# tf.contrib.distributions.WishartFull

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|---|
| [![](https://www.tensorflow.org/images/GitHub-Mark-32px.png) View source on GitHub](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/contrib/distributions/python/ops/wishart.py#L581-L700) |

The matrix Wishart distribution on positive definite matrices.

    tf.contrib.distributions.WishartFull(
        df, scale, cholesky_input_output_matrices=False, validate_args=False,
        allow_nan_stats=True, name='WishartFull'
    )

This distribution is defined by a scalar degrees of freedom `df` and a
symmetric, positive definite scale matrix.

Evaluation of the pdf, determinant, and sampling are all `O(k^3)` operations
where `(k, k)` is the event space shape.

#### Mathematical Details

The probability density function (pdf) is,

    pdf(X; df, scale) = det(X)**(0.5 (df-k-1)) exp(-0.5 tr[inv(scale) X]) / Z
    Z = 2**(0.5 df k) |det(scale)|**(0.5 df) Gamma_k(0.5 df)

where:

- `df >= k` denotes the degrees of freedom,
- `scale` is a symmetric, positive definite, `k x k` matrix,
- `Z` is the normalizing constant, and,
- `Gamma_k` is the [multivariate Gamma function](https://en.wikipedia.org/wiki/Multivariate_gamma_function).

#### Examples

    import tensorflow_probability as tfp
    tfd = tfp.distributions

    # Initialize a single 3x3 Wishart with Full factored scale matrix and 5
    # degrees-of-freedom.(*)
    df = 5
    scale = ...  # Shape is [3, 3]; positive definite.
    dist = tfd.WishartFull(df=df, scale=scale)

    # Evaluate this on an observation in R^3, returning a scalar.
    x = ...  # A 3x3 positive definite matrix.
    dist.prob(x)  # Shape is [], a scalar.

    # Evaluate this on a two observations, each in R^{3x3}, returning a length two
    # Tensor.
    x = [x0, x1]  # Shape is [2, 3, 3].
    dist.prob(x)  # Shape is [2].

    # Initialize two 3x3 Wisharts with Full factored scale matrices.
    df = [5, 4]
    scale = ...  # Shape is [2, 3, 3].
    dist = tfd.WishartFull(df=df, scale=scale)

    # Evaluate this on four observations.
    x = [[x0, x1], [x2, x3]]  # Shape is [2, 2, 3, 3]; xi is positive definite.
    dist.prob(x)  # Shape is [2, 2].

    # (*) - To efficiently create a trainable covariance matrix, see the example
    #   in tfd.matrix_diag_transform.

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| ## Args ||
|---|---|
| `df` | `float` or `double` `Tensor`. Degrees of freedom, must be greater than or equal to dimension of the scale matrix. |
| `scale` | `float` or `double` `Tensor`. The symmetric positive definite scale matrix of the distribution. |
| `cholesky_input_output_matrices` | Python `bool`. Any function which whose input or output is a matrix assumes the input is Cholesky and returns a Cholesky factored matrix. Example `log_prob` input takes a Cholesky and `sample_n` returns a Cholesky when `cholesky_input_output_matrices=True`. |
| `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. |

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| ## Attributes ||
|---|---|
| `allow_nan_stats` | Python `bool` describing behavior when a stat is undefined. <br /> 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. |
| `batch_shape` | Shape of a single sample from a single event index as a `TensorShape`. <br /> May be partially defined or unknown. The batch dimensions are indexes into independent, non-identical parameterizations of this distribution. |
| `cholesky_input_output_matrices` | Boolean indicating if `Tensor` input/outputs are Cholesky factorized. |
| `df` | Wishart distribution degree(s) of freedom. |
| `dimension` | Dimension of underlying vector space. The `p` in `R^(p*p)`. |
| `dtype` | The `DType` of `Tensor`s handled by this `Distribution`. |
| `event_shape` | Shape of a single sample from a single batch as a `TensorShape`. <br /> May be partially defined or unknown. |
| `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. <br /> Currently this is one of the static instances [`distributions.FULLY_REPARAMETERIZED`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions#FULLY_REPARAMETERIZED) or [`distributions.NOT_REPARAMETERIZED`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions#NOT_REPARAMETERIZED). |
| `scale_operator` | Wishart distribution scale matrix as an Linear Operator. |
| `validate_args` | Python `bool` indicating possibly expensive checks are enabled. |

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## Methods

### `batch_shape_tensor`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L637-L654)

    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.

