# tfp.experimental.substrates.jax.distributions.Binomial

Binomial distribution.

Inherits From: `Distribution`

This distribution is parameterized by `probs`, a (batch of) probabilities for drawing a `1`, and `total_count`, the number of trials per draw from the Binomial.

#### Mathematical Details

The Binomial is a distribution over the number of `1`'s in `total_count` independent trials, with each trial having the same probability of `1`, i.e., `probs`.

The probability mass function (pmf) is,

``````pmf(k; n, p) = p**k (1 - p)**(n - k) / Z
Z = k! (n - k)! / n!
``````

where:

• `total_count = n`,
• `probs = p`,
• `Z` is the normalizing constant, and,
• `n!` is the factorial of `n`.

#### Examples

Create a single distribution, corresponding to 5 coin flips.

``````dist = Binomial(total_count=5., probs=.5)
``````

Create a single distribution (using logits), corresponding to 5 coin flips.

``````dist = Binomial(total_count=5., logits=0.)
``````

Creates 3 distributions with the third distribution most likely to have successes.

``````p = [.2, .3, .8]
# n will be broadcast to [4., 4., 4.], to match p.
dist = Binomial(total_count=4., probs=p)
``````

The distribution functions can be evaluated on counts.

``````# counts same shape as p.
counts = [1., 2, 3]
dist.prob(counts)  # Shape [3]

# p will be broadcast to [[.2, .3, .8], [.2, .3, .8]] to match counts.
counts = [[1., 2, 1], [2, 2, 4]]
dist.prob(counts)  # Shape [2, 3]

# p will be broadcast to shape [5, 7, 3] to match counts.
counts = [[...]]  # Shape [5, 7, 3]
dist.prob(counts)  # Shape [5, 7, 3]
``````

`total_count` Non-negative floating point tensor with shape broadcastable to `[N1,..., Nm]` with `m >= 0` and the same dtype as `probs` or `logits`. Defines this as a batch of `N1 x ... x Nm` different Binomial distributions. Its components should be equal to integer values.
`logits` Floating point tensor representing the log-odds of a positive event with shape broadcastable to `[N1,..., Nm]` `m >= 0`, and the same dtype as `total_count`. Each entry represents logits for the probability of success for independent Binomial distributions. Only one of `logits` or `probs` should be passed in.
`probs` Positive floating point tensor with shape broadcastable to `[N1,..., Nm]` `m >= 0`, `probs in [0, 1]`. Each entry represents the probability of success for independent Binomial distributions. Only one of `logits` or `probs` should be passed in.
`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.

`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.

`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.

`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.

`logits` Input argument `logits`.
`name` Name prepended to all ops created by this `Distribution`.
`parameters` Dictionary of parameters used to instantiate this `Distribution`.
`probs` Input argument `probs`.
`reparameterization_type` Describes how samples from the distribution are reparameterized.

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

`total_count` Number of trials.
`trainable_variables`

`validate_args` Python `bool` indicating possibly expensive checks are enabled.
`variables`

## Methods

### `batch_shape_tensor`

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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`

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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.
`**kwargs` Named arguments forwarded to subclass implementation.

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

### `copy`

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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`

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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.
`**kwargs` Named arguments forwarded to subclass implementation.

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`

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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)`, (Shannon) 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 (Shannon) cross entropy.

### `entropy`

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Shannon entropy in nats.

### `event_shape_tensor`

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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`

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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`

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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`

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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 (Shannon) cross entropy, and `H[.]` denotes (Shannon) 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`

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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.
`**kwargs` Named arguments forwarded to subclass implementation.

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

### `log_prob`

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Log probability density/mass function.

Additional documentation from `Binomial`:

For each batch member of counts `value`, `P[value]` is the probability that after sampling `self.total_count` draws from this Binomial distribution, the number of successes is `value`. Since different sequences of draws can result in the same counts, the probability includes a combinatorial coefficient.

Args
`value` `float` or `double` `Tensor`.
`name` Python `str` prepended to names of ops created by this function.
`**kwargs` Named arguments forwarded to subclass implementation.

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

### `log_survival_function`

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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.
`**kwargs` Named arguments forwarded to subclass implementation.

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

### `logits_parameter`

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Logits computed from non-`None` input arg (`probs` or `logits`).

View source

Mean.

### `mode`

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Mode.

Additional documentation from `Binomial`:

Note that when `(1 + total_count) * probs` is an integer, there are actually two modes. Namely, `(1 + total_count) * probs` and `(1 + total_count) * probs - 1` are both modes. Here we return only the larger of the two modes.

### `param_shapes`

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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`

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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`

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Probability density/mass function.

Additional documentation from `Binomial`:

For each batch member of counts `value`, `P[value]` is the probability that after sampling `self.total_count` draws from this Binomial distribution, the number of successes is `value`. Since different sequences of draws can result in the same counts, the probability includes a combinatorial coefficient.

Args
`value` `float` or `double` `Tensor`.
`name` Python `str` prepended to names of ops created by this function.
`**kwargs` Named arguments forwarded to subclass implementation.

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

### `probs_parameter`

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Probs computed from non-`None` input arg (`probs` or `logits`).

### `quantile`

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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.
`**kwargs` Named arguments forwarded to subclass implementation.

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

### `sample`

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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 or `tfp.util.SeedStream` instance, for seeding PRNG.
`name` name to give to the op.
`**kwargs` Named arguments forwarded to subclass implementation.

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

### `stddev`

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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.
`**kwargs` Named arguments forwarded to subclass implementation.

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

### `survival_function`

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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` Python `str` prepended to names of ops created by this function.
`**kwargs` Named arguments forwarded to subclass implementation.

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

### `variance`

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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` Python `str` prepended to names of ops created by this function.
`**kwargs` Named arguments forwarded to subclass implementation.

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

### `__getitem__`

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Slices the batch axes of this distribution, returning a new instance.

``````b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape  # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., -2:, 1::2]
b2.batch_shape  # => [3, 1, 5, 2, 4]

x = tf.random.stateless_normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.cholesky(cov)
loc = tf.random.stateless_normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape  # => [4, 5, 3]
mvn.event_shape  # => [2]
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape  # => [4, 2, 3, 1]
mvn2.event_shape  # => [2]
``````

Args
`slices` slices from the [] operator

Returns
`dist` A new `tfd.Distribution` instance with sliced parameters.

View source