# LostTech.TensorFlow : API Documentation

Type MultivariateNormalDiag

Namespace tensorflow.contrib.distributions

Interfaces IMultivariateNormalDiag

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` `scale` matrix; `covariance = scale @ scale.T` where `@` denotes matrix-multiplication.

#### Mathematical Details

The probability density function (pdf) is,

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

where:

* `loc` is a vector in `R^k`, * `scale` is a linear operator in `R^{k x k}`, `cov = scale @ scale.T`, * `Z` denotes the normalization constant, and, * `||y||**2` denotes the squared Euclidean norm of `y`.

A (non-batch) `scale` matrix is:

```none scale = diag(scale_diag + scale_identity_multiplier * ones(k)) ```

where:

* `scale_diag.shape = [k]`, and, * `scale_identity_multiplier.shape = []`.

If both `scale_diag` and `scale_identity_multiplier` are `None`, then `scale` is the Identity matrix.

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 as,

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

#### Examples
Show Example
```import tensorflow_probability as tfp
tfd = tfp.distributions  # Initialize a single 2-variate Gaussian.
mvn = tfd.MultivariateNormalDiag(
loc=[1., -1],
scale_diag=[1, 2.])  mvn.mean().eval()
# ==> [1., -1]  mvn.stddev().eval()
# ==> [1., 2]  # Evaluate this on an observation in `R^2`, returning a scalar.
mvn.prob([-1., 0]).eval()  # shape: []  # Initialize a 3-batch, 2-variate scaled-identity Gaussian.
mvn = tfd.MultivariateNormalDiag(
loc=[1., -1],
scale_identity_multiplier=[1, 2., 3])  mvn.mean().eval()  # shape: [3, 2]
# ==> [[1., -1]
#      [1, -1],
#      [1, -1]]  mvn.stddev().eval()  # shape: [3, 2]
# ==> [[1., 1],
#      [2, 2],
#      [3, 3]]  # Evaluate this on an observation in `R^2`, returning a length-3 vector.
mvn.prob([-1., 0]).eval()  # shape: [3]  # Initialize a 2-batch of 3-variate Gaussians.
mvn = tfd.MultivariateNormalDiag(
loc=[[1., 2, 3],
[11, 22, 33]]           # shape: [2, 3]
scale_diag=[[1., 2, 3],
[0.5, 1, 1.5]])  # shape: [2, 3]  # Evaluate this on a two observations, each in `R^3`, returning a length-2
# vector.
x = [[-1., 0, 1],
[-11, 0, 11.]]   # shape: [2, 3].
mvn.prob(x).eval()    # shape: [2] ```