# LostTech.TensorFlow : API Documentation

Type MultivariateNormalFullCovariance

Namespace tensorflow.contrib.distributions

Parent MultivariateNormalTriL

Interfaces IMultivariateNormalFullCovariance

The multivariate normal distribution on `R^k`.

The Multivariate Normal distribution is defined over `R^k` and parameterized by a (batch of) length-`k` `loc` vector (aka "mu") and a (batch of) `k x k` `covariance_matrix` matrices that are the covariance. This is different than the other multivariate normals, which are parameterized by a matrix more akin to the standard deviation.

#### Mathematical Details

The probability density function (pdf) is, with `@` as matrix multiplication,

```none pdf(x; loc, covariance_matrix) = exp(-0.5 y) / Z, y = (x - loc)^T @ inv(covariance_matrix) @ (x - loc) Z = (2 pi)**(0.5 k) |det(covariance_matrix)|**(0.5). ```

where:

* `loc` is a vector in `R^k`, * `covariance_matrix` is an `R^{k x k}` symmetric positive definite matrix, * `Z` denotes the normalization constant.

Additional leading dimensions (if any) in `loc` and `covariance_matrix` allow for batch dimensions.

The MultivariateNormal distribution is a member of the [location-scale family](https://en.wikipedia.org/wiki/Location-scale_family), i.e., it can be constructed e.g. as,

```none X ~ MultivariateNormal(loc=0, scale=1) # Identity scale, zero shift. scale = Cholesky(covariance_matrix) Y = scale @ X + loc ```

#### Examples
Show Example
```import tensorflow_probability as tfp
tfd = tfp.distributions  # Initialize a single 3-variate Gaussian.
mu = [1., 2, 3]
cov = [[ 0.36,  0.12,  0.06],
[ 0.12,  0.29, -0.13],
[ 0.06, -0.13,  0.26]]
mvn = tfd.MultivariateNormalFullCovariance(
loc=mu,
covariance_matrix=cov)  mvn.mean().eval()
# ==> [1., 2, 3]  # Covariance agrees with covariance_matrix.
mvn.covariance().eval()
# ==> [[ 0.36,  0.12,  0.06],
#      [ 0.12,  0.29, -0.13],
#      [ 0.06, -0.13,  0.26]]  # Compute the pdf of an observation in `R^3` ; return a scalar.
mvn.prob([-1., 0, 1]).eval()  # shape: []  # Initialize a 2-batch of 3-variate Gaussians.
mu = [[1., 2, 3],
[11, 22, 33]]              # shape: [2, 3]
covariance_matrix =...  # shape: [2, 3, 3], symmetric, positive definite.
mvn = tfd.MultivariateNormalFullCovariance(
loc=mu,
covariance=covariance_matrix)  # Compute the pdf of two `R^3` observations; return a length-2 vector.
x = [[-0.9, 0, 0.1],
[-10, 0, 9]]     # shape: [2, 3]
mvn.prob(x).eval()    # shape:  ```