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# tf.contrib.distributions.MultivariateNormalTriL

## Class `MultivariateNormalTriL`

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,

``````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:

``````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`.

The MultivariateNormal distribution is a member of the location-scale family, i.e., it can be constructed as,

``````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

``````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")))
``````

## `__init__`

View source

``````__init__(
loc=None,
scale_tril=None,
validate_args=False,
allow_nan_stats=True,
name='MultivariateNormalTriL'
)
``````

Construct Multivariate Normal distribution on `R^k`. (deprecated)

The `batch_shape` is the broadcast shape between `loc` and `scale` arguments.

The `event_shape` is given by last dimension of the matrix implied by `scale`. The last dimension of `loc` (if provided) must broadcast with this.

Recall that `covariance = scale @ scale.T`. A (non-batch) `scale` matrix is:

``````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`.

#### Args:

• `loc`: Floating-point `Tensor`. If this is set to `None`, `loc` is implicitly `0`. When specified, may have shape `[B1, ..., Bb, k]` where `b >= 0` and `k` is the event size.
• `scale_tril`: Floating-point, lower-triangular `Tensor` with non-zero diagonal elements. `scale_tril` has shape `[B1, ..., Bb, k, k]` where ```b >= 0``` and `k` is the event size.
• `validate_args`: Python `bool`, default `False`. When `True` distribution parameters are checked for validity despite possibly degrading runtime performance. When `False` invalid inputs may silently render incorrect outputs.
• `allow_nan_stats`: Python `bool`, default `True`. When `True`, statistics (e.g., mean, mode, variance) use the value "`NaN`" to indicate the result is undefined. When `False`, an exception is raised if one or more of the statistic's batch members are undefined.
• `name`: Python `str` name prefixed to Ops created by this class.

#### Raises:

• `ValueError`: if neither `loc` nor `scale_tril` are specified.

## Properties

### `allow_nan_stats`

Python `bool` describing behavior when a stat is undefined.

Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.

#### Returns:

• `allow_nan_stats`: Python `bool`.

### `batch_shape`

Shape of a single sample from a single event index as a `TensorShape`.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

#### Returns:

• `batch_shape`: `TensorShape`, possibly unknown.

### `bijector`

Function transforming x => y.

### `distribution`

Base distribution, p(x).

### `dtype`

The `DType` of `Tensor`s handled by this `Distribution`.

### `event_shape`

Shape of a single sample from a single batch as a `TensorShape`.

May be partially defined or unknown.

#### Returns:

• `event_shape`: `TensorShape`, possibly unknown.

### `loc`

The `loc` `Tensor` in `Y = scale @ X + loc`.

### `name`

Name prepended to all ops created by this `Distribution`.

### `parameters`

Dictionary of parameters used to instantiate this `Distribution`.

### `reparameterization_type`

Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances `distributions.FULLY_REPARAMETERIZED` or `distributions.NOT_REPARAMETERIZED`.

#### Returns:

An instance of `ReparameterizationType`.

### `scale`

The `scale` `LinearOperator` in `Y = scale @ X + loc`.

### `validate_args`

Python `bool` indicating possibly expensive checks are enabled.

## Methods

### `batch_shape_tensor`

View source

``````batch_shape_tensor(name='batch_shape_tensor')
``````

Shape of a single sample from a single event index as a 1-D `Tensor`.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

#### Args:

• `name`: name to give to the op

#### Returns:

• `batch_shape`: `Tensor`.

### `cdf`

View source

``````cdf(
value,
name='cdf'
)
``````

Cumulative distribution function.

Given random variable `X`, the cumulative distribution function `cdf` is:

``````cdf(x) := P[X <= x]
``````

#### Args:

• `value`: `float` or `double` `Tensor`.
• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `cdf`: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `copy`

View source

``````copy(**override_parameters_kwargs)
``````

Creates a deep copy of the distribution.

#### Args:

• `**override_parameters_kwargs`: String/value dictionary of initialization arguments to override with new values.

#### Returns:

• `distribution`: A new instance of `type(self)` initialized from the union of self.parameters and override_parameters_kwargs, i.e., `dict(self.parameters, **override_parameters_kwargs)`.

