# tfp.substrates.jax.distributions.JointDistributionCoroutine

Joint distribution parameterized by a distribution-making generator.

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. The `JointDistributionCoroutine` is specified by a generator that generates the elements of this collection.

#### Mathematical Details

The `JointDistributionCoroutine` 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 `JointDistributionCoroutine` is parameterized by a generator that yields `tfp.distributions.Distribution`-like instances.

Each element yielded implements the `i`-th full conditional distribution, `p(x[i] | x[:i])`. Within the generator, the return value from the yield is a sample from the distribution that may be used to construct subsequent yielded `Distribution`-like instances. This allows later instances to be conditional on earlier ones.

Name resolution: The names of `JointDistributionCoroutine` components may be specified by passing `name` arguments to distribution constructors ( `tfd.Normal(0., 1., name='x')). Components without an explicit name will be assigned a dummy name.

#### Vectorized sampling and model evaluation

When a joint distribution's `sample` method is called with a `sample_shape` (or the `log_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 passing `use_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-vectorized `tfd.JointDistributionCoroutine` model:

``````def model_fn():
x = yield tfd.JointDistributionCoroutine.Root(
tfd.Normal(0., tf.ones([3])))
y = yield tfd.JointDistributionCoroutine.Root(
tfd.Normal(0., 1.))
z = yield tfd.Normal(x[..., :2] + y[..., tf.newaxis], 1.)

can be written in auto-vectorized form as

```python
def model_fn():
x = yield tfd.Normal(0., tf.ones([3]))
y = yield tfd.Normal(0., 1.)
z = yield tfd.Normal(x[:2] + y, 1.)
``````

in which we were able to drop the specification of `Root` nodes and to avoid explicitly accounting for batch dimensions when indexing and slicing computed quantities in the third line.

Root annotations: When the `sample` method for a manually-vectorized `JointDistributionCoroutine` is called with a `sample_shape`, the `sample` method for each of the yielded distributions is called. The distributions that have been wrapped in the `JointDistributionCoroutine.Root` class will be called with `sample_shape` as the `sample_shape` argument, and the unwrapped distributions will be called with `()` as the `sample_shape` argument. It is the user's responsibility to ensure that each of the distributions generates samples with the specified sample size; generally this means applying `Root` wrappers around any distributions whose parameters are not already a function of other random variables. The `Root` annotation can be omitted if you never intend to use a `sample_shape` other than `()`.

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 nontrivial `sample_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 like `tfd.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. If `batch_ndims==0`, then the joint distribution has batch shape `[]`, and all component dimensions are treated as event shape. For example, the model

``````def model_fn():
x = yield tfd.Normal(0., tf.ones([3]))
y = yield tfd.Normal(x[..., tf.newaxis], tf.ones([3, 2]))
jd = tfd.JointDistributionCoroutine(model_fn, batch_ndims=0)
``````

creates a joint distribution with batch shape `[]` and event shape `([3], [3, 2])`. The log-density of a sample always has shape `batch_shape`, so this guarantees that `jd.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 setting `batch_ndims=1`, in which case `jd.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's `log_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 with `tfd.Independent`, as in this model:

``````def model_fn():
x = yield tfd.Normal(0., tf.ones([3]))
y = yield tfd.Independent(tfd.Normal(x[..., tf.newaxis], tf.ones([3, 2])),
reinterpreted_batch_ndims=1)
jd = tfd.JointDistributionCoroutine(model_fn, batch_ndims=None)
``````

Here the components both have batch shape `[3]`, so `jd.log_prob(jd.sample())` returns a value of shape `[3]`, just as in the `batch_ndims=1` case above. In fact, auto-batching semantics are equivalent to implicitly wrapping each component `dist` as ```tfd.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 of `jd.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.

#### Examples

``````tfd = tfp.distributions
def model():
global_log_rate = yield tfd.Normal(loc=0., scale=1.)
local_log_rates = yield tfd.Normal(loc=0., scale=tf.ones([20]))
observed_counts = yield tfd.Poisson(
rate=tf.exp(global_log_rate + local_log_rates))
joint = tfd.JointDistributionCoroutine(model,
use_vectorized_map=True,
batch_ndims=0)

print(joint.event_shape)
# ==> [[], [20], [20]]
print(joint.batch_shape)
# ==> []
xs = joint.sample()
print([x.shape for x in xs])
# ==> [[], [20], [20]]
lp = joint.log_prob(xs)
print(lp.shape)
# ==> []
``````

Note that the component distributions of this model would, by themselves, return batches of log-densities (because they are constructed with batch shape); the joint model implicitly sums over these to compute the single joint log-density.

