class

Cauchy

extendsDistribution

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

source edit

Cauchy distribution on $\mathbb{R}$ — heavy-tailed, all moments undefined.

The canonical example of a pathological heavy-tailed distribution: its tails decay so slowly that no integer moment (mean, variance, skew, kurtosis) is defined. The Cauchy is the Student's-t distribution with one degree of freedom and arises naturally as the ratio of two independent standard Normals.

Parameters

locTensor or float

Location parameter

x_0 \in \mathbb{R}

(median and mode). The "mean" does not exist.

scaleTensor or float

Scale parameter

\gamma > 0

(half-width at half-maximum).

validate_argsbool= None

If True, validate parameter constraints at construction time.

Notes

Probability density:

p(x; x_0, \gamma) = \frac{1}{\pi \gamma} \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]^{-1}

Cumulative distribution:

F(x; x_0, \gamma) = \frac{1}{\pi} \arctan\!\left(\frac{x - x_0}{\gamma}\right) + \frac{1}{2}

Properties (no moments — all are undefined or infinite):

Median / mode: $x_0$
Mean / variance: undefined (integrals diverge)
Entropy: $H[X] = \log(4 \pi \gamma)$

Sample mean does not converge — the strong law of large numbers fails for IID Cauchy variates. Any finite-sample average of Cauchy draws is itself Cauchy-distributed with the same parameters.

Stability: a sum of independent Cauchy variables remains Cauchy (it is an $\alpha$ -stable distribution with stability index 1).

Reparameterised sampling uses the inverse-CDF $X = x_0 + \gamma \tan(\pi(U - \tfrac{1}{2}))$ .

Examples

>>> import lucid
>>> from lucid.distributions import Cauchy
>>> d = Cauchy(loc=0.0, scale=1.0)
>>> d.rsample((4,))
Tensor([...])
>>> d.log_prob(lucid.tensor(0.0))
Tensor(-1.1447)

Used by 3

Constructors

dunder

init

→None

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

source edit

Construct a Cauchy distribution.

Parameters

locTensor | float

Location parameter (median and mode)

x_0 \in \mathbb{R}

scaleTensor | float

Scale (half-width at half-maximum) parameter

\gamma > 0

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Notes

The Cauchy distribution has PDF:

p(x; x_0, \gamma) = \frac{1}{\pi \gamma} \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]^{-1}

The Cauchy is the Student's t-distribution with one degree of freedom. Because all moments of order $\geq 1$ are undefined, it is a canonical example of a heavy-tailed distribution where the sample mean does not converge to any value.

Examples

>>> from lucid.distributions import Cauchy
>>> d = Cauchy(loc=0.0, scale=1.0)
>>> x = d.rsample((1000,))

Instance methods

cdf

→Tensor

cdf(value: Tensor)

source edit

Cumulative distribution function of the Cauchy distribution.

F(x; x_0, \gamma) = \frac{1}{\pi} \arctan\!\left(\frac{x - x_0}{\gamma}\right) + \frac{1}{2}

Parameters

valueTensor

Real values at which to evaluate the CDF.

Returns

Tensor

CDF values in $(0, 1)$ , shape batch_shape.

entropy

→Tensor

entropy()

source edit

Shannon entropy of the Cauchy distribution (in nats).

H(X) = \log(4 \pi \gamma)

Returns

Tensor

Entropy in nats, shape batch_shape.

Examples

>>> Cauchy(loc=0.0, scale=1.0).entropy()  # log(4π) ≈ 2.531
Tensor(2.5310)

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-density of value under the Cauchy distribution.

\log p(x; x_0, \gamma) = -\log \pi - \log \gamma - \log\!\left(1 + \left(\frac{x - x_0}{\gamma}\right)^2\right)

Parameters

valueTensor

Real-valued observations.

Returns

Tensor

Element-wise log-densities, shape batch_shape.

Examples

>>> Cauchy(loc=0.0, scale=1.0).log_prob(lucid.tensor(0.0))
Tensor(-1.1447)

rsample

→Tensor

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

source edit

Reparameterised sample via the inverse-CDF method.

Uses the quantile function of the Cauchy distribution:

X = x_0 + \gamma \tan\!\bigl(\pi (U - \tfrac{1}{2})\bigr)

where $U \sim \text{Uniform}(0, 1)$ . Gradients flow through both $x_0$ and $\gamma$ .

Parameters

sample_shapetuple[int, ...]= ()

Leading shape dimensions for the sample batch. Default is ().

