The multivariate normal distribution on `R^k`.

This distribution is defined by a 1-D mean `mu` and a Cholesky factor `chol`.
Providing the Cholesky factor allows for `O(k^2)` pdf evaluation and sampling,
and requires `O(k^2)` storage.

#### Mathematical details

The Cholesky factor `chol` defines the covariance matrix: `C = chol chol^T`.

The PDF of this distribution is then:

```
f(x) = (2 pi)^(-k/2) |det(C)|^(-1/2) exp(-1/2 (x - mu)^T C^{-1} (x - mu))
```

#### Examples

A single multi-variate Gaussian distribution is defined by a vector of means
of length `k`, and a covariance matrix of shape `k x k`.

Extra leading dimensions, if provided, allow for batches.

```python
# Initialize a single 3-variate Gaussian with diagonal covariance.
# Note, this would be more efficient with MultivariateNormalDiag.
mu = [1, 2, 3.]
chol = [[1, 0, 0], [0, 3, 0], [0, 0, 2]]
dist = tf.contrib.distributions.MultivariateNormalCholesky(mu, chol)

# Evaluate this on an observation in R^3, returning a scalar.
dist.pdf([-1, 0, 1])

# Initialize a batch of two 3-variate Gaussians.
mu = [[1, 2, 3], [11, 22, 33]]
chol = ...  # shape 2 x 3 x 3, lower triangular, positive diagonal.
dist = tf.contrib.distributions.MultivariateNormalCholesky(mu, chol)

# Evaluate this on a two observations, each in R^3, returning a length two
# tensor.
x = [[-1, 0, 1], [-11, 0, 11]]  # Shape 2 x 3.
dist.pdf(x)
```

Trainable (batch) Cholesky matrices can be created with
`tf.contrib.distributions.matrix_diag_transform()`
- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.__init__(mu, chol, validate_args=False, allow_nan_stats=True, name='MultivariateNormalCholesky')` {#MultivariateNormalCholesky.__init__}

Multivariate Normal distributions on `R^k`.

User must provide means `mu` and `chol` which holds the (batch) Cholesky
factors, such that the covariance of each batch member is `chol chol^T`.

##### Args:


*  <b>`mu`</b>: `(N+1)-D` floating point tensor with shape `[N1,...,Nb, k]`,
    `b >= 0`.
*  <b>`chol`</b>: `(N+2)-D` `Tensor` with same `dtype` as `mu` and shape
    `[N1,...,Nb, k, k]`.  The upper triangular part is ignored (treated as
    though it is zero), and the diagonal must be positive.
*  <b>`validate_args`</b>: `Boolean`, default `False`.  Whether to validate input
    with asserts.  If `validate_args` is `False`, and the inputs are
    invalid, correct behavior is not guaranteed.
*  <b>`allow_nan_stats`</b>: `Boolean`, default `True`.  If `False`, raise an
    exception if a statistic (e.g. mean/mode/etc...) is undefined for any
    batch member If `True`, batch members with valid parameters leading to
    undefined statistics will return NaN for this statistic.
*  <b>`name`</b>: The name to give Ops created by the initializer.

##### Raises:


*  <b>`TypeError`</b>: If `mu` and `chol` are different dtypes.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.allow_nan_stats` {#MultivariateNormalCholesky.allow_nan_stats}

Python boolean 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:


*  <b>`allow_nan_stats`</b>: Python boolean.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.batch_shape(name='batch_shape')` {#MultivariateNormalCholesky.batch_shape}

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

The product of the dimensions of the `batch_shape` is the number of
independent distributions of this kind the instance represents.

##### Args:


*  <b>`name`</b>: name to give to the op

##### Returns:


*  <b>`batch_shape`</b>: `Tensor`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.cdf(value, name='cdf', **condition_kwargs)` {#MultivariateNormalCholesky.cdf}

Cumulative distribution function.

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

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

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`cdf`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.copy(**override_parameters_kwargs)` {#MultivariateNormalCholesky.copy}

Creates a deep copy of the distribution.

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

##### Args:


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

##### Returns:


*  <b>`distribution`</b>: A new instance of `type(self)` intitialized from the union
    of self.parameters and override_parameters_kwargs, i.e.,
    `dict(self.parameters, **override_parameters_kwargs)`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.dtype` {#MultivariateNormalCholesky.dtype}

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


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.entropy(name='entropy')` {#MultivariateNormalCholesky.entropy}

Shannon entropy in nats.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.event_shape(name='event_shape')` {#MultivariateNormalCholesky.event_shape}

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

##### Args:


*  <b>`name`</b>: name to give to the op

##### Returns:


*  <b>`event_shape`</b>: `Tensor`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.get_batch_shape()` {#MultivariateNormalCholesky.get_batch_shape}

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

Same meaning as `batch_shape`. May be only partially defined.

##### Returns:


*  <b>`batch_shape`</b>: `TensorShape`, possibly unknown.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.get_event_shape()` {#MultivariateNormalCholesky.get_event_shape}

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

Same meaning as `event_shape`. May be only partially defined.

