Joint distribution parameterized by named distribution-making functions.
Inherits From: JointDistribution
, Distribution
oryx.distributions.JointDistributionNamed(
model,
batch_ndims=None,
use_vectorized_map=False,
validate_args=False,
experimental_use_kahan_sum=False,
name=None
)
This distribution enables both sampling and joint probability computation from a single model specification.
A joint distribution is a collection of possibly interdependent distributions.
Like JointDistributionSequential
, JointDistributionNamed
is parameterized
by several distribution-making functions. Unlike JointDistributionNamed
,
each distribution-making function must have its own key. Additionally every
distribution-making function's arguments must refer to only specified keys.
#### Mathematical Details
Internally JointDistributionNamed
implements the chain rule of probability.
That is, the probability function of a length-d
vector x
is,
p(x) = prod{ p(x[i] | x[:i]) : i = 0, ..., (d - 1) }
The JointDistributionNamed
is parameterized by a dict
(or namedtuple
or
collections.OrderedDict
) composed of either:
tfp.distributions.Distribution
-like instances or,callable
s which return atfp.distributions.Distribution
-like instance.The "conditioned on" elements are represented by the
callable
's required arguments; every argument must correspond to a key in the named distribution-making functions. Distribution-makers which are directly aDistribution
-like instance are allowed for convenience and semantically identical a zero argumentcallable
. When the maker takes no arguments it is preferable to directly provide the distribution instance.Name resolution:
The names of
JointDistributionNamed` components are simply the keys specified explicitly in the model definition.Examples
Consider the following generative model:
e ~ Exponential(rate=[100,120]) g ~ Gamma(concentration=e[0], rate=e[1]) n ~ Normal(loc=0, scale=2.) m ~ Normal(loc=n, scale=g) for i = 1, ..., 12: x[i] ~ Bernoulli(logits=m)
We can code this as:
tfd = tfp.distributions joint = tfd.JointDistributionNamed(dict( e= tfd.Exponential(rate=[100, 120]), g=lambda e: tfd.Gamma(concentration=e[0], rate=e[1]), n= tfd.Normal(loc=0, scale=2.), m=lambda n, g: tfd.Normal(loc=n, scale=g), x=lambda m: tfd.Sample(tfd.Bernoulli(logits=m), 12) ), batch_ndims=0, use_vectorized_map=True)
Notice the 1:1 correspondence between "math" and "code". Further, notice that unlike
JointDistributionSequential
, there is no need to put the distribution-making functions in topologically sorted order nor is it ever necessary to use dummy arguments to skip dependencies.x = joint.sample() # ==> A 5-element `dict` of Tensors representing a draw/realization from each # distribution. joint.log_prob(x) # ==> A scalar `Tensor` representing the total log prob under all five # distributions. joint.resolve_graph() # ==> (('e', ()), # ('g', ('e',)), # ('n', ()), # ('m', ('n', 'g')), # ('x', ('m',)))
Discussion
JointDistributionNamed
topologically sorts the distribution-making functions and calls each by feeding in all previously created dependencies. A distribution-maker must either be a:tfd.Distribution
-like instance (e.g.,e
andn
in the above example),Python
callable
(e.g.,g
,m
,x
in the above example).Regarding #1, an object is deemed "
tfd.Distribution
-like" if it has asample
,log_prob
, and distribution properties, e.g.,batch_shape
,event_shape
,dtype
.Regarding #2, in addition to using a function (or
lambda
), supplying a TFD "class
" is also permissible, this also being a "Pythoncallable
." For example, instead of writing:lambda loc, scale: tfd.Normal(loc=loc, scale=scale)
one could have simply writtentfd.Normal
.Notice that directly providing a
tfd.Distribution
-like instance means there cannot exist a (dynamic) dependency on other distributions; it is "independent" both "computationally" and "statistically." The same is self-evidently true of zero-argumentcallable
s.A distribution instance depends on other distribution instances through the distribution making function's required arguments. The distribution makers' arguments are parameterized by samples from the corresponding previously constructed distributions. ("Previous" in the sense of a topological sorting of dependencies.)
