class

HalfNormal

extendsDistribution

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

source edit

Half-Normal distribution — the absolute value of a zero-mean Normal.

If $X \sim \mathcal{N}(0, \sigma^2)$ then $|X| \sim \operatorname{HalfNormal}(\sigma)$ . The distribution is supported on $[0, \infty)$ and arises naturally as a scale prior in hierarchical Bayesian models (it is weakly informative while keeping probability mass away from zero).

Parameters

scaleTensor | float

Scale parameter

\sigma > 0

— the standard deviation of the underlying zero-mean Normal.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Attributes

scaleTensor

Scale parameter

\sigma

Notes

PDF:

p(x; \sigma) = \frac{\sqrt{2}}{\sigma\sqrt{\pi}} \exp\!\left(-\frac{x^2}{2\sigma^2}\right), \quad x \geq 0

Log-PDF:

\log p(x) = \log 2 + \log\mathcal{N}(x; 0, \sigma^2)

where the right-hand side is the log-density of the full Normal evaluated at $x$ .

Moments:

Mean: $E[X] = \sigma\sqrt{2/\pi}$
Variance: $\operatorname{Var}[X] = \sigma^2(1 - 2/\pi)$

Entropy:

H[X] = \tfrac{1}{2}\log\!\left(\frac{\pi e \sigma^2}{2}\right)

Reparameterised sampling takes abs of a Normal sample, so gradients flow through $\sigma$ unobstructed.

Examples

>>> import lucid
>>> from lucid.distributions import HalfNormal
>>> dist = HalfNormal(scale=1.0)
>>> samples = dist.rsample((300,))
>>> bool((samples >= 0.0).all())
True

Used by 2

Constructors

dunder

init

→None

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

source edit

Initialise a HalfNormal distribution.

Parameters

scaleTensor | float

Scale parameter

\sigma > 0

— the standard deviation of the underlying zero-mean Normal distribution.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Properties

prop

mean

→Tensor

mean: Tensor

source edit

Mean of the HalfNormal distribution: $E[X] = \sigma\sqrt{2/\pi}$ .

Returns

Tensor

Mean values of shape batch_shape.

prop

mode

→Tensor

mode: Tensor

source edit

Mode of the half-Normal distribution: 0, where the density peaks.

prop

variance

→Tensor

variance: Tensor

source edit

Variance of the HalfNormal distribution: $\operatorname{Var}[X] = \sigma^2(1 - 2/\pi)$ .

Returns

Tensor

Variance values of shape batch_shape.

Instance methods

cdf

→Tensor

cdf(value: Tensor)

source edit

Cumulative distribution function of the half-Normal distribution.

F(x) = 2\Phi(x/\sigma) - 1 = \operatorname{erf}\!\left(\frac{x}{\sigma\sqrt 2}\right), \quad x \geq 0

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

Examples

>>> HalfNormal(scale=1.0).cdf(lucid.tensor(1.0))
tensor(0.6827)

entropy

→Tensor

entropy()

source edit

Entropy of the HalfNormal distribution.

H[X] = \tfrac{1}{2}\log\!\left(\frac{\pi e \sigma^2}{2}\right)

Returns

Tensor

Entropy values of shape batch_shape (nats).

icdf

→Tensor

icdf(value: Tensor)

source edit

Quantile function of the half-Normal distribution.

Q(p) = \sigma \Phi^{-1}\!\left(\frac{1 + p}{2}\right)

Returns quantiles in $[0, \infty]$ .

Examples

>>> HalfNormal(scale=1.0).icdf(lucid.tensor(0.5))
tensor(0.6745)

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-probability density of the HalfNormal distribution.

\log p(x) = \log 2 + \log\mathcal{N}(x; 0, \sigma^2)

Parameters

valueTensor

Non-negative points

x \geq 0

at which to evaluate.

Returns

Tensor

Log-density values of the same shape as value; -inf below 0, where the density is 0.

rsample

→Tensor

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

source edit

Draw reparameterised samples by folding a Normal sample.

Computes abs of a Normal sample, so gradients propagate through $\sigma$ unobstructed.

Parameters

sample_shapetuple[int, ...]= ()

Leading shape of the output sample batch.

