class

MultivariateNormal

extendsDistribution

MultivariateNormal(loc: Tensor, covariance_matrix: Tensor | None = None, precision_matrix: Tensor | None = None, scale_tril: Tensor | None = None, validate_args: bool | None = None)

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Multivariate Normal (Gaussian) distribution in $\mathbb{R}^D$ .

MultivariateNormal(loc=μ, ...) defines the $D$ -dimensional Gaussian with mean vector $\mu$ and a covariance structure expressed in one of three equivalent forms. Internally all forms are converted to the lower-triangular Cholesky factor $L$ of $\Sigma$ (i.e. $\Sigma = L L^\top$ ), which is the numerically preferred representation for both sampling and log-probability evaluation.

Specify exactly one of:

covariance_matrix $\Sigma$ (positive-definite, $D \times D$ ),
precision_matrix $\Sigma^{-1}$ (positive-definite, inverted via Cholesky internally), or
scale_tril $L$ (lower-triangular with positive diagonal).

Parameters

locTensor

Mean vector

\mu \in \mathbb{R}^D

of shape (..., D).

covariance_matrixTensor | None= None

Full covariance matrix

\Sigma

of shape (..., D, D).

precision_matrixTensor | None= None

Precision matrix

\Sigma^{-1}

of shape (..., D, D).

scale_trilTensor | None= None

Lower-triangular Cholesky factor

L

of shape (..., D, D) with positive diagonal entries.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Attributes

locTensor

Mean vector

\mu

scale_trilTensor

Cholesky factor

L

(always populated, regardless of which parameterisation was used for construction).

Notes

PDF:

p(x; \mu, \Sigma) = (2\pi)^{-D/2} |\Sigma|^{-1/2} \exp\!\left(-\tfrac{1}{2}(x-\mu)^\top \Sigma^{-1} (x-\mu)\right)

Log-PDF (numerically stable via Cholesky, avoiding explicit inversion):

\log p(x) = -\tfrac{1}{2}\|L^{-1}(x-\mu)\|^2 - \sum_i \log L_{ii} - \tfrac{D}{2}\log(2\pi)

where $\|L^{-1}(x-\mu)\|^2$ is the squared Mahalanobis distance computed via triangular solve (no explicit matrix inversion).

Moments:

Mean: $E[X] = \mu$
Mode: $\mu$ (Gaussian is unimodal)
Marginal variances: diagonal of $\Sigma = L L^\top$

Entropy:

H[X] = \tfrac{D}{2}(1 + \log(2\pi)) + \sum_i \log L_{ii}

Reparameterised sampling uses the Cholesky factorisation:

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

Gradients propagate through $\mu$ and $L$ unobstructed.

Examples

>>> import lucid
>>> from lucid.distributions import MultivariateNormal
>>> # 2-d isotropic Gaussian
>>> dist = MultivariateNormal(
...     loc=lucid.zeros(2),
...     covariance_matrix=lucid.eye(2),
... )
>>> samples = dist.rsample((50,))
>>> samples.shape  # (50, 2)
(50, 2)
>>> # Log-prob at the mean (maximum)
>>> dist.log_prob(lucid.zeros(2))

Used by 2

Constructors

dunder

init

→None

__init__(loc: Tensor, covariance_matrix: Tensor | None = None, precision_matrix: Tensor | None = None, scale_tril: Tensor | None = None, validate_args: bool | None = None)

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Initialise a MultivariateNormal distribution.

Exactly one of covariance_matrix, precision_matrix, or scale_tril must be provided. All parameterisations are internally converted to the lower-triangular Cholesky factor L via lucid.linalg.cholesky or triangular inversion.

Parameters

locTensor

Mean vector

\mu \in \mathbb{R}^D

of shape (..., D).

covariance_matrixTensor | None= None

Full covariance matrix

\Sigma

of shape (..., D, D).

precision_matrixTensor | None= None

Precision matrix

\Sigma^{-1}

of shape (..., D, D).

scale_trilTensor | None= None

Lower-triangular Cholesky factor

L

of shape (..., D, D) with positive diagonal.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Raises

ValueError

If not exactly one of the covariance parameterisations is given.

Properties

prop

covariance_matrix

→Tensor

covariance_matrix: Tensor

source edit

Full covariance matrix $\Sigma = L L^\top$ .

Returns

Tensor

Positive-definite covariance matrix of shape (*batch_shape, D, D).

prop

mean

→Tensor

mean: Tensor

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Mean of the MultivariateNormal: $E[X] = \mu$ .

Returns

Tensor

Mean vector of shape (*batch_shape, D).

prop

mode

→Tensor

mode: Tensor

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Mode of the MultivariateNormal: equal to the mean $\mu$ .

The Gaussian is unimodal; its unique maximum is at the mean.

Returns

Tensor

Mode vector of shape (*batch_shape, D).

prop

variance

→Tensor

variance: Tensor

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Marginal variances — diagonal entries of $\Sigma = LL^\top$ .

Returns

Tensor

Variance vector of shape (*batch_shape, D).

Instance methods

entropy

→Tensor

entropy()

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Entropy of the MultivariateNormal distribution.

H[X] = \tfrac{D}{2}(1 + \log(2\pi)) + \sum_i \log L_{ii}

Returns

Tensor

Entropy values of shape batch_shape (nats).

log_prob

→Tensor

log_prob(value: Tensor)

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Log-probability density of the MultivariateNormal distribution.

Computed via the Cholesky factor to avoid explicit matrix inversion:

\log p(x) = -\tfrac{1}{2}\|L^{-1}(x-\mu)\|^2 - \sum_i \log L_{ii} - \tfrac{D}{2}\log(2\pi)

Parameters

valueTensor

Observation vectors of shape (*batch_shape, D).

Returns

Tensor

Log-density values of shape batch_shape.

rsample

→Tensor

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

source edit

Draw reparameterised samples via the Cholesky factorisation.

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

Gradients propagate through both $\mu$ and $L$ .

Parameters

sample_shapetuple[int, ...]= ()

Leading shape of the output sample batch.

Returns

Tensor

Samples of shape (*sample_shape, *batch_shape, D).

>>> import lucid >>> from lucid.distributions import MultivariateNormal >>> # 2-d isotropic Gaussian >>> dist = MultivariateNormal( ... loc=lucid.zeros(2), ... covariance_matrix=lucid.eye(2), ... ) >>> samples = dist.rsample((50,)) >>> samples.shape # (50, 2) (50, 2) >>> # Log-prob at the mean (maximum) >>> dist.log_prob(lucid.zeros(2))

dunder

init

→None

__init__(loc: Tensor, covariance_matrix: Tensor | None = None, precision_matrix: Tensor | None = None, scale_tril: Tensor | None = None, validate_args: bool | None = None)

source edit

Initialise a MultivariateNormal distribution.

Parameters

locTensor

Mean vector

\mu \in \mathbb{R}^D

of shape (..., D).

covariance_matrixTensor | None= None

Full covariance matrix

\Sigma

of shape (..., D, D).

precision_matrixTensor | None= None

Precision matrix

\Sigma^{-1}

of shape (..., D, D).

scale_trilTensor | None= None

Lower-triangular Cholesky factor

L

of shape (..., D, D) with positive diagonal.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Raises

ValueError

If not exactly one of the covariance parameterisations is given.