##### Returns:


*  <b>`event_shape`</b>: `TensorShape`, possibly unknown.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.is_continuous` {#MultivariateNormalCholesky.is_continuous}




- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.is_reparameterized` {#MultivariateNormalCholesky.is_reparameterized}




- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.log_cdf(value, name='log_cdf', **condition_kwargs)` {#MultivariateNormalCholesky.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:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`logcdf`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.log_pdf(value, name='log_pdf', **condition_kwargs)` {#MultivariateNormalCholesky.log_pdf}

Log probability density function.

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`log_prob`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.

##### Raises:


*  <b>`TypeError`</b>: if not `is_continuous`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.log_pmf(value, name='log_pmf', **condition_kwargs)` {#MultivariateNormalCholesky.log_pmf}

Log probability mass function.

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`log_pmf`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.

##### Raises:


*  <b>`TypeError`</b>: if `is_continuous`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.log_prob(value, name='log_prob', **condition_kwargs)` {#MultivariateNormalCholesky.log_prob}

Log probability density/mass function (depending on `is_continuous`).


Additional documentation from `_MultivariateNormalOperatorPD`:

`x` is a batch vector with compatible shape if `x` 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:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`log_prob`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.log_sigma_det(name='log_sigma_det')` {#MultivariateNormalCholesky.log_sigma_det}

Log of determinant of covariance matrix.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.log_survival_function(value, name='log_survival_function', **condition_kwargs)` {#MultivariateNormalCholesky.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:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:

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


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.mean(name='mean')` {#MultivariateNormalCholesky.mean}

Mean.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.mode(name='mode')` {#MultivariateNormalCholesky.mode}

Mode.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.mu` {#MultivariateNormalCholesky.mu}




- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.name` {#MultivariateNormalCholesky.name}

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


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.param_shapes(cls, sample_shape, name='DistributionParamShapes')` {#MultivariateNormalCholesky.param_shapes}

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

Subclasses should override static method `_param_shapes`.

##### Args:


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

##### Returns:

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


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.param_static_shapes(cls, sample_shape)` {#MultivariateNormalCholesky.param_static_shapes}

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

##### Args:


*  <b>`sample_shape`</b>: `TensorShape` or python list/tuple. Desired shape of a call
    to `sample()`.

##### Returns:

  `dict` of parameter name to `TensorShape`.

##### Raises:


*  <b>`ValueError`</b>: if `sample_shape` is a `TensorShape` and is not fully defined.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.parameters` {#MultivariateNormalCholesky.parameters}

Dictionary of parameters used to instantiate this `Distribution`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.pdf(value, name='pdf', **condition_kwargs)` {#MultivariateNormalCholesky.pdf}

Probability density function.

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`prob`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.

##### Raises:


*  <b>`TypeError`</b>: if not `is_continuous`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.pmf(value, name='pmf', **condition_kwargs)` {#MultivariateNormalCholesky.pmf}

Probability mass function.

##### Args:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`pmf`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.

##### Raises:


*  <b>`TypeError`</b>: if `is_continuous`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.prob(value, name='prob', **condition_kwargs)` {#MultivariateNormalCholesky.prob}

Probability density/mass function (depending on `is_continuous`).


Additional documentation from `_MultivariateNormalOperatorPD`:

`x` is a batch vector with compatible shape if `x` 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:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`prob`</b>: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with
    values of type `self.dtype`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.sample(sample_shape=(), seed=None, name='sample', **condition_kwargs)` {#MultivariateNormalCholesky.sample}

Generate samples of the specified shape.

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

##### Args:


*  <b>`sample_shape`</b>: 0D or 1D `int32` `Tensor`. Shape of the generated samples.
*  <b>`seed`</b>: Python integer seed for RNG
*  <b>`name`</b>: name to give to the op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`samples`</b>: a `Tensor` with prepended dimensions `sample_shape`.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.sample_n(n, seed=None, name='sample_n', **condition_kwargs)` {#MultivariateNormalCholesky.sample_n}

Generate `n` samples.

##### Args:


*  <b>`n`</b>: `Scalar` `Tensor` of type `int32` or `int64`, the number of
    observations to sample.
*  <b>`seed`</b>: Python integer seed for RNG
*  <b>`name`</b>: name to give to the op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:


*  <b>`samples`</b>: a `Tensor` with a prepended dimension (n,).

##### Raises:


*  <b>`TypeError`</b>: if `n` is not an integer type.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.sigma` {#MultivariateNormalCholesky.sigma}

Dense (batch) covariance matrix, if available.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.sigma_det(name='sigma_det')` {#MultivariateNormalCholesky.sigma_det}

Determinant of covariance matrix.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.std(name='std')` {#MultivariateNormalCholesky.std}

Standard deviation.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.survival_function(value, name='survival_function', **condition_kwargs)` {#MultivariateNormalCholesky.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:


*  <b>`value`</b>: `float` or `double` `Tensor`.
*  <b>`name`</b>: The name to give this op.
*  <b>`**condition_kwargs`</b>: Named arguments forwarded to subclass implementation.

##### Returns:

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


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.validate_args` {#MultivariateNormalCholesky.validate_args}

Python boolean indicated possibly expensive checks are enabled.


- - -

#### `tf.contrib.distributions.MultivariateNormalCholesky.variance(name='variance')` {#MultivariateNormalCholesky.variance}

Variance.


