class

Pareto

extendsDistribution

Pareto(scale: Tensor | float, alpha: Tensor | float, validate_args: bool | None = None)

source edit

Pareto distribution — a heavy-tailed power-law family on $[x_m, \infty)$ .

Pareto(scale=x_m, alpha=α) models quantities that follow a power-law tail. The 80/20 rule (Pareto principle) is a heuristic consequence of this distribution with $\alpha \approx 1.16$ .

Parameters

scaleTensor | float

Scale parameter

x_m > 0

— the minimum possible value of the random variable (the lower bound of the support).

alphaTensor | float

Shape (tail-index) parameter

\alpha > 0

. Smaller values produce heavier tails. The mean exists only for

\alpha > 1

; the variance only for

\alpha > 2

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Attributes

scaleTensor

Minimum value

x_m

alphaTensor

Tail index

\alpha

Notes

PDF:

p(x; x_m, \alpha) = \frac{\alpha \, x_m^\alpha}{x^{\alpha+1}}, \quad x \geq x_m

Log-PDF:

\log p(x) = \log\alpha + \alpha\log x_m - (\alpha+1)\log x

Moments (when defined):

Mean ( $\alpha > 1$ ): $E[X] = \alpha x_m / (\alpha - 1)$
Variance ( $\alpha > 2$ ): $\operatorname{Var}[X] = x_m^2 \alpha / ((\alpha-1)^2 (\alpha-2))$

Entropy:

H[X] = \log(x_m / \alpha) + 1/\alpha + 1

Reparameterised sampling uses the inverse-CDF trick: if $U \sim \operatorname{Uniform}(0, 1)$ then $X = x_m (1-U)^{-1/\alpha} \sim \operatorname{Pareto}(x_m, \alpha)$ .

Examples

>>> import lucid
>>> from lucid.distributions import Pareto
>>> dist = Pareto(scale=1.0, alpha=2.0)
>>> samples = dist.rsample((200,))
>>> bool(samples.min() >= 1.0)  # always >= scale
True

Used by 2

Constructors

dunder

init

→None

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

source edit

Initialise a Pareto distribution.

Parameters

scaleTensor | float

Scale parameter

x_m > 0

— the minimum value of the support (lower bound of the distribution).

alphaTensor | float

Shape (tail-index) parameter

\alpha > 0

. Smaller values produce heavier tails.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Properties

prop

mean

→Tensor

mean: Tensor

source edit

Mean of the Pareto distribution: $E[X] = \alpha x_m / (\alpha - 1)$ .

Defined only for $\alpha > 1$ ; returns inf or nan for $\alpha \leq 1$ .

Returns

Tensor

Mean values of shape batch_shape.

prop

mode

→Tensor

mode: Tensor

source edit

Mode of the Pareto distribution: the scale $x_m$ , where the density is largest.

prop

support

→Constraint

support: Constraint

source edit

Support of the Pareto distribution: $[x_m, \infty)$ .

Returns

Constraint

greater_than_eq(scale) — the bound is the scale tensor, so each batch member is held to its own $x_m$ .

prop

variance

→Tensor

variance: Tensor

source edit

Variance of the Pareto distribution.

$\operatorname{Var}[X] = x_m^2 \alpha / ((\alpha-1)^2 (\alpha-2))$

Defined only for $\alpha > 2$ .

Returns

Tensor

Variance values of shape batch_shape.

Instance methods

cdf

→Tensor

cdf(value: Tensor)

source edit

Cumulative distribution function of the Pareto distribution.

F(x) = 1 - \left(\frac{x_m}{x}\right)^{\alpha}, \quad x \geq x_m

The Exponential(alpha) CDF at $\ln(x / x_m)$ ; 0 below $x_m$ .

Returns probabilities in $[0, 1]$ .

Examples

>>> Pareto(scale=1.0, alpha=2.0).cdf(lucid.tensor(2.0))  # 1 - 1/4
tensor(0.75)

entropy

→Tensor

entropy()

source edit

Entropy of the Pareto distribution.

H[X] = \log(x_m / \alpha) + 1/\alpha + 1

Returns

Tensor

Entropy values of shape batch_shape (nats).

icdf

→Tensor

icdf(value: Tensor)

source edit

Quantile function of the Pareto distribution.

Q(p) = x_m (1 - p)^{-1/\alpha}

Returns quantiles in $[x_m, \infty]$ .

Examples

>>> Pareto(scale=1.0, alpha=2.0).icdf(lucid.tensor(0.75))
tensor(2.)

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-probability density of the Pareto distribution.

\log p(x) = \log\alpha + \alpha\log x_m - (\alpha+1)\log x

Parameters

valueTensor

Points

x \geq x_m

at which to evaluate the density.

Returns

Tensor

Log-density values of the same shape as value.

rsample

→Tensor

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

source edit

Draw reparameterised samples via the inverse-CDF trick.

Uses $X = x_m (1 - U)^{-1/\alpha}$ where $U \sim \operatorname{Uniform}(0, 1)$ . Gradients propagate through both scale and alpha.

Parameters

sample_shapetuple[int, ...]= ()

Leading shape of the output sample batch.