Vectorized sampling and model evaluation
When a joint distribution's
sample
method is called with asample_shape
(or thelog_prob
method is called on an input with multiple sample dimensions) the model must be equipped to handle additional batch dimensions. This may be done manually, or automatically by passinguse_vectorized_map=True
. Manual vectorization has historically been the default, but we now recommend that most users enable automatic vectorization unless they are affected by a specific issue; some known issues are listed below.When using manually-vectorized joint distributions, each operation in the model must account for the possibility of batch dimensions in Distributions and their samples. By contrast, auto-vectorized models need only describe a single sample from the joint distribution; any batch evaluation is automated as required using
tf.vectorized_map
(vmap
in JAX). In many cases this allows for significant simplications. For example, the following manually-vectorizedtfd.JointDistributionSequential
model:model = tfd.JointDistributionSequential([ tfd.Normal(0., tf.ones([3])), tfd.Normal(0., 1.), lambda y, x: tfd.Normal(x[..., :2] + y[..., tf.newaxis], 1.) ])
can be written in auto-vectorized form as
model = tfd.JointDistributionSequential([ tfd.Normal(0., tf.ones([3])), tfd.Normal(0., 1.), lambda y, x: tfd.Normal(x[:2] + y, 1.) ], use_vectorized_map=True)
in which we were able to avoid explicitly accounting for batch dimensions when indexing and slicing computed quantities in the third line.
Known limitations of automatic vectorization:
- A small fraction of TensorFlow ops are unsupported; models that use an unsupported op will raise an error and must be manually vectorized.
- Sampling large batches may be slow under automatic vectorization because
TensorFlow's stateless samplers are currently converted using a
non-vectorized
while_loop
. This limitation applies only in TensorFlow; vectorized samplers in JAX should be approximately as fast as manually vectorized code. Calling
sample_distributions
with nontrivialsample_shape
will raise an error if the model contains any distributions that are not registered as CompositeTensors (TFP's basic distributions are usually fine, but support for wrapper distributions liketfd.Sample
is a work in progress).Batch semantics and (log-)densities
tl;dr: pass
batch_ndims=0
unless you have a good reason not to.Joint distributions now support 'auto-batching' semantics, in which the distribution's batch shape is derived by broadcasting the leftmost
batch_ndims
dimensions of its components' batch shapes. All remaining dimensions are considered to form a single 'event' of the joint distribution. Ifbatch_ndims==0
, then the joint distribution has batch shape[]
, and all component dimensions are treated as event shape. For example, the modeljd = tfd.JointDistributionSequential([ tfd.Normal(0., tf.ones([3])), lambda x: tfd.Normal(x[..., tf.newaxis], tf.ones([3, 2])) ], batch_ndims=0)
creates a joint distribution with batch shape
[]
and event shape([3], [3, 2])
. The log-density of a sample always has shapebatch_shape
, so this guarantees thatjd.log_prob(jd.sample())
will evaluate to a scalar value. We could alternately construct a joint distribution with batch shape[3]
and event shape([], [2])
by settingbatch_ndims=1
, in which casejd.log_prob(jd.sample())
would evaluate to a value of shape[3]
.Setting
batch_ndims=None
recovers the 'classic' batch semantics (currently still the default for backwards-compatibility reasons), in which the joint distribution'slog_prob
is computed by naively summing log densities from the component distributions. Since these component densities have shapes equal to the batch shapes of the individual components, to avoid broadcasting errors it is usually necessary to construct the components with identical batch shapes. For example, the component distributions in the model above have batch shapes of[3]
and[3, 2]
respectively, which would raise an error if summed directly, but can be aligned by wrapping withtfd.Independent
, as in this model:jd = tfd.JointDistributionSequential([ tfd.Normal(0., tf.ones([3])), lambda x: tfd.Independent(tfd.Normal(x[..., tf.newaxis], tf.ones([3, 2])), reinterpreted_batch_ndims=1) ], batch_ndims=None)
Here the components both have batch shape
[3]
, sojd.log_prob(jd.sample())
returns a value of shape[3]
, just as in thebatch_ndims=1
case above. In fact, auto-batching semantics are equivalent to implicitly wrapping each componentdist
astfd.Independent(dist, reinterpreted_batch_ndim=(dist.batch_shape.ndims - jd.batch_ndims))
; the only vestigial difference is that under auto-batching semantics, the joint distribution has a single batch shape[3]
, while under the classic semantics the value ofjd.batch_shape
is a structure of the component batch shapes([3], [3])
. Such structured batch shapes will be deprecated in the future, since they are inconsistent with the definition of batch shapes used elsewhere in TFP.References
[1] Dan Piponi, Dave Moore, and Joshua V. Dillon. Joint distributions for TensorFlow Probability. arXiv preprint arXiv:2001.11819_,
If every element of model
is a CompositeTensor
or a callable, the resulting JointDistributionNamed
is a CompositeTensor
. Otherwise, a non-CompositeTensor
_JointDistributionNamed
instance is created.