``````ds, xs = joint.sample_distributions()
print([d.event_shape for d in ds])
# ==> [[], [], []] != model.event_shape
print([d.batch_shape for d in ds])
# ==> [[], [20], [20]] != model.batch_shape
print([d.log_prob(x).shape for (d, x) in zip(ds, xs)])
# ==> [[], [20], [20]]
``````

For improved readability of sampled values, the yielded distributions can also be named:

``````tfd = tfp.distributions
def model():
global_log_rate = yield tfd.Normal(
loc=0., scale=1., name='global_log_rate')
local_log_rates = yield tfd.Normal(
loc=0., scale=tf.ones([20]), name='local_log_rates')
observed_counts = yield tfd.Poisson(
rate=tf.exp(global_log_rate + local_log_rates), name='observed_counts')
joint = tfd.JointDistributionCoroutine(model,
use_vectorized_map=True,
batch_ndims=0)

print(joint.event_shape)
# ==> StructTuple(global_log_rate=[], local_log_rates=[20],
#      observed_counts=[20])
print(joint.batch_shape)
# ==> []
xs = joint.sample()
print(['{}: {}'.format(k, x.shape) for k, x in xs._asdict().items()])
# ==> global_log_scale: []
#     local_log_rates: [20]
#     observed_counts: [20]
lp = joint.log_prob(xs)
print(lp.shape)
# ==> []

# Passing via `kwargs` also works.
lp = joint.log_prob(**xs._asdict())
# Or:
lp = joint.log_prob(
global_log_scale=...,
local_log_rates=...,
observed_counts=...,
)
``````

If any of the yielded distributions are not explicitly named, they will automatically be given a name of the form `var#` where `#` is the index of the associated distribution. E.g. the first yielded distribution will have a default name of `var0`.

#### References

[1] Dan Piponi, Dave Moore, and Joshua V. Dillon. Joint distributions for TensorFlow Probability. arXiv preprint arXiv:2001.11819_, 2020. https://arxiv.org/abs/2001.11819

`model` A generator that yields a sequence of `tfd.Distribution`-like instances.
`sample_dtype` Samples from this distribution will be structured like `tf.nest.pack_sequence_as(sample_dtype, list_)`. `sample_dtype` is only used for `tf.nest.pack_sequence_as` structuring of outputs, never casting (which is the responsibility of the component distributions). Default value: `None` (i.e. `namedtuple`).
`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., `JointDistributionCoroutine`).

`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` 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 `Tensor`s handled by this `Distribution`.
`event_shape` Shape of a single sample from a single batch as a `TensorShape`.

May be partially defined or unknown.

`experimental_shard_axis_names` Indicates whether part distributions have active shard axis names.
`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`

`use_vectorized_map`

`validate_args` Python `bool` indicating possibly expensive checks are enabled.
`variables`

## Child Classes

`class Root`

## 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
`name` name to give to the op

Returns
`batch_shape` `Tensor`.

### `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`.

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.

### `event_shape_tensor`

View source

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`

View source

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`

View source

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`

View source

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`

View source

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`

View source

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`

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.

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,

``````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`

View source

Log probability density/mass function.

Args
`value` `list` of `Tensor`s 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 `Tensor`s 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.

### `parameter_properties`

View source

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.ParameterProperties`dict mapping constructor argument names to`ParameterProperties` instances.

Raises
`NotImplementedError` if the distribution class does not implement `_parameter_properties`.

### `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,

``````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`

View source

Probability density/mass function.

Args
`value` `list` of `Tensor`s 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 `Tensor`s 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`.

### `sample`

View source

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 like `type(model)` used to parameterize other dependent ("downstream") distribution-making functions. Using `None` 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`

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` 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`

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`.

### `unnormalized_log_prob`

View source

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`

View source

Unnormalized log probability density/mass function.

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

Returns
`unnormalized_log_prob_parts` a `tuple` of `Tensor`s representing the `unnormalized_log_prob` for each `distribution_fn` evaluated at each corresponding `value`.

### `unnormalized_prob_parts`

View source

Unnormalized probability density/mass function.

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

Returns
`unnormalized_prob_parts` a `tuple` of `Tensor`s representing the `unnormalized_prob` for each `distribution_fn` evaluated at each corresponding `value`.

### `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.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__`

View source

[{ "type": "thumb-down", "id": "missingTheInformationINeed", "label":"Missing the information I need" },{ "type": "thumb-down", "id": "tooComplicatedTooManySteps", "label":"Too complicated / too many steps" },{ "type": "thumb-down", "id": "outOfDate", "label":"Out of date" },{ "type": "thumb-down", "id": "samplesCodeIssue", "label":"Samples / code issue" },{ "type": "thumb-down", "id": "otherDown", "label":"Other" }]
[{ "type": "thumb-up", "id": "easyToUnderstand", "label":"Easy to understand" },{ "type": "thumb-up", "id": "solvedMyProblem", "label":"Solved my problem" },{ "type": "thumb-up", "id": "otherUp", "label":"Other" }]