Laplace¶

class paddle.distribution. Laplace ( loc, scale ) [source]

Creates a Laplace distribution parameterized by loc and scale.

Mathematical details

The probability density function (pdf) is

\[pdf(x; \mu, \sigma) = \frac{1}{2 * \sigma} * e^{\frac{-|x - \mu|}{\sigma}}\]

In the above equation:

\(loc = \mu\): is the location parameter.
\(scale = \sigma\): is the scale parameter.

Parameters

loc (scalar|Tensor) – The mean of the distribution.
scale (scalar|Tensor) – The scale of the distribution.

Examples

           >>> import paddle
>>> paddle.seed(2023)
>>> 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=[], dtype=float32, place=Place(cpu), stop_gradient=True,
    1.31554604)

          

property mean

Mean of distribution.

Returns: The mean value.
Return type: Tensor

property stddev

Standard deviation.

The stddev is

\[stddev = \sqrt{2} * \sigma\]

In the above equation:

\(scale = \sigma\): is the scale parameter.

Returns: The std value.
Return type: Tensor

property variance

Variance of distribution.

The variance is

\[variance = 2 * \sigma^2\]

In the above equation:

\(scale = \sigma\): is the scale parameter.

Returns: The variance value.
Return type: Tensor

log_prob ( value ) log_prob¶

Log probability density/mass function.

The log_prob is

\[log\_prob(value) = \frac{-log(2 * \sigma) - |value - \mu|}{\sigma}\]

In the above equation:

\(loc = \mu\): is the location parameter.
\(scale = \sigma\): is the scale parameter.

Parameters: value (Tensor|Scalar) – The input value, can be a scalar or a tensor.
Returns: The log probability, whose data type is same with value.
Return type: Tensor

Examples

             >>> 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=[], dtype=float32, place=Place(cpu), stop_gradient=True,
        -0.79314721)

            

entropy ( ) entropy¶

Entropy of Laplace distribution.

The entropy is:

\[entropy() = 1 + log(2 * \sigma)\]

In the above equation:

\(scale = \sigma\): is the scale parameter.

Returns: The entropy of distribution.

Examples

             >>> import paddle

>>> m = paddle.distribution.Laplace(paddle.to_tensor(0.0), paddle.to_tensor(1.0))
>>> m.entropy()
Tensor(shape=[], dtype=float32, place=Place(cpu), stop_gradient=True,
        1.69314718)

            

cdf ( value ) cdf¶

Cumulative distribution function.

The cdf is

\[cdf(value) = 0.5 - 0.5 * sign(value - \mu) * e^\frac{-|(\mu - \sigma)|}{\sigma}\]

In the above equation:

\(loc = \mu\): is the location parameter.
\(scale = \sigma\): is the scale parameter.

Parameters: value (Tensor) – The value to be evaluated.
Returns: The cumulative probability of value.
Return type: Tensor

Examples

             >>> 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=[], dtype=float32, place=Place(cpu), stop_gradient=True,
        0.54758132)

            

icdf ( value ) icdf¶

Inverse Cumulative distribution function.

The icdf is

\[cdf^{-1}(value)= \mu - \sigma * sign(value - 0.5) * ln(1 - 2 * |value-0.5|)\]

In the above equation:

\(loc = \mu\): is the location parameter.
\(scale = \sigma\): is the scale parameter.

Parameters: value (Tensor) – The value to be evaluated.
Returns: The cumulative probability of value.
Return type: Tensor

Examples

             >>> 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=[], dtype=float32, place=Place(cpu), stop_gradient=True,
        -1.60943794)

            

sample ( shape=() ) sample¶

Generate samples of the specified shape.

Parameters: shape (tuple[int]) – The shape of generated samples.
Returns: A sample tensor that fits the Laplace distribution.
Return type: Tensor

Examples

             >>> import paddle
>>> paddle.seed(2023)
>>> 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=[], dtype=float32, place=Place(cpu), stop_gradient=True,
    1.31554604)

            

rsample ( shape ) rsample¶

Reparameterized sample.

Parameters: shape (tuple[int]) – The shape of generated samples.
Returns: A sample tensor that fits the Laplace distribution.
Return type: Tensor

Examples

             >>> import paddle
>>> paddle.seed(2023)
>>> 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,
    [[1.31554604]])

            

property batch_shape

Returns batch shape of distribution

Returns: batch shape
Return type: Sequence[int]

property event_shape

Returns event shape of distribution

Returns: event shape
Return type: Sequence[int]

prob ( value ) prob¶

Probability density/mass function evaluated at value.

Parameters: value (Tensor) – value which will be evaluated

probs ( value ) probs¶: Probability density/mass function.

Note

This method will be deprecated in the future, please use prob instead.

kl_divergence ( other ) [source] kl_divergence¶

Calculate the KL divergence KL(self || other) with two Laplace instances.

The kl_divergence between two Laplace distribution is

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

\[ratio = \frac{\sigma_0}{\sigma_1}\]

\[diff = \mu_1 - \mu_0\]

In the above equation:

\(loc = \mu\): is the location parameter of self.
\(scale = \sigma\): is the scale parameter of self.
\(loc = \mu_1\): is the location parameter of the reference Laplace distribution.
\(scale = \sigma_1\): is the scale parameter of the reference Laplace distribution.
\(ratio\): is the ratio between the two distribution.
\(diff\): is the difference between the two distribution.

Parameters: other (Laplace) – An instance of Laplace.
Returns: The kl-divergence between two laplace distributions.
Return type: Tensor

Examples

             >>> 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])

            

Laplace¶

log_prob¶

entropy¶

cdf¶

icdf¶

sample¶

rsample¶

prob¶

probs¶

kl_divergence¶