tf.contrib.distributions.Beta

class tf.contrib.distributions.Beta

See the guide: Statistical Distributions (contrib) > Univariate (scalar) distributions

Beta distribution.

This distribution is parameterized by a and b which are shape parameters.

Mathematical details

The Beta is a distribution over the interval (0, 1). The distribution has hyperparameters a and b and probability mass function (pdf):

pdf(x) = 1 / Beta(a, b) * x^(a - 1) * (1 - x)^(b - 1)

where Beta(a, b) = Gamma(a) * Gamma(b) / Gamma(a + b) is the beta function.

This class provides methods to create indexed batches of Beta distributions. One entry of the broadcasted shape represents of a and b represents one single Beta distribution. When calling distribution functions (e.g. dist.pdf(x)), a, b and x are broadcast to the same shape (if possible). Every entry in a/b/x corresponds to a single Beta distribution.

Examples

Creates 3 distributions. The distribution functions can be evaluated on x.

a = [1, 2, 3]
b = [1, 2, 3]
dist = Beta(a, b)

# x same shape as a.
x = [.2, .3, .7]
dist.pdf(x)  # Shape [3]

# a/b will be broadcast to [[1, 2, 3], [1, 2, 3]] to match x.
x = [[.1, .4, .5], [.2, .3, .5]]
dist.pdf(x)  # Shape [2, 3]

# a/b will be broadcast to shape [5, 7, 3] to match x.
x = [[...]]  # Shape [5, 7, 3]
dist.pdf(x)  # Shape [5, 7, 3]

Creates a 2-batch of 3-class distributions.

a = [[1, 2, 3], [4, 5, 6]]  # Shape [2, 3]
b = 5  # Shape []
dist = Beta(a, b)

# x will be broadcast to [[.2, .3, .9], [.2, .3, .9]] to match a/b.
x = [.2, .3, .9]
dist.pdf(x)  # Shape [2]

Properties

a

Shape parameter.

a_b_sum

Sum of parameters.

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:

• allow_nan_stats: Python boolean.

b

Shape parameter.

dtype

The DType of Tensors handled by this Distribution.

is_continuous

is_reparameterized

name

Name prepended to all ops created by this Distribution.

parameters

Dictionary of parameters used to instantiate this Distribution.

validate_args

Python boolean indicated possibly expensive checks are enabled.

Methods

__init__(a, b, validate_args=False, allow_nan_stats=True, name='Beta')

Initialize a batch of Beta distributions.

Args:

• a: Positive floating point tensor with shape broadcastable to [N1,..., Nm] m >= 0. Defines this as a batch of N1 x ... x Nm different Beta distributions. This also defines the dtype of the distribution.
• b: Positive floating point tensor with shape broadcastable to [N1,..., Nm] m >= 0. Defines this as a batch of N1 x ... x Nm different Beta distributions.
• validate_args: Boolean, default False. Whether to assert valid values for parameters a, b, and x in prob and log_prob. If False and inputs are invalid, correct behavior is not guaranteed.
• allow_nan_stats: 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.
• name: The name to prefix Ops created by this distribution class.

Examples:

# Define 1-batch.
dist = Beta(1.1, 2.0)

# Define a 2-batch.
dist = Beta([1.0, 2.0], [4.0, 5.0])

batch_shape(name='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:

• name: name to give to the op

Returns:

• batch_shape: Tensor.

cdf(value, name='cdf', **condition_kwargs)

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. **condition_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(**override_parameters_kwargs)

Creates a deep copy of the distribution.

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

Args:

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

Returns:

• distribution: 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).

entropy(name='entropy')

Shannon entropy in nats.

event_shape(name='event_shape')

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.

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:

• batch_shape: TensorShape, possibly unknown.

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:

• event_shape: TensorShape, possibly unknown.

is_scalar_batch(name='is_scalar_batch')

Indicates that batch_shape == [].

Args:

• name: The name to give this op.

Returns:

• is_scalar_batch: Boolean scalar Tensor.

is_scalar_event(name='is_scalar_event')

Indicates that event_shape == [].

Args:

• name: The name to give this op.

Returns:

• is_scalar_event: Boolean scalar Tensor.

log_cdf(value, name='log_cdf', **condition_kwargs)

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.

Additional documentation from Beta:

Note that the argument x must be a non-negative floating point tensor whose shape can be broadcast with self.a and self.b. For fixed leading dimensions, the last dimension represents counts for the corresponding Beta distribution in self.a and self.b. x is only legal if 0 < x < 1.

Args:

• value: float or double Tensor.
• name: The name to give this op. **condition_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_pdf(value, name='log_pdf', **condition_kwargs)

Log probability density function.

Args:

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

Raises:

• TypeError: if not is_continuous.

log_pmf(value, name='log_pmf', **condition_kwargs)

Log probability mass function.

Args:

• value: float or double Tensor.
• name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

Raises:

• TypeError: if is_continuous.

log_prob(value, name='log_prob', **condition_kwargs)

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

Args:

• value: float or double Tensor.
• name: The name to give this op. **condition_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(value, name='log_survival_function', **condition_kwargs)

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. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

mean(name='mean')

Mean.

mode(name='mode')

Mode.

Additional documentation from Beta:

Note that the mode for the Beta distribution is only defined when a > 1, b > 1. This returns the mode when a > 1 and b > 1, and NaN otherwise. If self.allow_nan_stats is False, an exception will be raised rather than returning NaN.

param_shapes(cls, sample_shape, name='DistributionParamShapes')

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

Subclasses should override static 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(cls, sample_shape)

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

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.

pdf(value, name='pdf', **condition_kwargs)

Probability density function.

Args:

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

Raises:

• TypeError: if not is_continuous.

pmf(value, name='pmf', **condition_kwargs)

Probability mass function.

Args:

• value: float or double Tensor.
• name: The name to give this op. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

Raises:

• TypeError: if is_continuous.

prob(value, name='prob', **condition_kwargs)

Probability density/mass function (depending on is_continuous).

Additional documentation from Beta:

Note that the argument x must be a non-negative floating point tensor whose shape can be broadcast with self.a and self.b. For fixed leading dimensions, the last dimension represents counts for the corresponding Beta distribution in self.a and self.b. x is only legal if 0 < x < 1.

Args:

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

sample(sample_shape=(), seed=None, name='sample', **condition_kwargs)

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. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

• samples: a Tensor with prepended dimensions sample_shape.

std(name='std')

Standard deviation.

survival_function(value, name='survival_function', **condition_kwargs)

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. **condition_kwargs: Named arguments forwarded to subclass implementation.

Returns:

Tensorof shapesample_shape(x) + self.batch_shapewith values of typeself.dtype`.

variance(name='variance')

Variance.

Defined in tensorflow/contrib/distributions/python/ops/beta.py.