Returns

Tensor

Non-negative samples of shape (*sample_shape, *batch_shape).

class

HalfNormal

extendsDistribution

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

source edit

Half-Normal distribution — the absolute value of a zero-mean Normal.

Parameters

scaleTensor | float

Scale parameter

\sigma > 0

— the standard deviation of the underlying zero-mean Normal.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Attributes

scaleTensor

Scale parameter

\sigma

Notes

PDF:

p(x; \sigma) = \frac{\sqrt{2}}{\sigma\sqrt{\pi}} \exp\!\left(-\frac{x^2}{2\sigma^2}\right), \quad x \geq 0

Log-PDF:

\log p(x) = \log 2 + \log\mathcal{N}(x; 0, \sigma^2)

where the right-hand side is the log-density of the full Normal evaluated at $x$ .

Moments:

Mean: $E[X] = \sigma\sqrt{2/\pi}$
Variance: $\operatorname{Var}[X] = \sigma^2(1 - 2/\pi)$

Entropy:

H[X] = \tfrac{1}{2}\log\!\left(\frac{\pi e \sigma^2}{2}\right)

Reparameterised sampling takes abs of a Normal sample, so gradients flow through $\sigma$ unobstructed.

Examples

>>> import lucid
>>> from lucid.distributions import HalfNormal
>>> dist = HalfNormal(scale=1.0)
>>> samples = dist.rsample((300,))
>>> bool((samples >= 0.0).all())
True

Used by 2

Constructors

dunder

init

→None

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

source edit

Initialise a HalfNormal distribution.

Parameters

scaleTensor | float

Scale parameter

\sigma > 0

— the standard deviation of the underlying zero-mean Normal distribution.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Properties

prop

mean

→Tensor

mean: Tensor

source edit

Mean of the HalfNormal distribution: $E[X] = \sigma\sqrt{2/\pi}$ .

Returns

Tensor

Mean values of shape batch_shape.

prop

mode

→Tensor

mode: Tensor

source edit

Mode of the half-Normal distribution: 0, where the density peaks.

prop

variance

→Tensor

variance: Tensor

source edit

Variance of the HalfNormal distribution: $\operatorname{Var}[X] = \sigma^2(1 - 2/\pi)$ .

Returns

Tensor

Variance values of shape batch_shape.

Instance methods

cdf

→Tensor

cdf(value: Tensor)

source edit

Cumulative distribution function of the half-Normal distribution.

F(x) = 2\Phi(x/\sigma) - 1 = \operatorname{erf}\!\left(\frac{x}{\sigma\sqrt 2}\right), \quad x \geq 0

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

Examples

>>> HalfNormal(scale=1.0).cdf(lucid.tensor(1.0))
tensor(0.6827)

entropy

→Tensor

entropy()

source edit

Entropy of the HalfNormal distribution.

H[X] = \tfrac{1}{2}\log\!\left(\frac{\pi e \sigma^2}{2}\right)

Returns

Tensor

Entropy values of shape batch_shape (nats).

icdf

→Tensor

icdf(value: Tensor)

source edit

Quantile function of the half-Normal distribution.

Q(p) = \sigma \Phi^{-1}\!\left(\frac{1 + p}{2}\right)

Returns quantiles in $[0, \infty]$ .

Examples

>>> HalfNormal(scale=1.0).icdf(lucid.tensor(0.5))
tensor(0.6745)

log_prob

→Tensor

log_prob(value: Tensor)

source edit

Log-probability density of the HalfNormal distribution.

\log p(x) = \log 2 + \log\mathcal{N}(x; 0, \sigma^2)

Parameters

valueTensor

Non-negative points

x \geq 0

at which to evaluate.

Returns

Tensor

Log-density values of the same shape as value; -inf below 0, where the density is 0.

rsample

→Tensor

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

source edit

Draw reparameterised samples by folding a Normal sample.

Computes abs of a Normal sample, so gradients propagate through $\sigma$ unobstructed.

Parameters

sample_shapetuple[int, ...]= ()

Leading shape of the output sample batch.

Returns

Tensor

Non-negative samples of shape (*sample_shape, *batch_shape).