# tfp.experimental.substrates.numpy.distributions.JointDistributionSequential

Joint distribution parameterized by distribution-making functions.

Inherits From: JointDistribution, Distribution

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 tf.keras.Sequential, the JointDistributionSequential can be specified via a list of functions (each responsible for making a tfp.distributions.Distribution-like instance). Unlike tf.keras.Sequential, each function can depend on the output of all previous elements rather than only the immediately previous.

#### Mathematical Details

The JointDistributionSequential 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 JointDistributionSequential is parameterized by a list comprised of either:

1. tfp.distributions.Distribution-like instances or,
2. callables which return a tfp.distributions.Distribution-like instance.

Each list element implements the i-th full conditional distribution, p(x[i] | x[:i]). The "conditioned on" elements are represented by the callable's required arguments. Directly providing a Distribution-like instance is a convenience and is semantically identical a zero argument callable.

Denote the i-th callables non-default arguments as args[i]. Since the callable is the conditional manifest, 0 <= len(args[i]) <= i - 1. When len(args[i]) < i - 1, the callable only depends on a subset of the previous distributions, specifically those at indexes: range(i - 1, i - 1 - num_args[i], -1). (See "Examples" and "Discussion" for why the order is reversed.)

Name resolution: The names ofJointDistributionSequentialcomponents are defined by explicitnamearguments passed to distributions (tfd.Normal(0., 1., name='x')) and/or by the argument names in distribution-making functions (lambda x: tfd.Normal(x., 1.)). Both approaches may be used in the same distribution, as long as they are consistent; referring to a single component by multiple names will raise aValueError`. Unnamed components will be assigned a dummy name.

#### Examples

tfd = tfp.distributions

# 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)

# In TFP, we can write this as:
joint = tfd.JointDistributionSequential([
tfd.Independent(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".)

x = joint.sample()
# ==> A length-5 list 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

JointDistributionSequential builds each distribution in list order; list items must be either a:

1. tfd.Distribution-like instance (e.g., e and n), or a
2. Python callable (e.g., g, m, x).

Regarding #1, an object is deemed "tfd.Distribution-like" if it has a sample, 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 "Python callable." For example, instead of writing: lambda loc, scale: tfd.Normal(loc=loc, scale=scale) one could have simply written tfd.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-argument callables.

A distribution instance depends on other distribution instances through the distribution making function's required arguments. If the distribution maker has k required arguments then the JointDistributionSequential calls the maker with samples produced by the previous k distributions.

model Python list of either tfd.Distribution instances and/or lambda functions which take the k previous distributions and returns a new tfd.Distribution instance.
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_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.

dtype The DType of Tensors handled by this Distribution.
event_shape Shape of a single sample from a single batch as a TensorShape.

May be partially defined or unknown.

model

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 tfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.

trainable_variables

validate_args Python bool indicating possibly expensive checks are enabled.
variables

## Methods

### batch_shape_tensor

View source

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
sample_shape The sample shape under which to evaluate the joint distribution. Sample shape at root (toplevel) nodes may affect the batch or event shapes of child nodes.
name name to give to the op

Returns
batch_shape Tensor representing batch shape of each distribution in model.

### 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.

Shannon entropy in nats.

### event_shape_tensor

View source

Shape of a single sample from a single batch as a 1-D int32 Tensor.

Args
sample_shape The sample shape under which to evaluate the joint distribution. Sample shape at root (toplevel) nodes may affect the batch or event shapes of child nodes.
name name to give to the op

Returns
event_shape tuple of Tensors representing the event_shape for each distribution in model.

### 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 for each distribution in model.

### 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
`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.

View source

Mean.

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
`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

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 Python integer seed for generating random numbers.
value list of Tensors 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 Tensors with prepended dimensions sample_shape for each of distribution_fn.

### 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().

### __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.stateless_normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.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.

View source