class

LogNormal

extendsDistribution

LogNormal(loc: Tensor | float, scale: Tensor | float, validate_args: bool | None = None)

source edit

Log-Normal distribution: $X = \exp(Y)$ where $Y \sim \mathcal{N}(\mu, \sigma^2)$ .

If a random variable $X$ has a Log-Normal distribution then its natural logarithm $\ln X$ is Normally distributed. The probability density function is

p(x \mid \mu, \sigma) = \frac{1}{x\,\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right), \quad x > 0

Parameters

locTensor or float

Mean

\mu

of the underlying Normal

\ln X

scaleTensor or float

Standard deviation

\sigma > 0

of the underlying Normal.

validate_argsbool or None= None

Enable constraint validation. Default None.

Attributes

locTensor

Mean of the latent Normal.

scaleTensor

Standard deviation of the latent Normal.

Notes

The Log-Normal arises naturally whenever a quantity is the product of many independent positive factors (multiplicative growth), just as the Normal arises from additive contributions. Applications include particle-size distributions, financial asset prices, and reaction times.

Mean and variance of $X = e^Y$ :

\mathbb{E}[X] = e^{\mu + \sigma^2/2}, \qquad \text{Var}[X] = (e^{\sigma^2} - 1)\,e^{2\mu + \sigma^2}

Note that both grow super-exponentially in $\sigma$ .

Mode:

\text{mode}(X) = e^{\mu - \sigma^2}

The mode is always less than the mean, reflecting right-skewness.

Examples

>>> import lucid.distributions as dist
>>> d = dist.LogNormal(loc=0.0, scale=1.0)
>>> x = d.rsample((100,))
>>> (x > 0).all()   # support is strictly positive
Tensor(True)

Used by 2

Constructors

dunder

init

→None

__init__(loc: Tensor | float, scale: Tensor | float, validate_args: bool | None = None)

source edit

Construct a Log-Normal distribution.

Parameters

locTensor or float

Mean of the latent Normal

\ln X

scaleTensor or float

Standard deviation of the latent Normal,

\sigma > 0

validate_argsbool or None= None

Validate parameter constraints on construction.

Properties

prop

mean

→Tensor

mean: Tensor

source edit

Mean of the Log-Normal distribution.

\mathbb{E}[X] = e^{\mu + \sigma^2/2}

Returns

Tensor

Shape batch_shape.

prop

mode

→Tensor

mode: Tensor

source edit

Mode of the Log-Normal distribution.

\text{mode}(X) = e^{\mu - \sigma^2}

The mode is strictly less than the mean, reflecting the right-skewed nature of the distribution.

Returns

Tensor

Shape batch_shape.

prop

variance

→Tensor

variance: Tensor

source edit

Variance of the Log-Normal distribution.

\text{Var}[X] = (e^{\sigma^2} - 1)\,e^{2\mu + \sigma^2}

Returns

Tensor

Shape batch_shape.

Instance methods

entropy

→Tensor

entropy()

source edit

Shannon differential entropy of the Log-Normal distribution.

H[X] = \mu + H[\mathcal{N}(0,\sigma^2)] = \mu + \frac{1}{2} + \frac{1}{2}\ln(2\pi) + \ln\sigma

where the extra $\mu$ term accounts for the Jacobian of the exponential transformation.

Returns

Tensor

Entropy values of shape batch_shape (in nats).

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-probability density of the Log-Normal distribution.

By the change-of-variables formula:

\log p(x) = \log p_{\mathcal{N}}(\ln x) - \ln x

where $p_{\mathcal{N}}$ is the density of the underlying Normal distribution.

Parameters

valueTensor

Strictly positive observation(s)

x > 0

Returns

Tensor

Log-density at value.

rsample

→Tensor

rsample(sample_shape: tuple[int, ...] = ())

source edit

Draw reparameterised samples.

Samples are obtained by exponentiating Normal samples from the underlying Normal base distribution:

X = \exp(\mu + \sigma \varepsilon), \quad \varepsilon \sim \mathcal{N}(0,1)

Parameters

sample_shapetuple[int, ...]= ()

Shape prefix for the batch of samples.