Returns

Tensor

Samples $\geq x_m$ of shape (*sample_shape, *batch_shape).

class

Pareto

extendsDistribution

Pareto(scale: Tensor | float, alpha: Tensor | float, validate_args: bool | None = None)

source edit

Pareto distribution — a heavy-tailed power-law family on $[x_m, \infty)$ .

Pareto(scale=x_m, alpha=α) models quantities that follow a power-law tail. The 80/20 rule (Pareto principle) is a heuristic consequence of this distribution with $\alpha \approx 1.16$ .

Parameters

scaleTensor | float

Scale parameter

x_m > 0

— the minimum possible value of the random variable (the lower bound of the support).

alphaTensor | float

Shape (tail-index) parameter

\alpha > 0

. Smaller values produce heavier tails. The mean exists only for

\alpha > 1

; the variance only for

\alpha > 2

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Attributes

scaleTensor

Minimum value

x_m

alphaTensor

Tail index

\alpha

Notes

PDF:

p(x; x_m, \alpha) = \frac{\alpha \, x_m^\alpha}{x^{\alpha+1}}, \quad x \geq x_m

Log-PDF:

\log p(x) = \log\alpha + \alpha\log x_m - (\alpha+1)\log x

Moments (when defined):

Mean ( $\alpha > 1$ ): $E[X] = \alpha x_m / (\alpha - 1)$
Variance ( $\alpha > 2$ ): $\operatorname{Var}[X] = x_m^2 \alpha / ((\alpha-1)^2 (\alpha-2))$

Entropy:

H[X] = \log(x_m / \alpha) + 1/\alpha + 1

Reparameterised sampling uses the inverse-CDF trick: if $U \sim \operatorname{Uniform}(0, 1)$ then $X = x_m (1-U)^{-1/\alpha} \sim \operatorname{Pareto}(x_m, \alpha)$ .

Examples

>>> import lucid
>>> from lucid.distributions import Pareto
>>> dist = Pareto(scale=1.0, alpha=2.0)
>>> samples = dist.rsample((200,))
>>> bool(samples.min() >= 1.0)  # always >= scale
True

Used by 2

Constructors

dunder

init

→None

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

source edit

Initialise a Pareto distribution.

Parameters

scaleTensor | float

Scale parameter

x_m > 0

— the minimum value of the support (lower bound of the distribution).

alphaTensor | float

Shape (tail-index) parameter

\alpha > 0

. Smaller values produce heavier tails.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Properties

prop

mean

→Tensor

mean: Tensor

source edit

Mean of the Pareto distribution: $E[X] = \alpha x_m / (\alpha - 1)$ .

Defined only for $\alpha > 1$ ; returns inf or nan for $\alpha \leq 1$ .

Returns

Tensor

Mean values of shape batch_shape.

prop

mode

→Tensor

mode: Tensor

source edit

Mode of the Pareto distribution: the scale $x_m$ , where the density is largest.

prop

support

→Constraint

support: Constraint

source edit

Support of the Pareto distribution: $[x_m, \infty)$ .

Returns

Constraint

greater_than_eq(scale) — the bound is the scale tensor, so each batch member is held to its own $x_m$ .

prop

variance

→Tensor

variance: Tensor

source edit

Variance of the Pareto distribution.

$\operatorname{Var}[X] = x_m^2 \alpha / ((\alpha-1)^2 (\alpha-2))$

Defined only for $\alpha > 2$ .

Returns

Tensor

Variance values of shape batch_shape.

Instance methods

cdf

→Tensor

cdf(value: Tensor)

source edit

Cumulative distribution function of the Pareto distribution.

F(x) = 1 - \left(\frac{x_m}{x}\right)^{\alpha}, \quad x \geq x_m

The Exponential(alpha) CDF at $\ln(x / x_m)$ ; 0 below $x_m$ .

Returns probabilities in $[0, 1]$ .

Examples

>>> Pareto(scale=1.0, alpha=2.0).cdf(lucid.tensor(2.0))  # 1 - 1/4
tensor(0.75)

entropy

→Tensor

entropy()

source edit

Entropy of the Pareto distribution.

H[X] = \log(x_m / \alpha) + 1/\alpha + 1

Returns

Tensor

Entropy values of shape batch_shape (nats).

icdf

→Tensor

icdf(value: Tensor)

source edit

Quantile function of the Pareto distribution.

Q(p) = x_m (1 - p)^{-1/\alpha}

Returns quantiles in $[x_m, \infty]$ .

Examples

>>> Pareto(scale=1.0, alpha=2.0).icdf(lucid.tensor(0.75))
tensor(2.)

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-probability density of the Pareto distribution.

\log p(x) = \log\alpha + \alpha\log x_m - (\alpha+1)\log x

Parameters

valueTensor

Points

x \geq x_m

at which to evaluate the density.

Returns

Tensor

Log-density values of the same shape as value.

rsample

→Tensor

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

source edit

Draw reparameterised samples via the inverse-CDF trick.

Uses $X = x_m (1 - U)^{-1/\alpha}$ where $U \sim \operatorname{Uniform}(0, 1)$ . Gradients propagate through both scale and alpha.

Parameters

sample_shapetuple[int, ...]= ()

Leading shape of the output sample batch.

Returns

Tensor

Samples $\geq x_m$ of shape (*sample_shape, *batch_shape).