# tf.compat.v1.distributions.Dirichlet

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|---|
| [![](https://www.tensorflow.org/images/GitHub-Mark-32px.png) View source on GitHub](https://github.com/tensorflow/tensorflow/blob/v2.13.1/tensorflow/python/ops/distributions/dirichlet.py#L44-L333) |

Dirichlet distribution.

Inherits From: [`Distribution`](https://www.tensorflow.org/versions/r2.13/api_docs/python/tf/compat/v1/distributions/Distribution)

    tf.compat.v1.distributions.Dirichlet(
        concentration,
        validate_args=False,
        allow_nan_stats=True,
        name='Dirichlet'
    )

The Dirichlet distribution is defined over the
[`(k-1)`-simplex](https://en.wikipedia.org/wiki/Simplex) using a positive,
length-`k` vector `concentration` (`k > 1`). The Dirichlet is identically the
Beta distribution when `k = 2`.

#### Mathematical Details

The Dirichlet is a distribution over the open `(k-1)`-simplex, i.e.,

    S^{k-1} = { (x_0, ..., x_{k-1}) in R^k : sum_j x_j = 1 and all_j x_j > 0 }.

The probability density function (pdf) is,

    pdf(x; alpha) = prod_j x_j**(alpha_j - 1) / Z
    Z = prod_j Gamma(alpha_j) / Gamma(sum_j alpha_j)

where:

- `x in S^{k-1}`, i.e., the `(k-1)`-simplex,
- `concentration = alpha = [alpha_0, ..., alpha_{k-1}]`, `alpha_j > 0`,
- `Z` is the normalization constant aka the [multivariate beta function](https://en.wikipedia.org/wiki/Beta_function#Multivariate_beta_function), and,
- `Gamma` is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function).

The `concentration` represents mean total counts of class occurrence, i.e.,

    concentration = alpha = mean * total_concentration

where `mean` in `S^{k-1}` and `total_concentration` is a positive real number
representing a mean total count.

Distribution parameters are automatically broadcast in all functions; see
examples for details.

> [!WARNING]
> **Warning:** Some components of the samples can be zero due to finite precision. This happens more often when some of the concentrations are very small. Make sure to round the samples to `np.finfo(dtype).tiny` before computing the density.

Samples of this distribution are reparameterized (pathwise differentiable).
The derivatives are computed using the approach described in
(Figurnov et al., 2018).

#### Examples

    import tensorflow_probability as tfp
    tfd = tfp.distributions

    # Create a single trivariate Dirichlet, with the 3rd class being three times
    # more frequent than the first. I.e., batch_shape=[], event_shape=[3].
    alpha = [1., 2, 3]
    dist = tfd.Dirichlet(alpha)

    dist.sample([4, 5])  # shape: [4, 5, 3]

    # x has one sample, one batch, three classes:
    x = [.2, .3, .5]   # shape: [3]
    dist.prob(x)       # shape: []

    # x has two samples from one batch:
    x = [[.1, .4, .5],
         [.2, .3, .5]]
    dist.prob(x)         # shape: [2]

    # alpha will be broadcast to shape [5, 7, 3] to match x.
    x = [[...]]   # shape: [5, 7, 3]
    dist.prob(x)  # shape: [5, 7]

    # Create batch_shape=[2], event_shape=[3]:
    alpha = [[1., 2, 3],
             [4, 5, 6]]   # shape: [2, 3]
    dist = tfd.Dirichlet(alpha)

    dist.sample([4, 5])  # shape: [4, 5, 2, 3]

    x = [.2, .3, .5]
    # x will be broadcast as [[.2, .3, .5],
    #                         [.2, .3, .5]],
    # thus matching batch_shape [2, 3].
    dist.prob(x)         # shape: [2]

Compute the gradients of samples w.r.t. the parameters:

    alpha = tf.constant([1.0, 2.0, 3.0])
    dist = tfd.Dirichlet(alpha)
    samples = dist.sample(5)  # Shape [5, 3]
    loss = tf.reduce_mean(tf.square(samples))  # Arbitrary loss function
    # Unbiased stochastic gradients of the loss function
    grads = tf.gradients(loss, alpha)

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| ## References ||
|---|---|
| Implicit Reparameterization Gradients: [Figurnov et al., 2018](http://papers.nips.cc/paper/7326-implicit-reparameterization-gradients) ([pdf](http://papers.nips.cc/paper/7326-implicit-reparameterization-gradients.pdf)) ||

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| ## Args ||
|---|---|
| `concentration` | Positive floating-point `Tensor` indicating mean number of class occurrences; aka "alpha". Implies `self.dtype`, and `self.batch_shape`, `self.event_shape`, i.e., if `concentration.shape = [N1, N2, ..., Nm, k]` then `batch_shape = [N1, N2, ..., Nm]` and `event_shape = [k]`. |
| `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. |
| `concentration` | Concentration parameter; expected counts for that coordinate. |
| `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/oryx/api_docs/python/oryx/distributions#FULLY_REPARAMETERIZED) or [`distributions.NOT_REPARAMETERIZED`](https://www.tensorflow.org/probability/oryx/api_docs/python/oryx/distributions#NOT_REPARAMETERIZED). |
| `total_concentration` | Sum of last dim of concentration parameter. |
| `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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L630-L647)

    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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L874-L891)

    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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L608-L624)

    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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L1087-L1124)

    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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L1139-L1162)

    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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L975-L978)

    entropy(
        name='entropy'
    )

Shannon entropy in nats.

### `event_shape_tensor`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L670-L684)

    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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L714-L726)

    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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L700-L712)

    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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L1168-L1194)

    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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L835-L856)

    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_prob`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L777-L788)

    log_prob(
        value, name='log_prob'
    )

Log probability density/mass function.

Additional documentation from `Dirichlet`:

> [!NOTE]
> **Note:** `value` must be a non-negative tensor with dtype `self.dtype` and be in the `(self.event_shape() - 1)`-simplex, i.e., `tf.reduce_sum(value, -1) = 1`. It must have a shape compatible with `self.batch_shape() + self.event_shape()`.

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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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L910-L932)

    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 ||
|---|---|
| `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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L984-L987)

    mean(
        name='mean'
    )

Mean.

### `mode`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L1130-L1133)

    mode(
        name='mode'
    )

Mode.

Additional documentation from `Dirichlet`:

> [!NOTE]
> **Note:** The mode is undefined when any `concentration <= 1`. If `self.allow_nan_stats` is `True`, `NaN` is used for undefined modes. If `self.allow_nan_stats` is `False` an exception is raised when one or more modes are undefined.

### `param_shapes`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L490-L509)

    @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 ||
|---|---|
| `dict` of parameter name to `Tensor` shapes. ||

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### `param_static_shapes`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L511-L548)

    @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 ||
|---|---|
| `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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L806-L817)

    prob(
        value, name='prob'
    )

Probability density/mass function.

Additional documentation from `Dirichlet`:

> [!NOTE]
> **Note:** `value` must be a non-negative tensor with dtype `self.dtype` and be in the `(self.event_shape() - 1)`-simplex, i.e., `tf.reduce_sum(value, -1) = 1`. It must have a shape compatible with `self.batch_shape() + self.event_shape()`.

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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 ||
|---|---|
| `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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L999-L1016)

    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 ||
|---|---|
| `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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L745-L759)

    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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### `stddev`

[View source](https://github.com/tensorflow/tensorflow/blob/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L1054-L1081)

    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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L950-L969)

    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/v2.13.1/tensorflow/python/ops/distributions/distribution.py#L1022-L1048)

    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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