tfp.distributions.JointDistributionSequentialAutoBatched

Joint distribution parameterized by distribution-making functions.

Inherits From: JointDistributionSequential, JointDistribution, Distribution

This class provides automatic vectorization and alternative semantics for tfd.JointDistributionNamed, which in many cases allows for simplifications in the model specification.

Automatic vectorization

Auto-vectorized variants of JointDistribution allow the user to avoid explicitly annotating a model's vectorization semantics. 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 using tf.vectorized_map as required. In many cases this allows for significant simplications. For example, the following manually-vectorized tfd.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.JointDistributionAutoBatchedSequential([
    tfd.Normal(0., tf.ones([3])),
    tfd.Normal(0., 1.),
    lambda y, x: tfd.Normal(x[:2] + y, 1.)
  ])

in which we were able to avoid explicitly accounting for batch dimensions when indexing and slicing computed quantities in the third line.

Alternative batch semantics

This class also provides alternative semantics for specifying a batch of independent (non-identical) joint distributions.

Instead of simply summing the log_probs of component distributions (which may have different shapes), it first reduces the component log_probs to ensure that jd.log_prob(jd.sample()) always returns a scalar, unless batch_ndims is explicitly set to a nonzero value (in which case the result will have the corresponding tensor rank).

The essential changes are:

  • An event of JointDistributionSequentialAutoBatched is the list of tensors produced by .sample(); thus, the event_shape is the list containing the shapes of sampled tensors. These combine both the event and batch dimensions of the component distributions. By contrast, the event shape of a base JointDistributions does not include batch dimensions of component distributions.
  • The batch_shape is a global property of the entire model, rather than a per-component property as in base JointDistributions. The global batch shape must be a prefix of the batch shapes of each component; the length of this prefix is specified by an optional argument batch_ndims. If batch_ndims is not specified, the model has batch shape [].

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.JointDistributionSequentialAutoBatched([
                 tfd.Exponential(rate=[100, 120]), 1,         # e
    lambda    e: tfd.Gamma(concentration=e[0], rate=e[1]),    # g
                 tfd.Normal(loc=0, scale=2.),                 # n
    lambda n, g: tfd.Normal(loc=n, scale=g)                   # m
    lambda    m: tfd.Sample(tfd.Bernoulli(logits=m), 12)      # x
])

Notice the 1:1 correspondence between "math" and "code". In a standard JointDistributionSequential, we would have wrapped the first variable as e = tfd.Independent(tfd.Exponential(rate=[100, 120]), reinterpreted_batch_ndims=1) to specify that log_prob of the Exponential should be a scalar, summing over both dimensions. We would also have had to extend indices as tfd.Gamma(concentration=e[..., 0], rate=e[..., 1]) to account for possible batch dimensions. Both of these behaviors are implicit in JointDistributionSequentialAutoBatched.

model A generator that yields a sequence of tfd.Distribution-like instances.
batch_ndims int Tensor number of batch dimensions. The batch_shapes of all component distributions must be such that the prefixes of length batch_ndims broadcast to a consistent joint batch shape. Default value: 0.
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: True.
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.
name The name for ops managed by the distribution. Default value: None (i.e., JointDistributionSequential).

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_ndims

batch_shape

dtype The DType of Tensors handled by this Distribution.
event_shape

model

name Name prepended to all ops created by this Distribution.
name_scope Returns a tf.name_scope instance for this class.
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 tfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.

submodules Sequence of all sub-modules.

Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on).

a = tf.Module()
b = tf.Module()
c = tf.Module()
a.b = b
b.c = c
list(a.submodules) == [b, c]
True
list(b.submodules) == [c]
True
list(c.submodules) == []
True

trainable_variables Sequence of trainable variables owned by this module and its submodules.

use_vectorized_map

validate_args Python bool indicating possibly expensive checks are enabled.
variables Sequence of variables owned by this module and its submodules.

Methods

batch_shape_tensor

View source

cdf

View source

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

View source

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

View source

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

View source

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

View source

Shannon entropy in nats.

Additional documentation from JointDistributionSequential:

Shannon entropy in nats.

event_shape_tensor

View source

Shape of a single sample from a single batch.

is_scalar_batch

View source

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

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

View source

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

View source

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

View source

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,

```python
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.

Note: not all `JointDistribution` subclasses support all calling styles;
for example, `JointDistributionNamed` does not support positional arguments
(aka "unnamed arguments") unless the provided model specifies an ordering of
variables (i.e., is an `collections.OrderedDict` or `collections.namedtuple`
rather than a plain `dict`).

Note: care is taken to resolve any potential ambiguity---this is generally
possible by inspecting the structure of the provided argument and "aligning"
it to the joint distribution output structure (defined by `jd.dtype`). For
example,

```python
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

View source

Log probability density/mass function.

Args
value list of Tensors in distribution_fn order for which we compute the log_prob_parts and to parameterize other ("downstream") distributions.
name name prepended to ops created by this function. Default value: "log_prob_parts".

Returns
log_prob_parts a tuple of Tensors representing the log_prob for each distribution_fn evaluated at each corresponding value.

log_survival_function

View source

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

View source

Mean.

mode

View source

Mode.

param_shapes

View source

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

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

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,

```python
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.

Note: not all `JointDistribution` subclasses support all calling styles;
for example, `JointDistributionNamed` does not support positional arguments
(aka "unnamed arguments") unless the provided model specifies an ordering of
variables (i.e., is an `collections.OrderedDict` or `collections.namedtuple`
rather than a plain `dict`).

Note: care is taken to resolve any potential ambiguity---this is generally
possible by inspecting the structure of the provided argument and "aligning"
it to the joint distribution output structure (defined by `jd.dtype`). For
example,

```python
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

View source

Log probability density/mass function.

Args
value list of Tensors in distribution_fn order for which we compute the prob_parts and to parameterize other ("downstream") distributions.
name name prepended to ops created by this function. Default value: "prob_parts".

Returns
prob_parts a tuple of Tensors representing the prob for each distribution_fn evaluated at each corresponding value.

quantile

View source

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

View source

Creates a tuple of tuples 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. (Nones 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

View source

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.

sample_distributions

View source

stddev

View source

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

View source

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.

variance

View source

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().

with_name_scope

Decorator to automatically enter the module name scope.

class MyModule(tf.Module):
  @tf.Module.with_name_scope
  def __call__(self, x):
    if not hasattr(self, 'w'):
      self.w = tf.Variable(tf.random.normal([x.shape[1], 3]))
    return tf.matmul(x, self.w)

Using the above module would produce tf.Variables and tf.Tensors whose names included the module name:

mod = MyModule()
mod(tf.ones([1, 2]))
<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>
mod.w
<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,
numpy=..., dtype=float32)>

Args
method The method to wrap.

Returns
The original method wrapped such that it enters the module's name scope.

__getitem__

View source

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.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.cholesky(cov)
loc = tf.random.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__

View source