### `covariance`

View source

``````covariance(name='covariance')
``````

Covariance.

Covariance is (possibly) defined only for non-scalar-event distributions.

For example, for a length-`k`, vector-valued distribution, it is calculated as,

``````Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]
``````

where `Cov` is a (batch of) `k x k` matrix, `0 <= (i, j) < k`, and `E` denotes expectation.

Alternatively, for non-vector, multivariate distributions (e.g., matrix-valued, Wishart), `Covariance` shall return a (batch of) matrices under some vectorization of the events, i.e.,

``````Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]
``````

where `Cov` is a (batch of) `k' x k'` matrices, `0 <= (i, j) < k' = reduce_prod(event_shape)`, and `Vec` is some function mapping indices of this distribution's event dimensions to indices of a length-`k'` vector.

#### Args:

• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `covariance`: Floating-point `Tensor` with shape `[B1, ..., Bn, k', k']` where the first `n` dimensions are batch coordinates and `k' = reduce_prod(self.event_shape)`.

### `cross_entropy`

View source

``````cross_entropy(
other,
name='cross_entropy'
)
``````

Computes the (Shannon) cross entropy.

Denote this distribution (`self`) by `P` and the `other` distribution by `Q`. Assuming `P, Q` are absolutely continuous with respect to one another and permit densities `p(x) dr(x)` and `q(x) dr(x)`, (Shanon) cross entropy is defined as:

``````H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)
``````

where `F` denotes the support of the random variable `X ~ P`.

#### Args:

• `other`: `tfp.distributions.Distribution` instance.
• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `cross_entropy`: `self.dtype` `Tensor` with shape `[B1, ..., Bn]` representing `n` different calculations of (Shanon) cross entropy.

### `entropy`

View source

``````entropy(name='entropy')
``````

Shannon entropy in nats.

### `event_shape_tensor`

View source

``````event_shape_tensor(name='event_shape_tensor')
``````

Shape of a single sample from a single batch as a 1-D int32 `Tensor`.

#### Args:

• `name`: name to give to the op

#### Returns:

• `event_shape`: `Tensor`.

### `is_scalar_batch`

View source

``````is_scalar_batch(name='is_scalar_batch')
``````

Indicates that `batch_shape == []`.

#### Args:

• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `is_scalar_batch`: `bool` scalar `Tensor`.

### `is_scalar_event`

View source

``````is_scalar_event(name='is_scalar_event')
``````

Indicates that `event_shape == []`.

#### Args:

• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `is_scalar_event`: `bool` scalar `Tensor`.

### `kl_divergence`

View source

``````kl_divergence(
other,
name='kl_divergence'
)
``````

Computes the Kullback--Leibler divergence.

Denote this distribution (`self`) by `p` and the `other` distribution by `q`. Assuming `p, q` are absolutely continuous with respect to reference measure `r`, the KL divergence is defined as:

``````KL[p, q] = E_p[log(p(X)/q(X))]
= -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
= H[p, q] - H[p]
``````

where `F` denotes the support of the random variable `X ~ p`, `H[., .]` denotes (Shanon) cross entropy, and `H[.]` denotes (Shanon) entropy.

#### Args:

• `other`: `tfp.distributions.Distribution` instance.
• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `kl_divergence`: `self.dtype` `Tensor` with shape `[B1, ..., Bn]` representing `n` different calculations of the Kullback-Leibler divergence.

### `log_cdf`

View source

``````log_cdf(
value,
name='log_cdf'
)
``````

Log cumulative distribution function.

Given random variable `X`, the cumulative distribution function `cdf` is:

``````log_cdf(x) := Log[ P[X <= x] ]
``````

Often, a numerical approximation can be used for `log_cdf(x)` that yields a more accurate answer than simply taking the logarithm of the `cdf` when `x << -1`.

#### Args:

• `value`: `float` or `double` `Tensor`.
• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `logcdf`: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `log_prob`

View source

``````log_prob(
value,
name='log_prob'
)
``````

Log probability density/mass function.