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| Args ||
|---|---|
| `name` | name to give to the op |

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| Returns ||
|---|---|
| `batch_shape` | `Tensor`. |

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### `cdf`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L881-L898)

    cdf(
        value, name='cdf'
    )

Cumulative distribution function.

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

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

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `cdf` | a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. |

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### `copy`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L615-L631)

    copy(
        **override_parameters_kwargs
    )

Creates a deep copy of the distribution.

> [!NOTE]
> **Note:** the copy distribution may continue to depend on the original initialization arguments.

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| Args ||
|---|---|
| `**override_parameters_kwargs` | String/value dictionary of initialization arguments to override with new values. |

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| 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)`. |

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### `covariance`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L1094-L1131)

    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.

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| Args ||
|---|---|
| `name` | Python `str` prepended to names of ops created by this function. |

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| 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)`. |

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### `cross_entropy`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L1146-L1169)

    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`.

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| Args ||
|---|---|
| `other` | [`tfp.distributions.Distribution`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Distribution) instance. |
| `name` | Python `str` prepended to names of ops created by this function. |

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

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### `entropy`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L982-L985)

    entropy(
        name='entropy'
    )

Shannon entropy in nats.

### `event_shape_tensor`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L677-L691)

    event_shape_tensor(
        name='event_shape_tensor'
    )

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

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| Args ||
|---|---|
| `name` | name to give to the op |

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| Returns ||
|---|---|
| `event_shape` | `Tensor`. |

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### `is_scalar_batch`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L721-L733)

    is_scalar_batch(
        name='is_scalar_batch'
    )

Indicates that `batch_shape == []`.

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| Args ||
|---|---|
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `is_scalar_batch` | `bool` scalar `Tensor`. |

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### `is_scalar_event`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L707-L719)

    is_scalar_event(
        name='is_scalar_event'
    )

Indicates that `event_shape == []`.

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| Args ||
|---|---|
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `is_scalar_event` | `bool` scalar `Tensor`. |

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### `kl_divergence`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L1175-L1201)

    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.

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| Args ||
|---|---|
| `other` | [`tfp.distributions.Distribution`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Distribution) instance. |
| `name` | Python `str` prepended to names of ops created by this function. |

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

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### `log_cdf`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L842-L863)

    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`.

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `logcdf` | a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. |

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### `log_normalization`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/contrib/distributions/python/ops/wishart.py#L417-L422)

    log_normalization(
        name='log_normalization'
    )

Computes the log normalizing constant, log(Z).

### `log_prob`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L784-L795)

    log_prob(
        value, name='log_prob'
    )

Log probability density/mass function.

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `log_prob` | a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. |

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### `log_survival_function`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L917-L939)

    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`.

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

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

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### `mean`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L991-L994)

    mean(
        name='mean'
    )

Mean.

### `mean_log_det`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/contrib/distributions/python/ops/wishart.py#L410-L415)

    mean_log_det(
        name='mean_log_det'
    )

Computes E\[log(det(X))\] under this Wishart distribution.

### `mode`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L1137-L1140)

    mode(
        name='mode'
    )

Mode.

### `param_shapes`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L497-L516)

    @classmethod
    param_shapes(
        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`.

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

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

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### `param_static_shapes`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L518-L555)

    @classmethod
    param_static_shapes(
        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.

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

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| Returns ||
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| `dict` of parameter name to `TensorShape`. ||

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| Raises ||
|---|---|
| `ValueError` | if `sample_shape` is a `TensorShape` and is not fully defined. |

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### `prob`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L813-L824)

    prob(
        value, name='prob'
    )

Probability density/mass function.

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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| Returns ||
|---|---|
| `prob` | a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. |

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### `quantile`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L1006-L1023)

    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

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

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| Returns ||
|---|---|
| `quantile` | a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. |

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### `sample`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L752-L766)

    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.

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| 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. |

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| Returns ||
|---|---|
| `samples` | a `Tensor` with prepended dimensions `sample_shape`. |

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### `scale`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/contrib/distributions/python/ops/wishart.py#L186-L191)

    scale()

Wishart distribution scale matrix.

### `stddev`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L1061-L1088)

    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`.

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| Args ||
|---|---|
| `name` | Python `str` prepended to names of ops created by this function. |

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

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### `survival_function`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L957-L976)

    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).

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| Args ||
|---|---|
| `value` | `float` or `double` `Tensor`. |
| `name` | Python `str` prepended to names of ops created by this function. |

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

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### `variance`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/distribution.py#L1029-L1055)

    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`.

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| Args ||
|---|---|
| `name` | Python `str` prepended to names of ops created by this function. |

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

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