Args | |
---|---|
model
|
Python dict , collections.OrderedDict , or namedtuple of
distribution-making functions each with required args corresponding
only to other keys.
|
batch_ndims
|
int Tensor number of batch dimensions. The batch_shape s
of all component distributions must be such that the prefixes of
length batch_ndims broadcast to a consistent joint batch shape.
Default value: None .
|
use_vectorized_map
|
Python bool . Whether to use tf.vectorized_map
to automatically vectorize evaluation of the model. This allows the
model specification to focus on drawing a single sample, which is often
simpler, but some ops may not be supported.
Default value: False .
|
validate_args
|
Python bool . Whether to validate input with asserts.
If validate_args is False , and the inputs are invalid,
correct behavior is not guaranteed.
Default value: False .
|
experimental_use_kahan_sum
|
Python bool . When True , we use Kahan
summation to aggregate independent underlying log_prob values, which
improves against the precision of a naive float32 sum. This can be
noticeable in particular for large dimensions in float32. See CPU caveat
on tfp.math.reduce_kahan_sum . This argument has no effect if
batch_ndims is None .
Default value: False .
|
name
|
The name for ops managed by the distribution.
Default value: None (i.e., "JointDistributionNamed" ).
|
Child Classes
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]
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
.
other
types with built-in registrations: JointDistributionNamed
, JointDistributionNamedAutoBatched
, JointDistributionSequential
, JointDistributionSequentialAutoBatched
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.
Additional documentation from _JointDistributionSequential
:
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 .
|
experimental_fit
@classmethod
experimental_fit( value, sample_ndims=1, validate_args=False, **init_kwargs )
Instantiates a distribution that maximizes the likelihood of x
.
Args | |
---|---|
value
|
a Tensor valid sample from this distribution family.
|
sample_ndims
|
Positive int Tensor number of leftmost dimensions of
value that index i.i.d. samples.
Default value: 1 .
|
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.
Default value: False .
|
**init_kwargs
|
Additional keyword arguments passed through to
cls.__init__ . These take precedence in case of collision with the
fitted parameters; for example,
tfd.Normal.experimental_fit([1., 1.], scale=20.) returns a Normal
distribution with scale=20. rather than the maximum likelihood
parameter scale=0. .
|
Returns | |
---|---|
maximum_likelihood_instance
|
instance of cls with parameters that
maximize the likelihood of value .
|
experimental_local_measure
experimental_local_measure(
value, backward_compat=False, **kwargs
)
Returns a log probability density together with a TangentSpace
.
A TangentSpace
allows us to calculate the correct push-forward
density when we apply a transformation to a Distribution
on
a strict submanifold of R^n (typically via a Bijector
in the
TransformedDistribution
subclass). The density correction uses
the basis of the tangent space.
Args | |
---|---|
value
|
float or double Tensor .
|
backward_compat
|
bool specifying whether to fall back to returning
FullSpace as the tangent space, and representing R^n with the standard
basis.
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
log_prob
|
a Tensor representing the log probability density, of shape
sample_shape(x) + self.batch_shape with values of type self.dtype .
|
tangent_space
|
a TangentSpace object (by default FullSpace )
representing the tangent space to the manifold at value .
|
Raises | |
---|---|
UnspecifiedTangentSpaceError if backward_compat is False and
the _experimental_tangent_space attribute has not been defined.
|
experimental_pin
experimental_pin(
*args, **kwargs
)
Pins some parts, returning an unnormalized distribution object.
