# tf.contrib.distributions.QuantizedDistribution

## Class QuantizedDistribution

Inherits From: Distribution

Distribution representing the quantization Y = ceiling(X).

#### Definition in terms of sampling.

1. Draw X
2. Set Y <-- ceiling(X)
3. If Y < low, reset Y <-- low
4. If Y > high, reset Y <-- high
5. Return Y


#### Definition in terms of the probability mass function.

Given scalar random variable X, we define a discrete random variable Y supported on the integers as follows:

P[Y = j] := P[X <= low],  if j == low,
:= P[X > high - 1],  j == high,
:= 0, if j < low or j > high,
:= P[j - 1 < X <= j],  all other j.


Conceptually, without cutoffs, the quantization process partitions the real line R into half open intervals, and identifies an integer j with the right endpoints:

R = ... (-2, -1](-1, 0](0, 1](1, 2](2, 3](3, 4] ...
j = ...      -1      0     1     2     3     4  ...


P[Y = j] is the mass of X within the jth interval. If low = 0, and high = 2, then the intervals are redrawn and j is re-assigned:

R = (-infty, 0](0, 1](1, infty)
j =          0     1     2


P[Y = j] is still the mass of X within the jth interval.

#### Caveats

Since evaluation of each P[Y = j] involves a cdf evaluation (rather than a closed form function such as for a Poisson), computations such as mean and entropy are better done with samples or approximations, and are not implemented by this class.

## 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.

### distribution

Base distribution, p(x).

### 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.

### 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__(
distribution,
low=None,
high=None,
validate_args=False,
name='QuantizedDistribution'
)


Construct a Quantized Distribution representing Y = ceiling(X).

Some properties are inherited from the distribution defining X. Example: allow_nan_stats is determined for this QuantizedDistribution by reading the distribution.

#### Args:

• distribution: The base distribution class to transform. Typically an instance of Distribution.
• low: Tensor with same dtype as this distribution and shape able to be added to samples. Should be a whole number. Default None. If provided, base distribution's prob should be defined at low.
• high: Tensor with same dtype as this distribution and shape able to be added to samples. Should be a whole number. Default None. If provided, base distribution's prob should be defined at high - 1. high must be strictly greater than low.
• 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.
• name: Python str name prefixed to Ops created by this class.

#### Raises:

• TypeError: If dist_cls is not a subclass of Distribution or continuous.
• NotImplementedError: If the base distribution does not implement cdf.

### 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]


Additional documentation from QuantizedDistribution:

For whole numbers y,

cdf(y) := P[Y <= y]
= 1, if y >= high,
= 0, if y < low,
= P[X <= y], otherwise.


Since Y only has mass at whole numbers, P[Y <= y] = P[Y <= floor(y)]. This dictates that fractional y are first floored to a whole number, and then above definition applies.

The base distribution's cdf method must be defined on y - 1.

#### Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.

#### 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.

#### 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: Python str prepended to names of ops created by this function.

#### 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

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.

#### Returns:

• cross_entropy: self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of (Shanon) cross entropy.

### entropy

entropy(name='entropy')


Shannon entropy in nats.

### 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: Python str prepended to names of ops created by this function.

#### Returns:

• is_scalar_batch: bool scalar Tensor.

### is_scalar_event

is_scalar_event(name='is_scalar_event')


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

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.

#### Returns:

• kl_divergence: self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of the Kullback-Leibler divergence.

### 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.

Additional documentation from QuantizedDistribution:

For whole numbers y,

cdf(y) := P[Y <= y]
= 1, if y >= high,
= 0, if y < low,
= P[X <= y], otherwise.


Since Y only has mass at whole numbers, P[Y <= y] = P[Y <= floor(y)]. This dictates that fractional y are first floored to a whole number, and then above definition applies.

The base distribution's log_cdf method must be defined on y - 1.

#### Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.

#### 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.

Additional documentation from QuantizedDistribution:

For whole numbers y,

P[Y = y] := P[X <= low],  if y == low,
:= P[X > high - 1],  y == high,
:= 0, if j < low or y > high,
:= P[y - 1 < X <= y],  all other y.


The base distribution's log_cdf method must be defined on y - 1. If the base distribution has a log_survival_function method results will be more accurate for large values of y, and in this case the log_survival_function must also be defined on y - 1.

#### Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.

#### 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.

Additional documentation from QuantizedDistribution:

For whole numbers y,

survival_function(y) := P[Y > y]
= 0, if y >= high,
= 1, if y < low,
= P[X <= y], otherwise.


Since Y only has mass at whole numbers, P[Y <= y] = P[Y <= floor(y)]. This dictates that fractional y are first floored to a whole number, and then above definition applies.

The base distribution's log_cdf method must be defined on y - 1.

#### Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.

#### 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.

Additional documentation from QuantizedDistribution:

For whole numbers y,

P[Y = y] := P[X <= low],  if y == low,
:= P[X > high - 1],  y == high,
:= 0, if j < low or y > high,
:= P[y - 1 < X <= y],  all other y.


The base distribution's cdf method must be defined on y - 1. If the base distribution has a survival_function method, results will be more accurate for large values of y, and in this case the survival_function must also be defined on y - 1.

#### Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.

#### 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: Python str prepended to names of ops created by this function.

#### 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: Python str prepended to names of ops created by this function.

#### 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).


Additional documentation from QuantizedDistribution:

For whole numbers y,

survival_function(y) := P[Y > y]
= 0, if y >= high,
= 1, if y < low,
= P[X <= y], otherwise.


Since Y only has mass at whole numbers, P[Y <= y] = P[Y <= floor(y)]. This dictates that fractional y are first floored to a whole number, and then above definition applies.

The base distribution's cdf method must be defined on y - 1.

#### Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.

#### 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: Python str prepended to names of ops created by this function.

#### Returns:

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