LogNormal¶

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

The LogNormal distribution with location loc and scale parameters.

\[ \begin{align}\begin{aligned}X \sim Normal(\mu, \sigma)\\Y = exp(X) \sim LogNormal(\mu, \sigma)\end{aligned}\end{align} \]

Due to LogNormal distribution is based on the transformation of Normal distribution, we call that \(Normal(\mu, \sigma)\) is the underlying distribution of \(LogNormal(\mu, \sigma)\)

Mathematical details

The probability density function (pdf) is

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

In the above equation:

\(loc = \mu\): is the means of the underlying Normal distribution.
\(scale = \sigma\): is the stddevs of the underlying Normal distribution.

Parameters

loc (int|float|list|tuple|numpy.ndarray|Tensor) – The means of the underlying Normal distribution.
scale (int|float|list|tuple|numpy.ndarray|Tensor) – The stddevs of the underlying Normal distribution.

Examples

           >>> import paddle
>>> from paddle.distribution import LogNormal

>>> # Define a single scalar LogNormal distribution.
>>> dist = LogNormal(loc=0., scale=3.)
>>> # Define a batch of two scalar valued LogNormals.
>>> # The underlying Normal of first has mean 1 and standard deviation 11, the underlying Normal of second 2 and 22.
>>> dist = LogNormal(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 LogNormals.
>>> # Their underlying Normal have mean 1, but different standard deviations.
>>> dist = LogNormal(loc=1., scale=[11., 22.])

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

>>> lognormal_a = LogNormal([0.], [1.])
>>> lognormal_b = LogNormal([0.5], [2.])
>>> sample = lognormal_a.sample((2, ))
>>> # a random tensor created by lognormal distribution with shape: [2, 1]
>>> entropy = lognormal_a.entropy()
>>> print(entropy)
Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
    [1.41893852])
>>> lp = lognormal_a.log_prob(value_tensor)
>>> print(lp)
Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
    [-0.72069150])
>>> p = lognormal_a.probs(value_tensor)
>>> print(p)
Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
    [0.48641577])
>>> kl = lognormal_a.kl_divergence(lognormal_b)
>>> print(kl)
Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
    [0.34939718])

          

property mean

Mean of lognormal distribuion.

Returns: mean value.
Return type: Tensor

property variance

Variance of lognormal distribution.

Returns: variance value.
Return type: Tensor

entropy ( ) entropy¶

Shannon entropy in nats.

The entropy is

\[entropy(\sigma) = 0.5 \log (2 \pi e \sigma^2) + \mu\]

In the above equation:

\(loc = \mu\): is the mean of the underlying Normal distribution.
\(scale = \sigma\): is the stddevs of the underlying Normal distribution.

Returns: Shannon entropy of lognormal distribution.
Return type: Tensor

probs ( value ) probs¶

Probability density/mass function.

Parameters: value (Tensor) – The input tensor.
Returns: probability.The data type is same with value .
Return type: Tensor

kl_divergence ( other ) [source] kl_divergence¶

The KL-divergence between two lognormal distributions.

The probability density function (pdf) 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:

\(loc = \mu_0\): is the means of current underlying Normal distribution.
\(scale = \sigma_0\): is the stddevs of current underlying Normal distribution.
\(loc = \mu_1\): is the means of other underlying Normal distribution.
\(scale = \sigma_1\): is the stddevs of other underlying Normal distribution.
\(ratio\): is the ratio of scales.
\(diff\): is the difference between means.

Parameters: other (LogNormal) – instance of LogNormal.
Returns: kl-divergence between two lognormal distributions.
Return type: Tensor

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]

log_prob ( value ) log_prob¶

The log probability evaluated at value.

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

prob ( value ) prob¶

Probability density/mass function evaluated at value.

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

rsample ( shape=() ) rsample¶

Reparameterized sample from TransformedDistribution.

Parameters: shape (Sequence[int], optional) – The sample shape. Defaults to ().
Returns: The sample result.
Return type: [Tensor]

sample ( shape=() ) sample¶

Sample from TransformedDistribution.

Parameters: shape (Sequence[int], optional) – The sample shape. Defaults to ().
Returns: The sample result.
Return type: [Tensor]