Distribution representing the quantization Y = ceiling(X)
.
Inherits From: Distribution
oryx.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.
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.math.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.Shift(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
that broadcasts to that of samples but does not result in additional
batch dimensions after broadcasting. 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
that broadcasts to that of samples but does not result in additional
batch dimensions after broadcasting. 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 Tensor s 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
|
trainable_variables
|
|
validate_args
|
Python bool indicating possibly expensive checks are enabled.
|
variables
|
Methods
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', **kwargs
)
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.
|
**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
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', **kwargs
)
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.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
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)
, (Shannon)
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 (Shannon) cross entropy.
|
entropy
entropy(
name='entropy', **kwargs
)
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 .
|
experimental_default_event_space_bijector
experimental_default_event_space_bijector(
*args, **kwargs
)
Bijector mapping the reals (R**n) to the event space of the distribution.
Distributions with continuous support may implement
_default_event_space_bijector
which returns a subclass of
tfp.bijectors.Bijector
that maps R**n to the distribution's event space.
For example, the default bijector for the Beta
distribution
is tfp.bijectors.Sigmoid()
, which maps the real line to [0, 1]
, the
support of the Beta
distribution. The default bijector for the
CholeskyLKJ
distribution is tfp.bijectors.CorrelationCholesky
, which
maps R^(k * (k-1) // 2) to the submanifold of k x k lower triangular
matrices with ones along the diagonal.
The purpose of experimental_default_event_space_bijector
is
to enable gradient descent in an unconstrained space for Variational
Inference and Hamiltonian Monte Carlo methods. Some effort has been made to
choose bijectors such that the tails of the distribution in the
unconstrained space are between Gaussian and Exponential.
For distributions with discrete event space, or for which TFP currently
lacks a suitable bijector, this function returns None
.
Args | |
---|---|
*args
|
Passed to implementation _default_event_space_bijector .
|
**kwargs
|
Passed to implementation _default_event_space_bijector .
|
Returns | |
---|---|
event_space_bijector
|
Bijector instance or None .
|
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 (Shannon) cross entropy, and H[.]
denotes (Shannon) 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
log_cdf(
value, name='log_cdf', **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
.
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.
|
**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_prob
log_prob(
value, name='log_prob', **kwargs
)
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.
|
**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
log_survival_function(
value, name='log_survival_function', **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
.
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.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .
|
mean
mean(
name='mean', **kwargs
)
Mean.
mode
mode(
name='mode', **kwargs
)
Mode.
param_shapes
@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
@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.
|
parameter_properties
@classmethod
parameter_properties( dtype=tf.float32, num_classes=None )
Returns a dict mapping constructor arg names to property annotations.
This dict should include an entry for each of the distribution's
Tensor
-valued constructor arguments.
Args | |
---|---|
dtype
|
Optional float dtype to assume for continuous-valued parameters.
Some constraining bijectors require advance knowledge of the dtype
because certain constants (e.g., tfb.Softplus.low ) must be
instantiated with the same dtype as the values to be transformed.
|
num_classes
|
Optional int Tensor number of classes to assume when
inferring the shape of parameters for categorical-like distributions.
Otherwise ignored.
|
Returns | |
---|---|
parameter_properties
|
A
str -> tfp.python.internal.parameter_properties.ParameterPropertiesdict mapping constructor argument names to ParameterProperties`
instances.
|
prob
prob(
value, name='prob', **kwargs
)
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.
|
**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 .
|
quantile
quantile(
value, name='quantile', **kwargs
)
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.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
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', **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 or tfp.util.SeedStream instance, for seeding PRNG.
|
name
|
name to give to the op. |
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
samples
|
a Tensor with prepended dimensions sample_shape .
|
stddev
stddev(
name='stddev', **kwargs
)
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.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
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', **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).
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.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .
|
variance
variance(
name='variance', **kwargs
)
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.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
variance
|
Floating-point Tensor with shape identical to
batch_shape + event_shape , i.e., the same shape as self.mean() .
|
__getitem__
__getitem__(
slices
)
Slices the batch axes of this distribution, returning a new instance.
b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., -2:, 1::2]
b2.batch_shape # => [3, 1, 5, 2, 4]
x = tf.random.stateless_normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.linalg.cholesky(cov)
loc = tf.random.stateless_normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape # => [4, 5, 3]
mvn.event_shape # => [2]
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape # => [4, 2, 3, 1]
mvn2.event_shape # => [2]
Args | |
---|---|
slices
|
slices from the [] operator |
Returns | |
---|---|
dist
|
A new tfd.Distribution instance with sliced parameters.
|
__iter__
__iter__()