class

InverseGamma

extendsDistribution

InverseGamma(concentration: Tensor | float, rate: Tensor | float, validate_args: bool | None = None)

source edit

Inverse-Gamma distribution on $(0, \infty)$ .

Distribution of the reciprocal of a Gamma random variable: if $Y \sim \mathrm{Gamma}(\alpha, \beta)$ then $X = 1/Y \sim \mathrm{InverseGamma}(\alpha, \beta)$ . Widely used as the conjugate prior for the variance parameter of a Normal distribution with known mean, and as a prior on positive scale parameters more generally.

Parameters

concentrationTensor or float

Shape parameter

\alpha > 0

rateTensor or float

Rate (inverse-scale) parameter

\beta > 0

validate_argsbool= None

If True, validate parameter constraints at construction time.

Notes

Probability density on $x > 0$ :

p(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp\!\left(-\frac{\beta}{x}\right)

Moments (when defined):

\mathbb{E}[X] = \frac{\beta}{\alpha - 1}\;\;(\alpha > 1), \qquad \mathrm{Var}[X] = \frac{\beta^2}{(\alpha - 1)^2 (\alpha - 2)} \;\;(\alpha > 2)

Mode: $\beta / (\alpha + 1)$ .

Conjugacy: with a $\mathrm{Normal}(\mu, \sigma^2)$ likelihood and known mean $\mu$ , the posterior over $\sigma^2$ is again Inverse-Gamma with updated parameters $\alpha' = \alpha + n/2$ , $\beta' = \beta + \tfrac{1}{2}\sum_i (x_i - \mu)^2$ .

Because Lucid's lucid.distributions.Gamma sampler is rejection-based, has_rsample is False here and sample returns detached samples obtained by reciprocating a Gamma draw.

Examples

>>> import lucid
>>> from lucid.distributions import InverseGamma
>>> d = InverseGamma(concentration=3.0, rate=2.0)
>>> d.mean  # β / (α - 1) = 1.0
tensor(1.)
>>> d.sample((4,)).shape
(4,)

Used by 2

Constructors

dunder

init

→None

__init__(concentration: Tensor | float, rate: Tensor | float, validate_args: bool | None = None)

source edit

Initialise an Inverse-Gamma distribution.

Parameters

concentrationTensor | float

Shape parameter

\alpha > 0

rateTensor | float

Rate (inverse-scale) parameter

\beta > 0

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Properties

prop

mean

→Tensor

mean: Tensor

source edit

Defined for concentration > 1: β / (α − 1).

prop

mode

→Tensor

mode: Tensor

source edit

Mode of the inverse-Gamma distribution: $\beta / (\alpha + 1)$ .

prop

variance

→Tensor

variance: Tensor

source edit

Defined for concentration > 2: β² / ((α − 1)² (α − 2)).

Instance methods

cdf

→Tensor

cdf(value: Tensor)

source edit

Cumulative distribution function of the inverse-Gamma distribution.

F(x; \alpha, \beta) = Q(\alpha, \beta / x), \quad x > 0

the regularized upper incomplete gamma (lucid.special.gammaincc) — $1/X$ is Gamma(alpha, beta), and $X \leq x$ is $1/X \geq 1/x$ . Evaluated as $Q$ itself, not as $1 - P$ , so the small left tail keeps its relative precision.

Returns probabilities in $[0, 1]$ ; 0 at and below 0.

Raises

NotImplementedError

From backward, when concentration requires a gradient, as in the reference framework.

Examples

>>> InverseGamma(concentration=1.0, rate=1.0).cdf(lucid.tensor(1.0))  # 1/e
tensor(0.3679)

entropy

→Tensor

entropy()

source edit

α + log β + log Γ(α) − (1 + α) · ψ(α).

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-probability density of the Inverse-Gamma distribution.

Parameters

valueTensor

Point(s)

x > 0

at which to evaluate the density.

Returns

Tensor

Log-density $\alpha\log\beta - \log\Gamma(\alpha) - (\alpha+1)\log x - \beta/x$ .

sample

→Tensor

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

source edit

Sample by taking the reciprocal of a Gamma sample.

class

InverseGamma

extendsDistribution

InverseGamma(concentration: Tensor | float, rate: Tensor | float, validate_args: bool | None = None)

source edit

Inverse-Gamma distribution on $(0, \infty)$ .

Parameters

concentrationTensor or float

Shape parameter

\alpha > 0

rateTensor or float

Rate (inverse-scale) parameter

\beta > 0

validate_argsbool= None

If True, validate parameter constraints at construction time.

Notes

Probability density on $x > 0$ :

p(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp\!\left(-\frac{\beta}{x}\right)

Moments (when defined):

\mathbb{E}[X] = \frac{\beta}{\alpha - 1}\;\;(\alpha > 1), \qquad \mathrm{Var}[X] = \frac{\beta^2}{(\alpha - 1)^2 (\alpha - 2)} \;\;(\alpha > 2)

Mode: $\beta / (\alpha + 1)$ .

Because Lucid's lucid.distributions.Gamma sampler is rejection-based, has_rsample is False here and sample returns detached samples obtained by reciprocating a Gamma draw.

Examples

>>> import lucid
>>> from lucid.distributions import InverseGamma
>>> d = InverseGamma(concentration=3.0, rate=2.0)
>>> d.mean  # β / (α - 1) = 1.0
tensor(1.)
>>> d.sample((4,)).shape
(4,)

Used by 2

Constructors

dunder

init

→None

__init__(concentration: Tensor | float, rate: Tensor | float, validate_args: bool | None = None)

source edit

Initialise an Inverse-Gamma distribution.

Parameters

concentrationTensor | float

Shape parameter

\alpha > 0

rateTensor | float

Rate (inverse-scale) parameter

\beta > 0

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Properties

prop

mean

→Tensor

mean: Tensor

source edit

Defined for concentration > 1: β / (α − 1).

prop

mode

→Tensor

mode: Tensor

source edit

Mode of the inverse-Gamma distribution: $\beta / (\alpha + 1)$ .

prop

variance

→Tensor

variance: Tensor

source edit

Defined for concentration > 2: β² / ((α − 1)² (α − 2)).

Instance methods

cdf

→Tensor

cdf(value: Tensor)

source edit

Cumulative distribution function of the inverse-Gamma distribution.

F(x; \alpha, \beta) = Q(\alpha, \beta / x), \quad x > 0

Returns probabilities in $[0, 1]$ ; 0 at and below 0.

Raises

NotImplementedError

From backward, when concentration requires a gradient, as in the reference framework.

Examples

>>> InverseGamma(concentration=1.0, rate=1.0).cdf(lucid.tensor(1.0))  # 1/e
tensor(0.3679)

entropy

→Tensor

entropy()

source edit

α + log β + log Γ(α) − (1 + α) · ψ(α).

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-probability density of the Inverse-Gamma distribution.

Parameters

valueTensor

Point(s)

x > 0

at which to evaluate the density.

Returns

Tensor

Log-density $\alpha\log\beta - \log\Gamma(\alpha) - (\alpha+1)\log x - \beta/x$ .

sample

→Tensor

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

source edit

Sample by taking the reciprocal of a Gamma sample.