The calling convention is much like other JointDistribution
methods (e.g.
log_prob
), but with the difference that not all parts are required. In
this respect, the behavior is similar to that of the sample
function's
value
argument.
Examples:
# Given the following joint distribution:
jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1., name='z'),
tfd.Normal(0., 1., name='y'),
lambda y, z: tfd.Normal(y + z, 1., name='x')
], validate_args=True)
# The following calls are all permissible and produce
# `JointDistributionPinned` objects behaving identically.
PartialXY = collections.namedtuple('PartialXY', 'x,y')
PartialX = collections.namedtuple('PartialX', 'x')
assert (jd.experimental_pin(x=2.).pins ==
jd.experimental_pin(x=2., z=None).pins ==
jd.experimental_pin(dict(x=2.)).pins ==
jd.experimental_pin(dict(x=2., y=None)).pins ==
jd.experimental_pin(PartialXY(x=2., y=None)).pins ==
jd.experimental_pin(PartialX(x=2.)).pins ==
jd.experimental_pin(None, None, 2.).pins ==
jd.experimental_pin([None, None, 2.]).pins)
Args | |
---|---|
*args
|
Positional arguments: a value structure or component values (see above). |
**kwargs
|
Keyword arguments: a value structure or component values (see
above). May also include name , specifying a Python string name for ops
generated by this method.
|
Returns | |
---|---|
pinned
|
a tfp.experimental.distributions.JointDistributionPinned with
the given values pinned.
|
experimental_sample_and_log_prob
experimental_sample_and_log_prob(
sample_shape=(), seed=None, name='sample_and_log_prob', **kwargs
)
Samples from this distribution and returns the log density of the sample.
The default implementation simply calls sample
and log_prob
:
def _sample_and_log_prob(self, sample_shape, seed, **kwargs):
x = self.sample(sample_shape=sample_shape, seed=seed, **kwargs)
return x, self.log_prob(x, **kwargs)
However, some subclasses may provide more efficient and/or numerically stable implementations.
Args | |
---|---|
sample_shape
|
integer Tensor desired shape of samples to draw.
Default value: () .
|
seed
|
PRNG seed; see tfp.random.sanitize_seed for details.
Default value: None .
|
name
|
name to give to the op.
Default value: 'sample_and_log_prob' .
|
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
samples
|
a Tensor , or structure of Tensor s, with prepended dimensions
sample_shape .
|
log_prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
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 for each distribution in model .
|
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 for each distribution in model .
|
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.
other
types with built-in registrations: JointDistributionNamed
, JointDistributionNamedAutoBatched
, JointDistributionSequential
, JointDistributionSequentialAutoBatched
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
.
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(
*args, **kwargs
)
Log probability density/mass function.
The measure methods of JointDistribution
(log_prob
, prob
, etc.)
can be called either by passing a single structure of tensors or by using
named args for each part of the joint distribution state. For example,
jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1.),
lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.log_prob(sample) ==
jd.log_prob(value=sample) ==
jd.log_prob(z, x) ==
jd.log_prob(z=z, x=x) ==
jd.log_prob(z, x=x))
# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.log_prob(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.log_prob(**sample)
JointDistribution
component distributions names are resolved via
jd._flat_resolve_names()
, which is implemented by each JointDistribution
subclass (see subclass documentation for details). Generally, for components
where a name was provided---
either explicitly as the name
argument to a distribution or as a key in a
dict-valued JointDistribution, or implicitly, e.g., by the argument name of
a JointDistributionSequential
distribution-making function---the provided
name will be used. Otherwise the component will receive a dummy name; these
may change without warning and should not be relied upon.
trivial_jd = tfd.JointDistributionSequential([tfd.Exponential(1.)])
trivial_jd.dtype # => [tf.float32]
trivial_jd.log_prob([4.])
# ==> Tensor with shape `[]`.
lp = trivial_jd.log_prob(4.)
# ==> Tensor with shape `[]`.
Notice that in the first call, [4.]
is interpreted as a list of one
scalar while in the second call the input is a scalar. Hence both inputs
result in identical scalar outputs. If we wanted to pass an explicit
vector to the Exponential
component---creating a vector-shaped batch
of log_prob
s---we could instead write
trivial_jd.log_prob(np.array([4]))
.
