MultivariateNormal¶

class paddle.distribution. MultivariateNormal ( loc: float | Tensor, covariance_matrix: Tensor | None = None, precision_matrix: Tensor | None = None, scale_tril: Tensor | None = None ) [source]

The Multivariate Normal distribution is a type multivariate continuous distribution defined on the real set, with parameter: loc and any one of the following parameters characterizing the variance: covariance_matrix, precision_matrix, scale_tril.

Mathematical details

The probability density function (pdf) is

$$p(X ;\mu, \Sigma) = \frac{1}{\sqrt{(2\pi)^k |\Sigma|}} \exp(-\frac{1}{2}(X - \mu)^{\intercal} \Sigma^{-1} (X - \mu))$$

In the above equation:

• $X$: is a k-dim random vector.

• $loc = \mu$: is the k-dim mean vector.

• $covariance_matrix = \Sigma$: is the k-by-k covariance matrix.

Parameters

• loc (int|float|Tensor) – The mean of Multivariate Normal distribution. If the input data type is int or float, the data type of loc will be convert to a 1-D Tensor the paddle global default dtype.

• covariance_matrix (Tensor|None) – The covariance matrix of Multivariate Normal distribution. The data type of covariance_matrix will be convert to be the same as the type of loc.

• precision_matrix (Tensor|None) – The inverse of the covariance matrix. The data type of precision_matrix will be convert to be the same as the type of loc.

• scale_tril (Tensor|None) – The cholesky decomposition (lower triangular matrix) of the covariance matrix. The data type of scale_tril will be convert to be the same as the type of loc.

Examples

>>> import paddle
>>> from paddle.distribution import MultivariateNormal
>>> paddle.set_device("cpu")
>>> paddle.seed(100)

>>> rv = MultivariateNormal(loc=paddle.to_tensor([2.,5.]), covariance_matrix=paddle.to_tensor([[2.,1.],[1.,2.]]))

>>> print(rv.sample([3, 2]))
Tensor(shape=[3, 2, 2], dtype=float32, place=Place(cpu), stop_gradient=True,
[[[-0.00339603,  4.31556797],
  [ 2.01385283,  4.63553190]],
 [[ 0.10132277,  3.11323833],
  [ 2.37435842,  3.56635118]],
 [[ 2.89701366,  5.10602522],
  [-0.46329355,  3.14768648]]])

>>> print(rv.mean)
Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
[2., 5.])

>>> print(rv.variance)
Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
[1.99999988, 2.        ])

>>> print(rv.entropy())
Tensor(shape=[], dtype=float32, place=Place(cpu), stop_gradient=True,
3.38718319)

>>> rv1 = MultivariateNormal(loc=paddle.to_tensor([2.,5.]), covariance_matrix=paddle.to_tensor([[2.,1.],[1.,2.]]))
>>> rv2 = MultivariateNormal(loc=paddle.to_tensor([-1.,3.]), covariance_matrix=paddle.to_tensor([[3.,2.],[2.,3.]]))
>>> print(rv1.kl_divergence(rv2))
Tensor(shape=[], dtype=float32, place=Place(cpu), stop_gradient=True,
1.55541301)

property batch_shape : Sequence[int]

Returns batch shape of distribution

Returns

batch shape

Return type

Sequence[int]

property event_shape : Sequence[int]

Returns event shape of distribution

Returns

event shape

Return type

Sequence[int]

probs ( value: Tensor ) → Tensor

probs¶

Probability density/mass function.

Note

This method will be deprecated in the future, please use prob instead.

property mean : Tensor

Mean of Multivariate Normal distribution.

Returns

mean value.

Return type

Tensor

property variance : Tensor

Variance of Multivariate Normal distribution.

Returns

variance value.

Return type

Tensor

sample ( shape: Sequence[int] = [] ) → Tensor

sample¶

Generate Multivariate Normal samples of the specified shape. The final shape would be sample_shape + batch_shape + event_shape.

Parameters

shape (Sequence[int], optional) – Prepended shape of the generated samples.

Returns

Tensor, Sampled data with shape sample_shape + batch_shape + event_shape. The data type is the same as self.loc.

rsample ( shape: Sequence[int] = [] ) → Tensor

rsample¶

Generate Multivariate Normal samples of the specified shape. The final shape would be sample_shape + batch_shape + event_shape.

Parameters

shape (Sequence[int], optional) – Prepended shape of the generated samples.

Returns

Tensor, Sampled data with shape sample_shape + batch_shape + event_shape. The data type is the same as self.loc.

log_prob ( value: Tensor ) → Tensor

log_prob¶

Log probability density function.

Parameters

value (Tensor) – The input tensor.

Returns

log probability. The data type is the same as self.loc.

Return type

Tensor

prob ( value: Tensor ) → Tensor

prob¶

Probability density function.

Parameters

value (Tensor) – The input tensor.

Returns

probability. The data type is the same as self.loc.

Return type

Tensor

entropy ( ) → Tensor

entropy¶

Shannon entropy in nats.

The entropy is

$$\mathcal{H}(X) = \frac{n}{2} \log(2\pi) + \log {\det A} + \frac{n}{2}$$

In the above equation:

• $\Omega$: is the support of the distribution.

Returns

Tensor, Shannon entropy of Multivariate Normal distribution. The data type is the same as self.loc.

kl_divergence ( other: MultivariateNormal ) → Tensor [source]

kl_divergence¶

The KL-divergence between two poisson distributions with the same batch_shape and event_shape.

The probability density function (pdf) is

$$KL\_divergence(\lambda_1, \lambda_2) = \log(\det A_2) - \log(\det A_1) -\frac{n}{2} +\frac{1}{2}[tr [\Sigma_2^{-1} \Sigma_1] + (\mu_1 - \mu_2)^{\intercal} \Sigma_2^{-1} (\mu_1 - \mu_2)]$$

Parameters

other (MultivariateNormal) – instance of Multivariate Normal.

Returns

Tensor, kl-divergence between two Multivariate Normal distributions. The data type is the same as self.loc.