%matplotlib inline 
import matplotlib.pyplot as plt
import numpy as np
def laplace_function(x, lambda_):
 return (1/(2*lambda_)) * np.e**(-1*(np.abs(x)/lambda_))
x = np.linspace(-5,5,10000)
y1 = [laplace_function(x_,1) for x_ in x]
y2 = [laplace_function(x_,2) for x_ in x]
y3 = [laplace_function(x_,0.5) for x_ in x]
plt.plot(x, y1, color='r', label="lambda:1")
plt.plot(x, y2, color='g', label="lambda:2")
plt.plot(x, y3, color='b', label="lambda:0.5")
plt.title("Laplace distribution")
plt.legend()
plt.show()

 self.loc

 (2 ** 0.5) * self.scale;

 self.stddev.pow(2)

 self.rsample(shape)

 self.loc - self.scale * u.sign() * paddle.log1p(-u.abs())

 self.log_prob(value).exp()

 -paddle.log(2 * self.scale) - paddle.abs(value - self.loc) / self.scale

 1 + paddle.log(2 * self.scale)

 0.5 - 0.5 * (value - self.loc).sign() * paddle.expm1(-(value - self.loc).abs() / self.scale)

 self.loc - self.scale * (value - 0.5).sign() * paddle.log1p(-2 * (value - 0.5).abs())

 (self.scale * paddle.exp(paddle.abs(self.loc - other.loc) / self.scale) + paddle.abs(self.loc - other.loc)) / other.scale + paddle.log(other.scale / self.scale) - 1