Args | |
---|---|
*args
|
Positional arguments: a value structure or component values
(see above).
|
**kwargs
|
Keyword arguments: a value structure or component values
(see above). May also include name , specifying a Python string name
for ops generated by this method.
|
Returns | |
---|---|
log_prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
log_prob_parts
log_prob_parts(
*args, **kwargs
)
Log probability density/mass function.
The measure methods of JointDistribution
(log_prob
, prob
, etc.)
can be called either by passing a single structure of tensors or by using
named args for each part of the joint distribution state. For example,
jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1.),
lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.log_prob_parts(sample) ==
jd.log_prob_parts(value=sample) ==
jd.log_prob_parts(z, x) ==
jd.log_prob_parts(z=z, x=x) ==
jd.log_prob_parts(z, x=x))
# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.log_prob_parts(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.log_prob_parts(**sample)
JointDistribution
component distributions names are resolved via
jd._flat_resolve_names()
, which is implemented by each JointDistribution
subclass (see subclass documentation for details). Generally, for components
where a name was provided---
either explicitly as the name
argument to a distribution or as a key in a
dict-valued JointDistribution, or implicitly, e.g., by the argument name of
a JointDistributionSequential
distribution-making function---the provided
name will be used. Otherwise the component will receive a dummy name; these
may change without warning and should not be relied upon.
trivial_jd = tfd.JointDistributionSequential([tfd.Exponential(1.)])
trivial_jd.dtype # => [tf.float32]
trivial_jd.log_prob_parts([4.])
# ==> Tensor with shape `[]`.
lp_parts = trivial_jd.log_prob_parts(4.)
# ==> Tensor with shape `[]`.
Notice that in the first call, [4.]
is interpreted as a list of one
scalar while in the second call the input is a scalar. Hence both inputs
result in identical scalar outputs. If we wanted to pass an explicit
vector to the Exponential
component---creating a vector-shaped batch
of log_prob_parts
s---we could instead write
trivial_jd.log_prob_parts(np.array([4]))
.
Args | |
---|---|
*args
|
Positional arguments: a value structure or component values
(see above).
|
**kwargs
|
Keyword arguments: a value structure or component values
(see above). May also include name , specifying a Python string name
for ops generated by this method.
|
Returns | |
---|---|
log_prob_parts
|
a self.dtype -like structure of Tensor s representing
the log_prob for each component distribution evaluated at each
corresponding value .
|
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
.
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.
Distribution subclasses are not required to implement
_parameter_properties
, so this method may raise NotImplementedError
.
Providing a _parameter_properties
implementation enables several advanced
features, including:
- Distribution batch slicing (
sliced_distribution = distribution[i:j]
). - Automatic inference of
_batch_shape
and_batch_shape_tensor
, which must otherwise be computed explicitly. - Automatic instantiation of the distribution within TFP's internal property tests.
- Automatic construction of 'trainable' instances of the distribution using appropriate bijectors to avoid violating parameter constraints. This enables the distribution family to be used easily as a surrogate posterior in variational inference.
In the future, parameter property annotations may enable additional
functionality; for example, returning Distribution instances from
tf.vectorized_map
.
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.
|
Raises | |
---|---|
NotImplementedError
|
if the distribution class does not implement
_parameter_properties .
|
prob
prob(
*args, **kwargs
)
Probability density/mass function.
The measure methods of JointDistribution
(log_prob
, prob
, etc.)
can be called either by passing a single structure of tensors or by using
named args for each part of the joint distribution state. For example,
jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1.),
lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.prob(sample) ==
jd.prob(value=sample) ==
jd.prob(z, x) ==
jd.prob(z=z, x=x) ==
jd.prob(z, x=x))
# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.prob(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.prob(**sample)
JointDistribution
component distributions names are resolved via
jd._flat_resolve_names()
, which is implemented by each JointDistribution
subclass (see subclass documentation for details). Generally, for components
where a name was provided---
either explicitly as the name
argument to a distribution or as a key in a
dict-valued JointDistribution, or implicitly, e.g., by the argument name of
a JointDistributionSequential
distribution-making function---the provided
name will be used. Otherwise the component will receive a dummy name; these
may change without warning and should not be relied upon.
trivial_jd = tfd.JointDistributionSequential([tfd.Exponential(1.)])
trivial_jd.dtype # => [tf.float32]
trivial_jd.prob([4.])