Returns

Tensor

Reparameterised samples of shape sample_shape + batch_shape.

Examples

>>> d = Cauchy(loc=0.0, scale=1.0)
>>> x = d.rsample((500,))

class

Cauchy

extendsDistribution

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

source edit

Cauchy distribution on $\mathbb{R}$ — heavy-tailed, all moments undefined.

Parameters

locTensor or float

Location parameter

x_0 \in \mathbb{R}

(median and mode). The "mean" does not exist.

scaleTensor or float

Scale parameter

\gamma > 0

(half-width at half-maximum).

validate_argsbool= None

If True, validate parameter constraints at construction time.

Notes

Probability density:

p(x; x_0, \gamma) = \frac{1}{\pi \gamma} \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]^{-1}

Cumulative distribution:

F(x; x_0, \gamma) = \frac{1}{\pi} \arctan\!\left(\frac{x - x_0}{\gamma}\right) + \frac{1}{2}

Properties (no moments — all are undefined or infinite):

Median / mode: $x_0$
Mean / variance: undefined (integrals diverge)
Entropy: $H[X] = \log(4 \pi \gamma)$

Sample mean does not converge — the strong law of large numbers fails for IID Cauchy variates. Any finite-sample average of Cauchy draws is itself Cauchy-distributed with the same parameters.

Stability: a sum of independent Cauchy variables remains Cauchy (it is an $\alpha$ -stable distribution with stability index 1).

Reparameterised sampling uses the inverse-CDF $X = x_0 + \gamma \tan(\pi(U - \tfrac{1}{2}))$ .

Examples

>>> import lucid
>>> from lucid.distributions import Cauchy
>>> d = Cauchy(loc=0.0, scale=1.0)
>>> d.rsample((4,))
Tensor([...])
>>> d.log_prob(lucid.tensor(0.0))
Tensor(-1.1447)

Used by 3

Constructors

dunder

init

→None

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

source edit

Construct a Cauchy distribution.

Parameters

locTensor | float

Location parameter (median and mode)

x_0 \in \mathbb{R}

scaleTensor | float

Scale (half-width at half-maximum) parameter

\gamma > 0

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Notes

The Cauchy distribution has PDF:

p(x; x_0, \gamma) = \frac{1}{\pi \gamma} \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]^{-1}

Examples

>>> from lucid.distributions import Cauchy
>>> d = Cauchy(loc=0.0, scale=1.0)
>>> x = d.rsample((1000,))

Instance methods

cdf

→Tensor

cdf(value: Tensor)

source edit

Cumulative distribution function of the Cauchy distribution.

F(x; x_0, \gamma) = \frac{1}{\pi} \arctan\!\left(\frac{x - x_0}{\gamma}\right) + \frac{1}{2}

Parameters

valueTensor

Real values at which to evaluate the CDF.

Returns

Tensor

CDF values in $(0, 1)$ , shape batch_shape.

entropy

→Tensor

entropy()

source edit

Shannon entropy of the Cauchy distribution (in nats).

H(X) = \log(4 \pi \gamma)

Returns

Tensor

Entropy in nats, shape batch_shape.

Examples

>>> Cauchy(loc=0.0, scale=1.0).entropy()  # log(4π) ≈ 2.531
Tensor(2.5310)

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-density of value under the Cauchy distribution.

\log p(x; x_0, \gamma) = -\log \pi - \log \gamma - \log\!\left(1 + \left(\frac{x - x_0}{\gamma}\right)^2\right)

Parameters

valueTensor

Real-valued observations.

Returns

Tensor

Element-wise log-densities, shape batch_shape.

Examples

>>> Cauchy(loc=0.0, scale=1.0).log_prob(lucid.tensor(0.0))
Tensor(-1.1447)

rsample

→Tensor

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

source edit

Reparameterised sample via the inverse-CDF method.

Uses the quantile function of the Cauchy distribution:

X = x_0 + \gamma \tan\!\bigl(\pi (U - \tfrac{1}{2})\bigr)

where $U \sim \text{Uniform}(0, 1)$ . Gradients flow through both $x_0$ and $\gamma$ .

Parameters

sample_shapetuple[int, ...]= ()

Leading shape dimensions for the sample batch. Default is ().

Returns

Tensor

Reparameterised samples of shape sample_shape + batch_shape.

Examples

>>> d = Cauchy(loc=0.0, scale=1.0)
>>> x = d.rsample((500,))