import numbers
import numpy as np
import paddle
from paddle.distribution import distribution
from paddle.fluid import framework as framework
class Laplace(distribution.Distribution):
    r"""
    Creates a Laplace distribution parameterized by :attr:`loc` and :attr:`scale`.
    Mathematical details
    The probability density function (pdf) is
 .. math::
 pdf(x; \mu, \sigma) = \frac{1}{2 * \sigma} * e^{\frac {-|x - \mu|}{\sigma}}
    In the above equation:
 * :math:`loc = \mu`: is the location parameter.
 * :math:`scale = \sigma`: is the scale parameter.
 Args:
 loc (scalar|Tensor): The mean of the distribution.
 scale (scalar|Tensor): The scale of the distribution.
 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
                        m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
 m.sample()  # Laplace distributed with loc=0, scale=1
                        # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True, 
                        # [3.68546247])
 """
    def __init__(self, loc, scale):
 if not isinstance(loc, (numbers.Real, framework.Variable)):
            raise TypeError(
 f"Expected type of loc is Real|Variable, but got {type(loc)}")
 if not isinstance(scale, (numbers.Real, framework.Variable)):
            raise TypeError(
 f"Expected type of scale is Real|Variable, but got {type(scale)}"
 )
 if isinstance(loc, numbers.Real):
            loc = paddle.full(shape=(), fill_value=loc)
 if isinstance(scale, numbers.Real):
            scale = paddle.full(shape=(), fill_value=scale)
 if (len(scale.shape) > 0 or len(loc.shape) > 0) and (loc.dtype
 == scale.dtype):
 self.loc, self.scale = paddle.broadcast_tensors([loc, scale])
 else:
 self.loc, self.scale = loc, scale
 super(Laplace, self).__init__(self.loc.shape)
    @property
    def mean(self):
 """Mean of distribution.
        Returns:
            Tensor: The mean value.
 """
 return self.loc
    @property
    def stddev(self):
 """Standard deviation.
        The stddev is 
 .. math::
 stddev = \sqrt{2} * \sigma
        In the above equation:
 * :math:`scale = \sigma`: is the scale parameter.
        Returns:
            Tensor: The std value.
 """
 return (2**0.5) * self.scale
    @property
    def variance(self):
 """Variance of distribution.
        The variance is 
 .. math::
            variance = 2 * \sigma^2
        In the above equation:
 * :math:`scale = \sigma`: is the scale parameter.
        Returns:
            Tensor: The variance value.
 """
 return self.stddev.pow(2)
    def _validate_value(self, value):
 """Argument dimension check for distribution methods such as `log_prob`,
 `cdf` and `icdf`. 
 Args:
 value (Tensor|Scalar): The input value, which can be a scalar or a tensor.
        Returns:
          loc, scale, value: The broadcasted loc, scale and value, with the same dimension and data type.
 """
 if isinstance(value, numbers.Real):
            value = paddle.full(shape=(), fill_value=value)
 if value.dtype != self.scale.dtype:
            value = paddle.cast(value, self.scale.dtype)
 if len(self.scale.shape) > 0 or len(self.loc.shape) > 0 or len(
 value.shape) > 0:
            loc, scale, value = paddle.broadcast_tensors(
 [self.loc, self.scale, value])
 else:
            loc, scale = self.loc, self.scale
 return loc, scale, value
    def log_prob(self, value):
 """Log probability density/mass function.
        The log_prob is
 .. math::
            log\_prob(value) = \frac{-log(2 * \sigma) - |value - \mu|}{\sigma}
        In the above equation:
 * :math:`loc = \mu`: is the location parameter.
 * :math:`scale = \sigma`: is the scale parameter.
 Args:
 value (Tensor|Scalar): The input value, can be a scalar or a tensor.
        Returns:
          Tensor: The log probability, whose data type is same with value.
        Examples:
 .. code-block:: python
 import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                            value = paddle.to_tensor([0.1])
 m.log_prob(value) 
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [-0.79314721])
 """
        loc, scale, value = self._validate_value(value)
 log_scale = -paddle.log(2 * scale)
 return (log_scale - paddle.abs(value - loc) / scale)
    def entropy(self):
 """Entropy of Laplace distribution.
        The entropy is:
 .. math::
 entropy() = 1 + log(2 * \sigma)
        In the above equation:
 * :math:`scale = \sigma`: is the scale parameter.
        Returns:
            The entropy of distribution.
        Examples:
 .. code-block:: python
 import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
 m.entropy()
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [1.69314718])
 """
 return 1 + paddle.log(2 * self.scale)
    def cdf(self, value):
 """Cumulative distribution function.
        The cdf is
 .. math::
 cdf(value) = 0.5 - 0.5 * sign(value - \mu) * e^\frac{-|(\mu - \sigma)|}{\sigma}
        In the above equation:
 * :math:`loc = \mu`: is the location parameter.
 * :math:`scale = \sigma`: is the scale parameter.
 Args:
 value (Tensor): The value to be evaluated.
        Returns:
            Tensor: The cumulative probability of value.
        Examples:
 .. code-block:: python
 import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                            value = paddle.to_tensor([0.1])
 m.cdf(value)
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [0.54758132])
 """
        loc, scale, value = self._validate_value(value)
 iterm = (0.5 * (value - loc).sign() *
 paddle.expm1(-(value - loc).abs() / scale))
 return 0.5 - iterm
    def icdf(self, value):
 """Inverse Cumulative distribution function.
        The icdf is 
 .. math::
 cdf^{-1}(value)= \mu - \sigma * sign(value - 0.5) * ln(1 - 2 * |value-0.5|)
        In the above equation:
 * :math:`loc = \mu`: is the location parameter.
 * :math:`scale = \sigma`: is the scale parameter.
 Args:
 value (Tensor): The value to be evaluated.
        Returns:
            Tensor: The cumulative probability of value.
        Examples:
 .. code-block:: python
 import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                            value = paddle.to_tensor([0.1])
 m.icdf(value)
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [-1.60943794])
 """
        loc, scale, value = self._validate_value(value)
        term = value - 0.5
 return (loc - scale * (term).sign() * paddle.log1p(-2 * term.abs()))
    def sample(self, shape=()):
 """Generate samples of the specified shape.
 Args:
 shape(tuple[int]): The shape of generated samples.
        Returns:
            Tensor: A sample tensor that fits the Laplace distribution.
        Examples:
 .. code-block:: python
 import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
 m.sample()  # Laplace distributed with loc=0, scale=1
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [3.68546247])
 """
 if not isinstance(shape, tuple):
            raise TypeError(
 f'Expected shape should be tuple[int], but got {type(shape)}')
 with paddle.no_grad():
 return self.rsample(shape)
    def rsample(self, shape):
 """Reparameterized sample.
 Args:
 shape(tuple[int]): The shape of generated samples.
        Returns:
            Tensor: A sample tensor that fits the Laplace distribution.
        Examples:
 .. code-block:: python
 import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
 m.rsample((1,))  # Laplace distributed with loc=0, scale=1
                            # Tensor(shape=[1, 1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [[0.04337667]])
 """
        eps = self._get_eps()
        shape = self._extend_shape(shape) or (1, )
        uniform = paddle.uniform(shape=shape,
                                 min=float(np.nextafter(-1, 1)) + eps / 2,
                                 max=1. - eps / 2,
 dtype=self.loc.dtype)
 if len(self.scale.shape) == 0 and len(self.loc.shape) == 0:
            loc, scale, uniform = paddle.broadcast_tensors(
 [self.loc, self.scale, uniform])
 else:
            loc, scale = self.loc, self.scale
 return (loc - scale * uniform.sign() * paddle.log1p(-uniform.abs()))
    def _get_eps(self):
 """
        Get the eps of certain data type.
        Note: 
            Since paddle.finfo is temporarily unavailable, we 
            use hard-coding style to get eps value.
        Returns:
            Float: An eps value by different data types.
 """
        eps = 1.19209e-07
 if (self.loc.dtype == paddle.float64
                or self.loc.dtype == paddle.complex128):
            eps = 2.22045e-16
 return eps
    def kl_divergence(self, other):
 """Calculate the KL divergence KL(self || other) with two Laplace instances.
        The kl_divergence between two Laplace distribution 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`: is the location parameter of self.
 * :math:`scale = \sigma`: is the scale parameter of self.
 * :math:`loc = \mu_1`: is the location parameter of the reference Laplace distribution.
 * :math:`scale = \sigma_1`: is the scale parameter of the reference Laplace distribution.
 * :math:`ratio`: is the ratio between the two distribution.
 * :math:`diff`: is the difference between the two distribution.
 Args:
 other (Laplace): An instance of Laplace.
        Returns:
            Tensor: The kl-divergence between two laplace distributions.
        Examples:
 .. code-block:: python
 import paddle
                            m1 = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                            m2 = paddle.distribution.Laplace(paddle.to_tensor([1.0]), paddle.to_tensor([0.5]))
                            m1.kl_divergence(m2)
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [1.04261160])
 """
 var_ratio = other.scale / self.scale
        t = paddle.abs(self.loc - other.loc)
        term1 = ((self.scale * paddle.exp(-t / self.scale) + t) / other.scale)
        term2 = paddle.log(var_ratio)
 return term1 + term2 - 1