class

Normal

extendsExponentialFamily

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

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Univariate Gaussian (Normal) distribution $\mathcal{N}(\mu, \sigma^2)$ .

The Normal distribution is arguably the most important distribution in probability and statistics. Its probability density function is the iconic bell curve:

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

where $\mu$ is the location (mean) and $\sigma > 0$ is the scale (standard deviation).

Parameters

locTensor or float

Mean

\mu

of the distribution. May be any real value. Can be a scalar, a 1-D tensor of means (producing a batch), or any higher-dimensional tensor.

scaleTensor or float

Standard deviation

\sigma > 0

. Must be strictly positive. Broadcast-compatible with loc.

validate_argsbool or None= None

Enable parameter and support validation. Default None.

Attributes

locTensor

Broadcast-expanded mean tensor.

scaleTensor

Broadcast-expanded standard-deviation tensor.

Notes

Location-scale family

The Normal distribution is closed under linear transformations: if $X \sim \mathcal{N}(\mu, \sigma^2)$ then $aX + b \sim \mathcal{N}(a\mu + b, a^2\sigma^2)$ . In particular the standard Normal $Z = (X - \mu)/\sigma$ satisfies $Z \sim \mathcal{N}(0, 1)$ .

Central Limit Theorem

By the CLT, the (rescaled) sum of $n$ i.i.d. random variables with finite mean and variance converges in distribution to a Normal. This explains why the Normal appears naturally across virtually all domains of science and engineering.

Reparameterisation

has_rsample = True — samples are drawn via the standard-Normal reparameterisation trick:

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

so gradients flow through $\mu$ and $\sigma$ .

Exponential family

The Normal is an exponential-family distribution with natural parameters $\eta_1 = \mu/\sigma^2$ , $\eta_2 = -1/(2\sigma^2)$ and sufficient statistics $T(x) = (x,\, x^2)$ .

Examples

>>> import lucid.distributions as dist
>>> d = dist.Normal(loc=0.0, scale=1.0)
>>> d.mean
Tensor(0.)
>>> d.variance
Tensor(1.)
>>> x = d.rsample((5,))   # 5 standard-Normal draws
>>> lp = d.log_prob(x)    # evaluate log-density at those points

Used by 3

Constructors

dunder

init

→None

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

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Construct a Normal distribution.

Parameters

locTensor or float

Mean

\mu

scaleTensor or float

Standard deviation

\sigma > 0

validate_argsbool or None= None

Validate parameter constraints on construction.

Properties

prop

mean

→Tensor

mean: Tensor

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Mean of the Normal distribution.

For $X \sim \mathcal{N}(\mu, \sigma^2)$ ,

\mathbb{E}[X] = \mu

Returns

Tensor

The loc parameter tensor (shape batch_shape).

prop

mode

→Tensor

mode: Tensor

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Mode of the Normal distribution.

The Normal density is symmetric and unimodal; its unique maximum is at $x = \mu$ .

\text{mode}(X) = \mu

Returns

Tensor

The loc parameter tensor (shape batch_shape).

prop

stddev

→Tensor

stddev: Tensor

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Standard deviation of the Normal distribution.

\text{std}(X) = \sigma

Overrides the base-class default (which would compute variance.sqrt()) for a cheaper, exact result.

Returns

Tensor

The scale parameter tensor (shape batch_shape).

prop

variance

→Tensor

variance: Tensor

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Variance of the Normal distribution.

\text{Var}[X] = \sigma^2

Returns

Tensor

Element-wise square of scale (shape batch_shape).

Instance methods

cdf

→Tensor

cdf(value: Tensor)

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Cumulative distribution function of the Normal distribution.

Expressed in terms of the Gauss error function:

F(x) = \frac{1}{2}\left[1 + \operatorname{erf} \!\left(\frac{x - \mu}{\sigma\sqrt{2}}\right)\right]

Parameters

valueTensor

Evaluation point(s)

x

Returns

Tensor

Probabilities in $(0, 1)$ .

entropy

→Tensor

entropy()

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Shannon differential entropy of the Normal distribution.

The Normal maximises entropy among all distributions with fixed mean and variance. The closed-form entropy is

H[X] = \frac{1}{2}\ln(2\pi e\sigma^2) = \frac{1}{2} + \frac{1}{2}\ln(2\pi) + \ln\sigma

Measured in nats.

Returns

Tensor

Entropy values of shape batch_shape.

icdf

→Tensor

icdf(value: Tensor)

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Inverse CDF (quantile function / probit function) of the Normal.

Q(p) = \mu + \sigma\,\sqrt{2}\,\operatorname{erfinv}(2p - 1)

Parameters

valueTensor

Probability values

p \in (0, 1)

Returns

Tensor

Corresponding quantiles $x = Q(p)$ .

log_prob

→Tensor

log_prob(value: Tensor)

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Log-probability density of the Normal distribution.

\log p(x \mid \mu, \sigma) = -\frac{(x - \mu)^2}{2\sigma^2} - \log \sigma - \frac{1}{2}\log(2\pi)

This is computed in a numerically stable form using log_scale rather than squaring and dividing.

Parameters

valueTensor

Observation(s)

x \in \mathbb{R}

Returns

Tensor

Log-density at value, shape broadcast(value, batch_shape).

rsample

→Tensor

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

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Draw reparameterised samples via the location-scale trick.

Samples are generated as

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

Because the stochasticity is isolated in $\varepsilon$ (which does not depend on the parameters), gradients flow back through both loc ( $\mu$ ) and scale ( $\sigma$ ).

Parameters

sample_shapetuple[int, ...]= ()

Shape prefix for the batch of samples.

Returns

Tensor

Samples of shape sample_shape + batch_shape, attached to the autograd graph.

>>> import lucid.distributions as dist >>> d = dist.Normal(loc=0.0, scale=1.0) >>> d.mean Tensor(0.) >>> d.variance Tensor(1.) >>> x = d.rsample((5,)) # 5 standard-Normal draws >>> lp = d.log_prob(x) # evaluate log-density at those points