contrib.distributions.Mixture

tf.contrib.distributions.Mixture

class tf.contrib.distributions.Mixture

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

See the guide: Statistical Distributions (contrib) > Mixture Models

Mixture distribution.

The Mixture object implements batched mixture distributions. The mixture model is defined by a Categorical distribution (the mixture) and a python list of Distribution objects.

Methods supported include log_prob, prob, mean, sample, and entropy_lower_bound.

Properties

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.

Returns:

• allow_nan_stats: Python bool.

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.

Returns:

• batch_shape: TensorShape, possibly unknown.

cat

components

dtype

The DType of Tensors handled by this Distribution.

event_shape

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

May be partially defined or unknown.

Returns:

• event_shape: TensorShape, possibly unknown.

name

Name prepended to all ops created by this Distribution.

num_components

parameters

Dictionary of parameters used to instantiate this Distribution.

reparameterization_type

Describes how samples from the distribution are reparameterized.

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

Returns:

An instance of ReparameterizationType.

validate_args

Python bool indicating possibly expensive checks are enabled.

Methods

__init__

__init__(
    cat,
    components,
    validate_args=False,
    allow_nan_stats=True,
    name='Mixture'
)

Initialize a Mixture distribution.

A Mixture is defined by a Categorical (cat, representing the mixture probabilities) and a list of Distribution objects all having matching dtype, batch shape, event shape, and continuity properties (the components).

The num_classes of cat must be possible to infer at graph construction time and match len(components).

Args:

• cat: A Categorical distribution instance, representing the probabilities of distributions.
• components: A list or tuple of Distribution instances. Each instance must have the same type, be defined on the same domain, and have matching event_shape and batch_shape.
• validate_args: Python bool, default False. If True, raise a runtime error if batch or event ranks are inconsistent between cat and any of the distributions. This is only checked if the ranks cannot be determined statically at graph construction time.
• 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: A name for this distribution (optional).

Raises:

• TypeError: If cat is not a Categorical, or components is not a list or tuple, or the elements of components are not instances of Distribution, or do not have matching dtype.
• ValueError: If components is an empty list or tuple, or its elements do not have a statically known event rank. If cat.num_classes cannot be inferred at graph creation time, or the constant value of cat.num_classes is not equal to len(components), or all components and cat do not have matching static batch shapes, or all components do not have matching static event shapes.

batch_shape_tensor

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.

Args:

• name: name to give to the op

Returns:

• batch_shape: Tensor.

cdf

cdf(
    value,
    name='cdf'
)

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.

Returns:

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

copy

copy(**override_parameters_kwargs)

Creates a deep copy of the distribution.

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

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

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.

Args:

• name: The name to give this op.

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).

entropy

entropy(name='entropy')

Shannon entropy in nats.

entropy_lower_bound

entropy_lower_bound(name='entropy_lower_bound')

A lower bound on the entropy of this mixture model.

The bound below is not always very tight, and its usefulness depends on the mixture probabilities and the components in use.

A lower bound is useful for ELBO when the Mixture is the variational distribution:

\( \log p(x) >= ELBO = \int q(z) \log p(x, z) dz + H[q] \)

where \( p \) is the prior distribution, \( q \) is the variational, and \( H[q] \) is the entropy of \( q \). If there is a lower bound \( G[q] \) such that \( H[q] \geq G[q] \) then it can be used in place of \( H[q] \).

For a mixture of distributions \( q(Z) = \sum_i c_i q_i(Z) \) with \( \sum_i c_i = 1 \), by the concavity of \( f(x) = -x \log x \), a simple lower bound is:

\( \begin{align} H[q] & = - \int q(z) \log q(z) dz \\ & = - \int (\sum_i c_i q_i(z)) \log(\sum_i c_i q_i(z)) dz \\ & \geq - \sum_i c_i \int q_i(z) \log q_i(z) dz \\ & = \sum_i c_i H[q_i] \end{align} \)

This is the term we calculate below for \( G[q] \).

Args:

• name: A name for this operation (optional).

Returns:

A lower bound on the Mixture's entropy.

event_shape_tensor

event_shape_tensor(name='event_shape_tensor')

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

is_scalar_batch(name='is_scalar_batch')

Indicates that batch_shape == [].

Args:

• name: The name to give this op.

Returns:

• is_scalar_batch: bool scalar Tensor.

is_scalar_event

is_scalar_event(name='is_scalar_event')

Indicates that event_shape == [].

Args:

• name: The name to give this op.

Returns:

• is_scalar_event: bool scalar Tensor.

log_cdf

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.

Args:

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

Returns:

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

log_prob

log_prob(
    value,
    name='log_prob'
)

Log probability density/mass function.

Args:

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

Returns:

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

log_survival_function

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.

Args:

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

Returns:

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

mean

mean(name='mean')

Mean.

mode

mode(name='mode')

Mode.

param_shapes

param_shapes(
    cls,
    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.

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

param_static_shapes(
    cls,
    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.

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

prob(
    value,
    name='prob'
)

Probability density/mass function.

Args:

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

Returns:

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

quantile

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

Args:

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

Returns:

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

sample

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.

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.

Returns:

• samples: a Tensor with prepended dimensions sample_shape.

stddev

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.

Args:

• name: The name to give this op.

Returns:

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

survival_function

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).

Args:

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

Returns:

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

variance

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.

Args:

• name: The name to give this op.

Returns:

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