# ==> Tensor with shape `[]`.
prob = trivial_jd.prob(4.)
# ==> Tensor with shape `[]`.
Notice that in the first call, [4.]
is interpreted as a list of one
scalar while in the second call the input is a scalar. Hence both inputs
result in identical scalar outputs. If we wanted to pass an explicit
vector to the Exponential
component---creating a vector-shaped batch
of prob
s---we could instead write
trivial_jd.prob(np.array([4]))
.
Args | |
---|---|
*args
|
Positional arguments: a value structure or component values
(see above).
|
**kwargs
|
Keyword arguments: a value structure or component values
(see above). May also include name , specifying a Python string name
for ops generated by this method.
|
Returns | |
---|---|
prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
prob_parts
prob_parts(
*args, **kwargs
)
Probability density/mass function.
The measure methods of JointDistribution
(log_prob
, prob
, etc.)
can be called either by passing a single structure of tensors or by using
named args for each part of the joint distribution state. For example,
jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1.),
lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.prob_parts(sample) ==
jd.prob_parts(value=sample) ==
jd.prob_parts(z, x) ==
jd.prob_parts(z=z, x=x) ==
jd.prob_parts(z, x=x))
# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.prob_parts(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.prob_parts(**sample)
JointDistribution
component distributions names are resolved via
jd._flat_resolve_names()
, which is implemented by each JointDistribution
subclass (see subclass documentation for details). Generally, for components
where a name was provided---
either explicitly as the name
argument to a distribution or as a key in a
dict-valued JointDistribution, or implicitly, e.g., by the argument name of
a JointDistributionSequential
distribution-making function---the provided
name will be used. Otherwise the component will receive a dummy name; these
may change without warning and should not be relied upon.
trivial_jd = tfd.JointDistributionSequential([tfd.Exponential(1.)])
trivial_jd.dtype # => [tf.float32]
trivial_jd.prob_parts([4.])
# ==> Tensor with shape `[]`.
p_parts = trivial_jd.prob_parts(4.)
# ==> Tensor with shape `[]`.
Notice that in the first call, [4.]
is interpreted as a list of one
scalar while in the second call the input is a scalar. Hence both inputs
result in identical scalar outputs. If we wanted to pass an explicit
vector to the Exponential
component---creating a vector-shaped batch
of prob_parts
s---we could instead write
trivial_jd.prob_parts(np.array([4]))
.
Args | |
---|---|
*args
|
Positional arguments: a value structure or component values
(see above).
|
**kwargs
|
Keyword arguments: a value structure or component values
(see above). May also include name , specifying a Python string name
for ops generated by this method.
|
Returns | |
---|---|
prob_parts
|
a self.dtype -like structure of Tensor s representing the
prob for each component distribution evaluated at each corresponding
value .
|
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 .
|
resolve_graph
resolve_graph(
distribution_names=None, leaf_name='x'
)
Creates a tuple
of tuple
s of dependencies.
This function is experimental. That said, we encourage its use
and ask that you report problems to tfprobability@tensorflow.org
.
Args | |
---|---|
distribution_names
|
list of str or None names corresponding to each
of model elements. (None s are expanding into the
appropriate str .)
|
leaf_name
|
str used when no maker depends on a particular
model element.
|
Returns | |
---|---|
graph
|
tuple of (str tuple) pairs representing the name of each
distribution (maker) and the names of its dependencies.
|
Example
d = tfd.JointDistributionSequential([
tfd.Independent(tfd.Exponential(rate=[100, 120]), 1),
lambda e: tfd.Gamma(concentration=e[..., 0], rate=e[..., 1]),
tfd.Normal(loc=0, scale=2.),
lambda n, g: tfd.Normal(loc=n, scale=g),
])
d.resolve_graph()
# ==> (
# ('e', ()),
# ('g', ('e',)),
# ('n', ()),
# ('x', ('n', 'g')),
# )
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.
