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tf.contrib.distributions.QuantizedDistribution

Class `QuantizedDistribution`

Inherits From: Distribution

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

See the guide: Statistical Distributions (contrib) > Transformed distributions

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.

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: 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.

Args:

other: tf.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 (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.

Args:

other: tf.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`

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().

© 2018 The TensorFlow Authors. All rights reserved.
Licensed under the Creative Commons Attribution License 3.0.
Code samples licensed under the Apache 2.0 License.
https://www.tensorflow.org/api_docs/python/tf/contrib/distributions/QuantizedDistribution

tf.contrib.distributions.QuantizedDistribution

Class QuantizedDistribution

Definition in terms of sampling.

Definition in terms of the probability mass function.

Caveats

Properties

allow_nan_stats

Returns:

batch_shape

Returns:

distribution

dtype

event_shape

Returns:

name

parameters

reparameterization_type

Returns:

validate_args

Methods

__init__

Args:

Raises:

batch_shape_tensor

Args:

Returns:

cdf

Args:

Returns:

copy

Args:

Returns:

covariance

Args:

Returns:

cross_entropy

Args:

Returns:

entropy

event_shape_tensor

Args:

Returns:

is_scalar_batch

Args:

Returns:

is_scalar_event

Args:

Returns:

kl_divergence

Args:

Returns:

log_cdf

Args:

Returns:

log_prob

Args:

Returns:

log_survival_function

Args:

Returns:

mean

mode

param_shapes

Args:

Returns:

param_static_shapes

Args:

Returns:

Raises:

prob

Args:

Returns:

quantile

Args:

Returns:

sample

Args:

Returns:

stddev

Args:

Class `QuantizedDistribution`

`allow_nan_stats`

`batch_shape`

`distribution`

`dtype`

`event_shape`

`name`

`parameters`

`reparameterization_type`

`validate_args`

`init`

`batch_shape_tensor`

`cdf`

`copy`

`covariance`

`cross_entropy`

`entropy`

`event_shape_tensor`

`is_scalar_batch`

`is_scalar_event`

`kl_divergence`

`log_cdf`

`log_prob`

`log_survival_function`

`mean`

`mode`

`param_shapes`

`param_static_shapes`

`prob`

`quantile`

`sample`

`stddev`

`survival_function`

`variance`