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tfp.experimental.substrates.numpy.distributions.JointDistributionSequentialAutoBatched

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Joint distribution parameterized by distribution-making functions.

Inherits From: JointDistributionSequential

tfp.experimental.substrates.numpy.distributions.JointDistributionSequentialAutoBatched(
    *args, **kwargs
)

This class provides alternate vectorization semantics for tfd.JointDistributionSequential, which in many cases eliminate the need to explicitly account for batch shapes in the model specification. 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 otherwise specified.

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. This behavior is implicit in JointDistributionSequentialAutoBatched.

Attributes:

  • allow_nan_stats: Python bool describing behavior when a stat is undefined.

    Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.

  • batch_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.

  • 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

__getitem__

View source

__getitem__(
    slices
)

Slices the batch axes of this distribution, returning a new instance.

b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape  # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., -2:, 1::2]
b2.batch_shape  # => [3, 1, 5, 2, 4]

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

__iter__()

batch_shape_tensor

View source

batch_shape_tensor(
    sample_shape=(), name='batch_shape_tensor'
)

cdf

View source

cdf(
    value, name='cdf', **kwargs
)

Cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

cdf(x) := P[X <= x]

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.
  • **kwargs: Named arguments forwarded to subclass implementation.

Returns:

  • cdf: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

copy

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copy(
    **override_parameters_kwargs
)

Creates a deep copy of the distribution.

Args:

  • **override_parameters_kwargs: String/value dictionary of initialization arguments to override with new values.

Returns:

  • distribution: A new instance of type(self) initialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

covariance

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covariance(
    name='covariance', **kwargs
)

Covariance.

Covariance is (possibly) defined only for non-scalar-event distributions.

For example, for a length-k, vector-valued distribution, it is calculated as,

Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]

where Cov is a (batch of) k x k matrix, 0 <= (i, j) < k, and E denotes expectation.

Alternatively, for non-vector, multivariate distributions (e.g., matrix-valued, Wishart), Covariance shall return a (batch of) matrices under some vectorization of the events, i.e.,

Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]

where Cov is a (batch of) k' x k' matrices, 0 <= (i, j) < k' = reduce_prod(event_shape), and Vec is some function mapping indices of this distribution's event dimensions to indices of a length-k' vector.

Args:

  • name: Python str prepended to names of ops created by this function.
  • **kwargs: Named arguments forwarded to subclass implementation.

Returns:

  • covariance: Floating-point Tensor with shape [B1, ..., Bn, k', k'] where the first n dimensions are batch coordinates and k' = reduce_prod(self.event_shape).

cross_entropy

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cross_entropy(
    other, name='cross_entropy'
)

Computes the (Shannon) cross entropy.

Denote this distribution (self) by P and the other distribution by Q. Assuming P, Q are absolutely continuous with respect to one another and permit densities p(x) dr(x) and q(x) dr(x), (Shannon) cross entropy is defined as:

H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)

where F denotes the support of the random variable X ~ P.

other types with built-in registrations: JointDistributionNamed, JointDistributionNamedAutoBatched, JointDistributionSequential, JointDistributionSequentialAutoBatched

Args:

Returns:

  • cross_entropy: self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of (Shannon) cross entropy.

entropy

View source

entropy(
    name='entropy', **kwargs
)

Shannon entropy in nats.

Additional documentation from JointDistributionSequential:

Shannon entropy in nats.

event_shape_tensor

View source

event_shape_tensor(
    sample_shape=(), name='event_shape_tensor'
)

is_scalar_batch

View source

is_scalar_batch(
    name='is_scalar_batch'
)

Indicates that batch_shape == [].

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • is_scalar_batch: bool scalar Tensor.

is_scalar_event

View source

is_scalar_event(
    name='is_scalar_event'
)

Indicates that event_shape == [].