Additional documentation from JointDistribution
:
kwargs
:
value
:Tensor
s structured liketype(model)
used to parameterize other dependent ("downstream") distribution-making functions. UsingNone
for any element will trigger a sample from the corresponding distribution. Default value:None
(i.e., draw a sample from each distribution).
Args | |
---|---|
sample_shape
|
0D or 1D int32 Tensor . Shape of the generated samples.
|
seed
|
PRNG seed; see tfp.random.sanitize_seed for details.
|
name
|
name to give to the op. |
**kwargs
|
Named arguments forwarded to subclass implementation. |
Returns | |
---|---|
samples
|
a Tensor with prepended dimensions sample_shape .
|
sample_distributions
sample_distributions(
sample_shape=(),
seed=None,
value=None,
name='sample_distributions',
**kwargs
)
Generate samples and the (random) distributions.
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
|
PRNG seed; see tfp.random.sanitize_seed for details.
|
value
|
list of Tensor s in distribution_fn order to use to
parameterize other ("downstream") distribution makers.
Default value: None (i.e., draw a sample from each distribution).
|
name
|
name prepended to ops created by this function.
Default value: "sample_distributions" .
|
**kwargs
|
This is an alternative to passing a value , and achieves the
same effect. Named arguments will be used to parameterize other
dependent ("downstream") distribution-making functions. If a value
argument is also provided, raises a ValueError.
|
Returns | |
---|---|
distributions
|
a tuple of Distribution instances for each of
distribution_fn .
|
samples
|
a tuple of Tensor s with prepended dimensions sample_shape
for each of distribution_fn .
|
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).
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 .
|
unnormalized_log_prob
unnormalized_log_prob(
*args, **kwargs
)
Unnormalized log probability density/mass function.
The measure methods of JointDistribution
(log_prob
, prob
, etc.)
can be called either by passing a single structure of tensors or by using
named args for each part of the joint distribution state. For example,
jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1.),
lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.unnormalized_log_prob(sample) ==
jd.unnormalized_log_prob(value=sample) ==
jd.unnormalized_log_prob(z, x) ==
jd.unnormalized_log_prob(z=z, x=x) ==
jd.unnormalized_log_prob(z, x=x))
# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.unnormalized_log_prob(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.unnormalized_log_prob(**sample)
JointDistribution
component distributions names are resolved via
jd._flat_resolve_names()
, which is implemented by each JointDistribution
subclass (see subclass documentation for details). Generally, for components
where a name was provided---
either explicitly as the name
argument to a distribution or as a key in a
dict-valued JointDistribution, or implicitly, e.g., by the argument name of
a JointDistributionSequential
distribution-making function---the provided
name will be used. Otherwise the component will receive a dummy name; these
may change without warning and should not be relied upon.
trivial_jd = tfd.JointDistributionSequential([tfd.Exponential(1.)])
trivial_jd.dtype # => [tf.float32]
trivial_jd.unnormalized_log_prob([4.])
# ==> Tensor with shape `[]`.
lp = trivial_jd.unnormalized_log_prob(4.)
# ==> Tensor with shape `[]`.
Notice that in the first call, [4.]
is interpreted as a list of one
scalar while in the second call the input is a scalar. Hence both inputs
result in identical scalar outputs. If we wanted to pass an explicit
vector to the Exponential
component---creating a vector-shaped batch
of unnormalized_log_prob
s---we could instead write
trivial_jd.unnormalized_log_prob(np.array([4]))
.
Args | |
---|---|
*args
|
Positional arguments: a value structure or component values
(see above).
|
**kwargs
|
Keyword arguments: a value structure or component values
(see above). May also include name , specifying a Python string name
for ops generated by this method.
|
Returns | |
---|---|
log_prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
unnormalized_log_prob_parts
unnormalized_log_prob_parts(
*args, **kwargs
)
Unnormalized log probability density/mass function.