Returns

Tensor

Strictly positive samples of shape sample_shape + batch_shape.

class

LogNormal

extendsDistribution

LogNormal(loc: Tensor | float, scale: Tensor | float, validate_args: bool | None = None)

source edit

Log-Normal distribution: $X = \exp(Y)$ where $Y \sim \mathcal{N}(\mu, \sigma^2)$ .

If a random variable $X$ has a Log-Normal distribution then its natural logarithm $\ln X$ is Normally distributed. The probability density function is

p(x \mid \mu, \sigma) = \frac{1}{x\,\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right), \quad x > 0

Parameters

locTensor or float

Mean

\mu

of the underlying Normal

\ln X

scaleTensor or float

Standard deviation

\sigma > 0

of the underlying Normal.

validate_argsbool or None= None

Enable constraint validation. Default None.

Attributes

locTensor

Mean of the latent Normal.

scaleTensor

Standard deviation of the latent Normal.

Notes

Mean and variance of $X = e^Y$ :

\mathbb{E}[X] = e^{\mu + \sigma^2/2}, \qquad \text{Var}[X] = (e^{\sigma^2} - 1)\,e^{2\mu + \sigma^2}

Note that both grow super-exponentially in $\sigma$ .

Mode:

\text{mode}(X) = e^{\mu - \sigma^2}

The mode is always less than the mean, reflecting right-skewness.

Examples

>>> import lucid.distributions as dist
>>> d = dist.LogNormal(loc=0.0, scale=1.0)
>>> x = d.rsample((100,))
>>> (x > 0).all()   # support is strictly positive
Tensor(True)

Used by 2

Constructors

dunder

init

→None

__init__(loc: Tensor | float, scale: Tensor | float, validate_args: bool | None = None)

source edit

Construct a Log-Normal distribution.

Parameters

locTensor or float

Mean of the latent Normal

\ln X

scaleTensor or float

Standard deviation of the latent Normal,

\sigma > 0

validate_argsbool or None= None

Validate parameter constraints on construction.

Properties

prop

mean

→Tensor

mean: Tensor

source edit

Mean of the Log-Normal distribution.

\mathbb{E}[X] = e^{\mu + \sigma^2/2}

Returns

Tensor

Shape batch_shape.

prop

mode

→Tensor

mode: Tensor

source edit

Mode of the Log-Normal distribution.

\text{mode}(X) = e^{\mu - \sigma^2}

The mode is strictly less than the mean, reflecting the right-skewed nature of the distribution.

Returns

Tensor

Shape batch_shape.

prop

variance

→Tensor

variance: Tensor

source edit

Variance of the Log-Normal distribution.

\text{Var}[X] = (e^{\sigma^2} - 1)\,e^{2\mu + \sigma^2}

Returns

Tensor

Shape batch_shape.

Instance methods

entropy

→Tensor

entropy()

source edit

Shannon differential entropy of the Log-Normal distribution.

H[X] = \mu + H[\mathcal{N}(0,\sigma^2)] = \mu + \frac{1}{2} + \frac{1}{2}\ln(2\pi) + \ln\sigma

where the extra $\mu$ term accounts for the Jacobian of the exponential transformation.

Returns

Tensor

Entropy values of shape batch_shape (in nats).

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-probability density of the Log-Normal distribution.

By the change-of-variables formula:

\log p(x) = \log p_{\mathcal{N}}(\ln x) - \ln x

where $p_{\mathcal{N}}$ is the density of the underlying Normal distribution.

Parameters

valueTensor

Strictly positive observation(s)

x > 0

Returns

Tensor

Log-density at value.

rsample

→Tensor

rsample(sample_shape: tuple[int, ...] = ())

source edit

Draw reparameterised samples.

Samples are obtained by exponentiating Normal samples from the underlying Normal base distribution:

X = \exp(\mu + \sigma \varepsilon), \quad \varepsilon \sim \mathcal{N}(0,1)

Parameters

sample_shapetuple[int, ...]= ()

Shape prefix for the batch of samples.

Returns

Tensor

Strictly positive samples of shape sample_shape + batch_shape.