# LostTech.TensorFlow : API Documentation

Type MultivariateNormalTriL

Namespace tensorflow.contrib.distributions

Interfaces IMultivariateNormalTriL

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 matrix in `R^{k x k}`, `covariance = 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 = scale_tril ```

where `scale_tril` is lower-triangular `k x k` matrix with non-zero diagonal, i.e., `tf.linalg.tensor_diag_part(scale_tril) != 0`.

Additional leading dimensions (if any) will index batches.

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 ```

Trainable (batch) lower-triangular matrices can be created with `tfp.distributions.matrix_diag_transform()` and/or `tfp.distributions.fill_triangular()`

#### 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]]
scale = tf.linalg.cholesky(cov)
# ==> [[ 0.6,  0. ,  0. ],
#      [ 0.2,  0.5,  0. ],
#      [ 0.1, -0.3,  0.4]])
mvn = tfd.MultivariateNormalTriL(
loc=mu,
scale_tril=scale)  mvn.mean().eval()
# ==> [1., 2, 3]  # Covariance agrees with cholesky(cov) parameterization.
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]
tril =...  # shape: [2, 3, 3], lower triangular, non-zero diagonal.
mvn = tfd.MultivariateNormalTriL(
loc=mu,
scale_tril=tril)  # 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:   # Instantiate a "learnable" MVN.
dims = 4
with tf.compat.v1.variable_scope("model"):
mvn = tfd.MultivariateNormalTriL(
loc=tf.compat.v1.get_variable(shape=[dims], dtype=tf.float32,
name="mu"),
scale_tril=tfd.fill_triangular(
tf.compat.v1.get_variable(shape=[dims * (dims + 1) / 2],
dtype=tf.float32, name="chol_Sigma"))) ```