# tf.contrib.distributions.MultivariateNormalFullCovariance

### `class tf.contrib.distributions.MultivariateNormalFullCovariance`

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,

```pdf(x; loc, covariance_matrix) = exp(-0.5 ||y||**2) / 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, and,
• `||y||**2` denotes the squared Euclidean norm of `y`.

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, i.e., it can be constructed e.g. as,

```X ~ MultivariateNormal(loc=0, scale=1)   # Identity scale, zero shift.
scale = Cholesky(covariance_matrix)
Y = scale @ X + loc
```

#### Examples

```ds = tf.contrib.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 = ds.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 = ds.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: [2]

```

## 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

### `__init__`

```__init__(
loc=None,
covariance_matrix=None,
validate_args=False,
allow_nan_stats=True,
name='MultivariateNormalFullCovariance'
)
```

Construct Multivariate Normal distribution on `R^k`.

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

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

A non-batch `covariance_matrix` matrix is a `k x k` symmetric positive definite matrix. In other words it is (real) symmetric with all eigenvalues strictly positive.

#### 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.
• `covariance_matrix`: Floating-point, symmetric positive definite `Tensor` of same `dtype` as `loc`. The strict upper triangle of `covariance_matrix` is ignored, so if `covariance_matrix` is not symmetric no error will be raised (unless `validate_args is True`). `covariance_matrix` 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 `covariance_matrix` are specified.

### `batch_shape_tensor`

```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`

```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`: The name to give this op.

#### Returns:

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

### `copy`

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

Creates a deep copy of the distribution.

Note: the copy distribution may continue to depend on the original initialization arguments.

#### 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`

```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`: The name to give this op.

#### 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)`.

### `entropy`

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

Shannon entropy in nats.

### `event_shape_tensor`

```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`

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

Indicates that `batch_shape == []`.

#### Args:

• `name`: The name to give this op.

#### Returns:

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

### `is_scalar_event`

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

Indicates that `event_shape == []`.

#### Args:

• `name`: The name to give this op.

#### Returns:

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

### `log_cdf`

```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`: The name to give this op.

#### Returns:

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

### `log_prob`

```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`: The name to give this op.

#### Returns:

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

### `log_survival_function`

```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`: The name to give this op.

#### Returns:

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

### `mean`

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

Mean.

### `mode`

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

Mode.

### `param_shapes`

```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`

```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`

```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`: The name to give this op.

#### Returns:

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

### `quantile`

```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`: The name to give this op.

#### Returns:

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

### `sample`

```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`

```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`: The name to give this op.

#### Returns:

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

### `survival_function`

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

Survival function.

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

```survival_function(x) = P[X > x]
= 1 - P[X <= x]
= 1 - cdf(x).
```

#### Args:

• `value`: `float` or `double` `Tensor`.
• `name`: The name to give this op.

#### Returns:

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

### `variance`

```variance(name='variance')
```

Variance.

Variance is defined as,

```Var = E[(X - E[X])**2]
```

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

#### Args:

• `name`: The name to give this op.

#### Returns:

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

https://www.tensorflow.org/api_docs/python/tf/contrib/distributions/MultivariateNormalFullCovariance

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