

    def kl_divergence(self, other):
        """The KL-divergence between two Categorical distributions.

        Args:
            other (Categorical): instance of Categorical. The data type is float32.

        Returns:
            Tensor: kl-divergence between two Categorical distributions.
        
        Examples:
            .. code-block:: python

                import paddle
                from paddle.distribution import Categorical

                paddle.seed(100) # on CPU device
                x = paddle.rand([6])
                print(x)
                # [0.5535528  0.20714243 0.01162981
                #  0.51577556 0.36369765 0.2609165 ]

                paddle.seed(200) # on CPU device
                y = paddle.rand([6])
                print(y)
                # [0.77663314 0.90824795 0.15685187
                #  0.04279523 0.34468332 0.7955718 ]

                cat = Categorical(x)
                cat2 = Categorical(y)

                cat.kl_divergence(cat2)
                # [0.071952]

        """
        name = self.name + '_kl_divergence'
        if not in_dygraph_mode():
            check_type(other, 'other', Categorical, 'kl_divergence')

        logits = self.logits - nn.reduce_max(self.logits, dim=-1, keep_dim=True)
        other_logits = other.logits - nn.reduce_max(
            other.logits, dim=-1, keep_dim=True)
        e_logits = ops.exp(logits)
        other_e_logits = ops.exp(other_logits)
        z = nn.reduce_sum(e_logits, dim=-1, keep_dim=True)
        other_z = nn.reduce_sum(other_e_logits, dim=-1, keep_dim=True)
        prob = e_logits / z
        kl = nn.reduce_sum(
            prob * (logits - nn.log(z) - other_logits + nn.log(other_z)),
            dim=-1,
            keep_dim=True,
            name=name)

        return kl

    def entropy(self):
        """Shannon entropy in nats.

        Returns:
            Tensor: Shannon entropy of Categorical distribution. The data type is float32.
        
        Examples:
            .. code-block:: python

                import paddle
                from paddle.distribution import Categorical

                paddle.seed(100) # on CPU device
                x = paddle.rand([6])
                print(x)
                # [0.5535528  0.20714243 0.01162981
                #  0.51577556 0.36369765 0.2609165 ]

                cat = Categorical(x)

                cat.entropy()
                # [1.77528]

        """
        name = self.name + '_entropy'
        logits = self.logits - nn.reduce_max(self.logits, dim=-1, keep_dim=True)
        e_logits = ops.exp(logits)
        z = nn.reduce_sum(e_logits, dim=-1, keep_dim=True)
        prob = e_logits / z

        neg_entropy = nn.reduce_sum(
            prob * (logits - nn.log(z)), dim=-1, keep_dim=True)
        entropy = nn.scale(neg_entropy, scale=-1.0, name=name)
        return entropy

    def probs(self, value):
        """Probabilities of the given category (``value``).

        If ``logits`` is 2-D or higher dimension, the last dimension will be regarded as 
        category, and the others represents the different distributions.
        At the same time, if ``vlaue`` is 1-D Tensor, ``value`` will be broadcast to the 
        same number of distributions as ``logits``.
        If ``value`` is not 1-D Tensor, ``value`` should have the same number distributions
        with ``logits. That is, ``value[:-1] = logits[:-1]``.

        Args:
            value (Tensor): The input tensor represents the selected category index.

        Returns:
            Tensor: probability according to the category index.
        
        Examples:
            .. code-block:: python

                import paddle
                from paddle.distribution import Categorical

                paddle.seed(100) # on CPU device
                x = paddle.rand([6])
                print(x)
                # [0.5535528  0.20714243 0.01162981
                #  0.51577556 0.36369765 0.2609165 ]

                cat = Categorical(x)

                value = paddle.to_tensor([2,1,3])
                cat.probs(value)
                # [0.00608027 0.108298 0.269656]

        """
        name = self.name + '_probs'

        dist_sum = nn.reduce_sum(self.logits, dim=-1, keep_dim=True)
        prob = self.logits / dist_sum

        shape = list(prob.shape)
        value_shape = list(value.shape)
        if len(shape) == 1:
            num_value_in_one_dist = np.prod(value_shape)
            index_value = nn.reshape(value, [num_value_in_one_dist, 1])
            index = index_value
        else:
            num_dist = np.prod(shape[:-1])
            num_value_in_one_dist = value_shape[-1]
            prob = nn.reshape(prob, [num_dist, shape[-1]])
            if len(value_shape) == 1:
                value = nn.expand(value, [num_dist])
                value_shape = shape[:-1] + value_shape
            index_value = nn.reshape(value, [num_dist, -1, 1])
            if shape[:-1] != value_shape[:-1]:
                raise ValueError(
                    "shape of value {} must match shape of logits {}".format(
                        str(value_shape[:-1]), str(shape[:-1])))

            index_prefix = nn.unsqueeze(
                arange(
                    num_dist, dtype=index_value.dtype), axes=-1)
            index_prefix = nn.expand(index_prefix, [1, num_value_in_one_dist])
            index_prefix = nn.unsqueeze(index_prefix, axes=-1)

            if index_value.dtype != index_prefix.dtype:
                tensor.cast(index_prefix, dtype=index_value.dtype)
            index = concat([index_prefix, index_value], axis=-1)

        # value is the category index to search for the corresponding probability.
        select_prob = gather_nd(prob, index)
        return nn.reshape(select_prob, value_shape, name=name)

    def log_prob(self, value):
        """Log probabilities of the given category. Refer to ``probs`` method.

        Args:
            value (Tensor): The input tensor represents the selected category index.

        Returns:
            Tensor: Log probability.
        
        Examples:
            .. code-block:: python

                import paddle
                from paddle.distribution import Categorical

                paddle.seed(100) # on CPU device
                x = paddle.rand([6])
                print(x)
                # [0.5535528  0.20714243 0.01162981
                #  0.51577556 0.36369765 0.2609165 ]

                cat = Categorical(x)

                value = paddle.to_tensor([2,1,3])
                cat.log_prob(value)
                # [-5.10271 -2.22287 -1.31061]

        """
        name = self.name + '_log_prob'

        return nn.log(self.probs(value), name=name)