LostTech.TensorFlow : API Documentation

Type MultivariateNormalTriL

Namespace tensorflow.contrib.distributions

Parent MultivariateNormalLinearOperator

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: [2]

# 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")))

Properties

Public properties

object allow_nan_stats get;

object allow_nan_stats_dyn get;

TensorShape batch_shape get;

object batch_shape_dyn get;

object bijector get;

object bijector_dyn get;

object distribution get;

object distribution_dyn get;

object dtype get;

object dtype_dyn get;

TensorShape event_shape get;

object event_shape_dyn get;

object loc get;

object loc_dyn get;

string name get;

object name_dyn get;

IDictionary<object, object> parameters get;

object parameters_dyn get;

object PythonObject get;

object reparameterization_type get;

object reparameterization_type_dyn get;

object scale get;

object scale_dyn get;

object validate_args get;

object validate_args_dyn get;