The measure methods of JointDistribution
(log_prob
, prob
, etc.)
can be called either by passing a single structure of tensors or by using
named args for each part of the joint distribution state. For example,
jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1.),
lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.unnormalized_log_prob_parts(sample) ==
jd.unnormalized_log_prob_parts(value=sample) ==
jd.unnormalized_log_prob_parts(z, x) ==
jd.unnormalized_log_prob_parts(z=z, x=x) ==
jd.unnormalized_log_prob_parts(z, x=x))
# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.unnormalized_log_prob_parts(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.unnormalized_log_prob_parts(**sample)
JointDistribution
component distributions names are resolved via
jd._flat_resolve_names()
, which is implemented by each JointDistribution
subclass (see subclass documentation for details). Generally, for components
where a name was provided---
either explicitly as the name
argument to a distribution or as a key in a
dict-valued JointDistribution, or implicitly, e.g., by the argument name of
a JointDistributionSequential
distribution-making function---the provided
name will be used. Otherwise the component will receive a dummy name; these
may change without warning and should not be relied upon.
trivial_jd = tfd.JointDistributionSequential([tfd.Exponential(1.)])
trivial_jd.dtype # => [tf.float32]
trivial_jd.unnormalized_log_prob_parts([4.])
# ==> Tensor with shape `[]`.
unnorm_lp_parts = trivial_jd.unnormalized_log_prob_parts(4.)
# ==> Tensor with shape `[]`.
Notice that in the first call, [4.]
is interpreted as a list of one
scalar while in the second call the input is a scalar. Hence both inputs
result in identical scalar outputs. If we wanted to pass an explicit
vector to the Exponential
component---creating a vector-shaped batch
of unnormalized_log_prob_parts
s---we could instead write
trivial_jd.unnormalized_log_prob_parts(np.array([4]))
.
Args | |
---|---|
*args
|
Positional arguments: a value structure or component values
(see above).
|
**kwargs
|
Keyword arguments: a value structure or component values
(see above). May also include name , specifying a Python string name
for ops generated by this method.
|
Returns | |
---|---|
unnormalized_log_prob_parts
|
a self.dtype -like structure of Tensor s
representing the unnormalized_log_prob for each component distribution
evaluated at each corresponding value .
|
unnormalized_prob_parts
unnormalized_prob_parts(
*args, **kwargs
)
Unnormalized probability density/mass function.
The measure methods of JointDistribution
(log_prob
, prob
, etc.)
can be called either by passing a single structure of tensors or by using
named args for each part of the joint distribution state. For example,
jd = tfd.JointDistributionSequential([
tfd.Normal(0., 1.),
lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.unnormalized_prob_parts(sample) ==
jd.unnormalized_prob_parts(value=sample) ==
jd.unnormalized_prob_parts(z, x) ==
jd.unnormalized_prob_parts(z=z, x=x) ==
jd.unnormalized_prob_parts(z, x=x))
# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.unnormalized_prob_parts(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.unnormalized_prob_parts(**sample)
JointDistribution
component distributions names are resolved via
jd._flat_resolve_names()
, which is implemented by each JointDistribution
subclass (see subclass documentation for details). Generally, for components
where a name was provided---
either explicitly as the name
argument to a distribution or as a key in a
dict-valued JointDistribution, or implicitly, e.g., by the argument name of
a JointDistributionSequential
distribution-making function---the provided
name will be used. Otherwise the component will receive a dummy name; these
may change without warning and should not be relied upon.
trivial_jd = tfd.JointDistributionSequential([tfd.Exponential(1.)])
trivial_jd.dtype # => [tf.float32]
trivial_jd.unnormalized_prob_parts([4.])
# ==> Tensor with shape `[]`.
unnorm_prob_parts = trivial_jd.unnormalized_prob_parts(4.)
# ==> Tensor with shape `[]`.
Notice that in the first call, [4.]
is interpreted as a list of one
scalar while in the second call the input is a scalar. Hence both inputs
result in identical scalar outputs. If we wanted to pass an explicit
vector to the Exponential
component---creating a vector-shaped batch
of unnormalized_prob_parts
s---we could instead write
trivial_jd.unnormalized_prob_parts(np.array([4]))
.
Args | |
---|---|
*args
|
Positional arguments: a value structure or component values
(see above).
|
**kwargs
|
Keyword arguments: a value structure or component values
(see above). May also include name , specifying a Python string name
for ops generated by this method.
|
Returns | |
---|---|
unnormalized_prob_parts
|
a self.dtype -like structure of Tensor s
representing the unnormalized_prob for each component distribution
evaluated at each corresponding value .
|
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__()