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • is_scalar_event: bool scalar Tensor for each distribution in model.

kl_divergence

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kl_divergence(
    other, name='kl_divergence'
)

Computes the Kullback--Leibler divergence.

Denote this distribution (self) by p and the other distribution by q. Assuming p, q are absolutely continuous with respect to reference measure r, the KL divergence is defined as:

KL[p, q] = E_p[log(p(X)/q(X))]
         = -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
         = H[p, q] - H[p]

where F denotes the support of the random variable X ~ p, H[., .] denotes (Shannon) cross entropy, and H[.] denotes (Shannon) entropy.

other types with built-in registrations: JointDistributionNamed, JointDistributionNamedAutoBatched, JointDistributionSequential, JointDistributionSequentialAutoBatched

Args:

Returns:

  • kl_divergence: self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of the Kullback-Leibler divergence.

log_cdf

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log_cdf(
    value, name='log_cdf', **kwargs
)

Log cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

log_cdf(x) := Log[ P[X <= x] ]

Often, a numerical approximation can be used for log_cdf(x) that yields a more accurate answer than simply taking the logarithm of the cdf when x << -1.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.
  • **kwargs: Named arguments forwarded to subclass implementation.

Returns:

  • logcdf: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_prob

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log_prob(
    *args, **kwargs
)

Log probability density/mass function.

The measure methods of `JointDistribution` (`log_prob`, `prob`, etc.)
can be called either by passing a single structure of tensors or by using
named args for each part of the joint distribution state. For example,
jd = tfd.JointDistributionSequential([
    tfd.Normal(0., 1.),
    lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.log_prob(sample) ==
        jd.log_prob(value=sample) ==
        jd.log_prob(z, x) ==
        jd.log_prob(z=z, x=x) ==
        jd.log_prob(z, x=x))

# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.log_prob(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.log_prob(**sample)
`JointDistribution` component distributions names are resolved via
`jd._flat_resolve_names()`, which is implemented by each `JointDistribution`
subclass (see subclass documentation for details). Generally, for components
where a name was provided---
either explicitly as the `name` argument to a distribution or as a key in a
dict-valued JointDistribution, or implicitly, e.g., by the argument name of
a `JointDistributionSequential` distribution-making function---the provided
name will be used. Otherwise the component will receive a dummy name; these
may change without warning and should not be relied upon.

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,
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_prob_parts(
    value, name='log_prob_parts'
)

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(
    value, name='log_survival_function', **kwargs
)

Log survival function.

Given random variable X, the survival function is defined:

log_survival_function(x) = Log[ P[X > x] ]
                         = Log[ 1 - P[X <= x] ]
                         = Log[ 1 - cdf(x) ]

Typically, different numerical approximations can be used for the log survival function, which are more accurate than 1 - cdf(x) when x >> 1.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.
  • **kwargs: Named arguments forwarded to subclass implementation.

Returns:

Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

mean

View source

mean(
    name='mean', **kwargs
)

Mean.

mode

View source

mode(
    name='mode', **kwargs
)

Mode.

param_shapes

View source

@classmethod
param_shapes(
    cls, sample_shape, name='DistributionParamShapes'
)

Shapes of parameters given the desired shape of a call to sample().

This is a class method that describes what key/value arguments are required to instantiate the given Distribution so that a particular shape is returned for that instance's call to sample().

Subclasses should override class method _param_shapes.

Args:

  • sample_shape: Tensor or python list/tuple. Desired shape of a call to sample().
  • name: name to prepend ops with.

Returns:

dict of parameter name to Tensor shapes.

param_static_shapes

View source

@classmethod
param_static_shapes(
    cls, sample_shape
)

param_shapes with static (i.e. TensorShape) shapes.

This is a class method that describes what key/value arguments are required to instantiate the given Distribution so that a particular shape is returned for that instance's call to sample(). Assumes that the sample's shape is known statically.

Subclasses should override class method _param_shapes to return constant-valued tensors when constant values are fed.

Args:

  • sample_shape: TensorShape or python list/tuple. Desired shape of a call to sample().

Returns:

dict of parameter name to TensorShape.

Raises:

  • ValueError: if sample_shape is a TensorShape and is not fully defined.

prob

View source

prob(
    *args, **kwargs
)

Probability density/mass function.

The measure methods of `JointDistribution` (`log_prob`, `prob`, etc.)
can be called either by passing a single structure of tensors or by using
named args for each part of the joint distribution state. For example,
jd = tfd.JointDistributionSequential([
    tfd.Normal(0., 1.),
    lambda z: tfd.Normal(z, 1.)
], validate_args=True)
jd.dtype
# ==> [tf.float32, tf.float32]
z, x = sample = jd.sample()
# The following calling styles are all permissable and produce the exactly
# the same output.
assert (jd.prob(sample) ==
        jd.prob(value=sample) ==
        jd.prob(z, x) ==
        jd.prob(z=z, x=x) ==
        jd.prob(z, x=x))

# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
jd.prob(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
jd.prob(**sample)
`JointDistribution` component distributions names are resolved via
`jd._flat_resolve_names()`, which is implemented by each `JointDistribution`
subclass (see subclass documentation for details). Generally, for components
where a name was provided---
either explicitly as the `name` argument to a distribution or as a key in a
dict-valued JointDistribution, or implicitly, e.g., by the argument name of
a `JointDistributionSequential` distribution-making function---the provided
name will be used. Otherwise the component will receive a dummy name; these
may change without warning and should not be relied upon.

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,
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

prob_parts(
    value, name='prob_parts'
)

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(
    value, name='quantile', **kwargs
)

Quantile function. Aka 'inverse cdf' or 'percent point function'.

Given random variable X and p in [0, 1], the quantile is:

quantile(p) := x such that P[X <= x] == p

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.
  • **kwargs: Named arguments forwarded to subclass implementation.

Returns:

  • quantile: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

resolve_graph

View source

resolve_graph(
    distribution_names=None, leaf_name='x'
)

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

sample(
    sample_shape=(), seed=None, name='sample', **kwargs
)

Generate samples of the specified shape.

Note that a call to sample() without arguments will generate a single sample.

Args:

  • sample_shape: 0D or 1D int32 Tensor. Shape of the generated samples.
  • seed: Python integer or tfp.util.SeedStream instance, for seeding PRNG.
  • name: name to give to the op.
  • **kwargs: Named arguments forwarded to subclass implementation.

Returns:

  • samples: a Tensor with prepended dimensions sample_shape.

sample_distributions

View source

sample_distributions(
    sample_shape=(), seed=None, value=None, name='sample_distributions'
)

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".

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

stddev(
    name='stddev', **kwargs
)

Standard deviation.

Standard deviation is defined as,

stddev = E[(X - E[X])**2]**0.5

where X is the random variable associated with this distribution, E denotes expectation, and stddev.shape = batch_shape + event_shape.

Args:

  • name: Python str prepended to names of ops created by this function.
  • **kwargs: Named arguments forwarded to subclass implementation.

Returns:

  • stddev: Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

survival_function

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survival_function(
    value, name='survival_function', **kwargs
)

Survival function.

Given random variable X, the survival function is defined:

survival_function(x) = P[X > x]
                     = 1 - P[X <= x]
                     = 1 - cdf(x).

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.
  • **kwargs: Named arguments forwarded to subclass implementation.

Returns:

Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

variance

View source

variance(
    name='variance', **kwargs
)

Variance.

Variance is defined as,

Var = E[(X - E[X])**2]

where X is the random variable associated with this distribution, E denotes expectation, and Var.shape = batch_shape + event_shape.

Args:

  • name: Python str prepended to names of ops created by this function.
  • **kwargs: Named arguments forwarded to subclass implementation.

Returns:

  • variance: Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().