# tf.contrib.distributions.Mixture

### class tf.contrib.distributions.Mixture

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 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.

### dtype

The DType of Tensors handled by this Distribution.

### 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__(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: Boolean, 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(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.

#### 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.

### 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(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.

#### 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.

Mode.

### 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).

#### 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.