Additional documentation from `MultivariateNormalLinearOperator`:

`value` is a batch vector with compatible shape if `value` is a `Tensor` whose shape can be broadcast up to either:

``````self.batch_shape + self.event_shape
``````

or

``````[M1, ..., Mm] + self.batch_shape + self.event_shape
``````

#### Args:

• `value`: `float` or `double` `Tensor`.
• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `log_prob`: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `log_survival_function`

View source

``````log_survival_function(
value,
name='log_survival_function'
)
``````

Log survival function.

Given random variable `X`, the survival function is defined:

``````log_survival_function(x) = Log[ P[X > x] ]
= Log[ 1 - P[X <= x] ]
= Log[ 1 - cdf(x) ]
``````

Typically, different numerical approximations can be used for the log survival function, which are more accurate than `1 - cdf(x)` when `x >> 1`.

#### Args:

• `value`: `float` or `double` `Tensor`.
• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

`Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `mean`

View source

``````mean(name='mean')
``````

Mean.

### `mode`

View source

``````mode(name='mode')
``````

Mode.

### `param_shapes`

View source

``````param_shapes(
cls,
sample_shape,
name='DistributionParamShapes'
)
``````

Shapes of parameters given the desired shape of a call to `sample()`.

This is a class method that describes what key/value arguments are required to instantiate the given `Distribution` so that a particular shape is returned for that instance's call to `sample()`.

Subclasses should override class method `_param_shapes`.

#### Args:

• `sample_shape`: `Tensor` or python list/tuple. Desired shape of a call to `sample()`.
• `name`: name to prepend ops with.

#### Returns:

`dict` of parameter name to `Tensor` shapes.

### `param_static_shapes`

View source

``````param_static_shapes(
cls,
sample_shape
)
``````

param_shapes with static (i.e. `TensorShape`) shapes.

This is a class method that describes what key/value arguments are required to instantiate the given `Distribution` so that a particular shape is returned for that instance's call to `sample()`. Assumes that the sample's shape is known statically.

Subclasses should override class method `_param_shapes` to return constant-valued tensors when constant values are fed.

#### Args:

• `sample_shape`: `TensorShape` or python list/tuple. Desired shape of a call to `sample()`.

#### Returns:

`dict` of parameter name to `TensorShape`.

#### Raises:

• `ValueError`: if `sample_shape` is a `TensorShape` and is not fully defined.

### `prob`

View source

``````prob(
value,
name='prob'
)
``````

Probability density/mass function.

Additional documentation from `MultivariateNormalLinearOperator`:

`value` is a batch vector with compatible shape if `value` is a `Tensor` whose shape can be broadcast up to either:

``````self.batch_shape + self.event_shape
``````

or

``````[M1, ..., Mm] + self.batch_shape + self.event_shape
``````

#### Args:

• `value`: `float` or `double` `Tensor`.
• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `prob`: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `quantile`

View source

``````quantile(
value,
name='quantile'
)
``````

Quantile function. Aka "inverse cdf" or "percent point function".

Given random variable `X` and `p in [0, 1]`, the `quantile` is:

``````quantile(p) := x such that P[X <= x] == p
``````

#### Args:

• `value`: `float` or `double` `Tensor`.
• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `quantile`: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `sample`

View source

``````sample(
sample_shape=(),
seed=None,
name='sample'
)
``````

Generate samples of the specified shape.

Note that a call to `sample()` without arguments will generate a single sample.

#### Args:

• `sample_shape`: 0D or 1D `int32` `Tensor`. Shape of the generated samples.
• `seed`: Python integer seed for RNG
• `name`: name to give to the op.

#### Returns:

• `samples`: a `Tensor` with prepended dimensions `sample_shape`.

### `stddev`

View source

``````stddev(name='stddev')
``````

Standard deviation.

Standard deviation is defined as,

``````stddev = E[(X - E[X])**2]**0.5
``````

where `X` is the random variable associated with this distribution, `E` denotes expectation, and `stddev.shape = batch_shape + event_shape`.

#### Args:

• `name`: Python `str` prepended to names of ops created by this function.

#### Returns:

• `stddev`: Floating-point `Tensor` with shape identical to `batch_shape + event_shape`, i.e., the same shape as `self.mean()`.

### `survival_function`

View source

``````survival_function(
value,
name='survival_function'
)
``````

Survival function.

Given random variable `X`, the survival function is defined: