tf.contrib.distributions.QuantizedDistribution

Distribution representing the quantization Y = ceiling(X).

Inherits From: Distribution

tf.contrib.distributions.QuantizedDistribution(
    distribution, low=None, high=None, validate_args=False,
    name='QuantizedDistribution'
)

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.

Examples

We illustrate a mixture of discretized logistic distributions [(Salimans et al., 2017)][1]. This is used, for example, for capturing 16-bit audio in WaveNet [(van den Oord et al., 2017)][2]. The values range in a 1-D integer domain of [0, 2**16-1], and the discretization captures P(x - 0.5 < X <= x + 0.5) for all x in the domain excluding the endpoints. The lowest value has probability P(X <= 0.5) and the highest value has probability P(2**16 - 1.5 < X).

Below we assume a wavenet function. It takes as input right-shifted audio samples of shape [..., sequence_length]. It returns a real-valued tensor of shape [..., num_mixtures * 3], i.e., each mixture component has a loc and scale parameter belonging to the logistic distribution, and a logits parameter determining the unnormalized probability of that component.

import tensorflow_probability as tfp
tfd = tfp.distributions
tfb = tfp.bijectors

net = wavenet(inputs)
loc, unconstrained_scale, logits = tf.split(net,
                                            num_or_size_splits=3,
                                            axis=-1)
scale = tf.nn.softplus(unconstrained_scale)

# Form mixture of discretized logistic distributions. Note we shift the
# logistic distribution by -0.5. This lets the quantization capture "rounding"
# intervals, `(x-0.5, x+0.5]`, and not "ceiling" intervals, `(x-1, x]`.
discretized_logistic_dist = tfd.QuantizedDistribution(
    distribution=tfd.TransformedDistribution(
        distribution=tfd.Logistic(loc=loc, scale=scale),
        bijector=tfb.AffineScalar(shift=-0.5)),
    low=0.,
    high=2**16 - 1.)
mixture_dist = tfd.MixtureSameFamily(
    mixture_distribution=tfd.Categorical(logits=logits),
    components_distribution=discretized_logistic_dist)

neg_log_likelihood = -tf.reduce_sum(mixture_dist.log_prob(targets))
train_op = tf.train.AdamOptimizer().minimize(neg_log_likelihood)

After instantiating mixture_dist, we illustrate maximum likelihood by calculating its log-probability of audio samples as target and optimizing.

References

[1]: Tim Salimans, Andrej Karpathy, Xi Chen, and Diederik P. Kingma. PixelCNN++: Improving the PixelCNN with discretized logistic mixture likelihood and other modifications. International Conference on Learning Representations, 2017. https://arxiv.org/abs/1701.05517 [2]: Aaron van den Oord et al. Parallel WaveNet: Fast High-Fidelity Speech Synthesis. arXiv preprint arXiv:1711.10433, 2017. https://arxiv.org/abs/1711.10433

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.

Attributes
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.
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.
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.
high	Highest value that quantization returns.
low	Lowest value that quantization returns.
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.
validate_args	Python bool indicating possibly expensive checks are enabled.

Methods

batch_shape_tensor

View source

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

View source

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

View source

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

View source

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

View source

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	tfp.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

View source

entropy(
    name='entropy'
)

Shannon entropy in nats.

event_shape_tensor

View source

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

View source

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

View source

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

View source

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	tfp.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

View source

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

View source

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

View source

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

View source

mean(
    name='mean'
)

Mean.

mode

View source

mode(
    name='mode'
)

Mode.

param_shapes

View source

@classmethod
param_shapes(
    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

View source

@classmethod
param_static_shapes(
    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

View source

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

View source

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

View source

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

View source

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

View source

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

View source

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().