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# tfp.experimental.distributions.JointDistributionPinned

A wrapper class for `JointDistribution` which pins, e.g., the evidence.

This object is experimental; the API may change without warning.

Think of this object as `functools.partial` for joint distributions. Sampling trims off pinned values (after specifying them as `jd.sample(value=pins)` to the underlying distribution). Log-density evaluates the joint probability of the given event and the pinned values.

This object represents an unnormalized probability density, and as such is not a `tfp.distributions.Distribution`, and lacks `sample` and `log_prob` methods. In their place, it provides:

• `unnormalized_log_prob`, `unnormalized_log_prob_parts`
• `sample_unpinned`, `sample_weighted`

Mathematically speaking, the object represents a joint probability density, `p(x, y)` where the `x` are pinned and the `y` are unpinned. Accordingly, it is also proportional to `p(y | x)`, up to a (generally) intractable normalizing constant `p(x)`, i.e. `p(x, y) = p(y | x) p(x)`.

A common use-case with probabilistic inference is writing out a generative model to explain some observed data:

``````jd = tfd.JointDistributionNamed(dict(
loc = yield tfd.Normal(0., 1.),
scale = yield tfd.Gamma(1., 1.),
obs = lambda loc, scale: tfd.Normal(loc, scale),
))
``````

Later, when we want to infer 'typical' values of `loc` and `scale` conditioned on some given `data`, we will often write:

``````def target_log_prob_fn(loc, scale):
return jd.log_prob(loc=loc, scale=scale, obs=data)
``````

This class enables one to write instead:

``````partial = tfde.JointDistributionPinned(jd, obs=data)
target_log_prob_fn = partial.unnormalized_log_prob
``````

Or, even more concisely `partial = jd.experimental_pin(obs=data)`.

This is nice, but it wasn't too hard to write out the `target_log_prob_fn` function explicitly.

Now, let's consider that for many inference and optimization methods, we may want to use a smooth change of variables to perform inference in the unconstrained space of real numbers. In some cases this transformation can be parameter-dependent. For example, if we want to unconstrain the support of `tfp.distributions.Uniform(-3., 2.)` to the real line, we might use `tfp.bijectors.Sigmoid(low=-3., high=2.)`. In support of such use cases, most distributions (including the `JointDistribution*` classes) provide a `experimental_default_event_space_bijector()` method.

When these transformations may be dependent on ancestral parts of a joint distribution, and some of those parameters may be pinned, it is helpful to have a utility class to bridge the gap and provide the multi-part bijective transform. This is the "raison d'etre" of this class.

The model below is somewhat contrived, but demonstrates the use-case.

``````tfd = tfp.distributions
tfde = tfp.experimental.distributions

n = 75
joint = tfd.JointDistributionNamed(dict(
upper = tfd.Uniform(.4, 1.5),
concentration = tfd.Gamma(1., .5),
corr = lambda concentration: tfd.CholeskyLKJ(
dim, concentration=concentration),
stddev = lambda upper: tfd.Sample(tfd.Uniform(.2, upper), dim),
obs = lambda corr, stddev: tfd.Sample(
tfd.MultivariateNormalTriL(
loc=tf.zeros([dim]), scale_tril=corr * stddev[..., tf.newaxis]),
n)
))
fixed_upper = 1.3
data = joint.sample(upper=fixed_upper)['obs']

pinned = tfde.JointDistributionPinned(joint, upper=fixed_upper, obs=data)
bij = pinned.experimental_default_event_space_bijector()
pulled_back_shape = bij.inverse_event_shape(pinned.event_shape)

# Fit an ensemble using SGD.
batch = 16
uniform_init = tf.nest.map_structure(
lambda s: tf.random.uniform(tf.concat([[batch], s], axis=0), -2., 2.),
pulled_back_shape)
vars = tf.nest.map_structure(tf.Variable, uniform_init)

@tf.function(autograph=False)
def one_step():
lp = pinned.unnormalized_log_prob(bij.forward(vars))

for _ in range(100):
one_step()

# Alternatively, sample using MCMC (currently aspirational):
initial_state = bij.forward(uniform_init)

kernel = tfp.mcmc.HamiltonianMonteCarlo(
target_log_prob_fn=pinned.unnormalized_log_prob,
step_size=.5, num_leapfrog_steps=4)
# **This line is currently aspirational**, to demonstrate the use-case.
kernel = tfp.mcmc.TransformedTransitionKernel(kernel, bij)
tfp.mcmc.sample_chain(10, kernel=kernel, current_state=initial_state)
``````

`distribution` A `tfp.distributions.JointDistribution`.
`*pins` A single object like the `value` argument that may be passed into `JointDistribution.sample` (some parts may be `None`), or a sequence of objects similar to such sequence as might be passed to `JointDistribution.log_prob`, but with the difference that some parts may be `None` (`log_prob` would require all parts be specified). More precisely, the user may pass (A) a single argument specifiying pins of one or more of the parts of the underlying `distribution` either by name (i.e. a `dict`, `namedtuple`) or by sequence ordering (`tuple`, `list`), or (B) a sequence of arguments which align with the model of the underlying distribution (which must be ordered). It is an error to use an unordered sequence of pins with an unordered model, e.g. a `tfp.distributions.JointDistributionNamed` constructed with a `dict` model (`collections.OrderedDict` is allowed).
`**named_pins` Named elements to pin. The names given must align with the part names defined by `distribution._flat_resolve_names()`, i.e. either the explicitly named parts of `tfp.distributions.JointDistributionNamed` or the `name` parameters passed to distributions constructed by the model given to `JointDistribution*`.

`distribution` The underlying distribution being partially pinned.
`dtype` DType of unpinned parts.
`event_shape` Statically resolvable event shapes of unpinned parts.
`pins` Dictionary of pins resolved to names.
`validate_args`

## Methods

### `event_shape_tensor`

View source

Dynamic/graph Tensor event shapes of unpinned parts.

### `experimental_default_event_space_bijector`

View source

A bijector to pull back unpinned values to unconstrained reals.

### `experimental_pin`

View source

Logical equivalent of `JointDistribution.experimental_pin`.

For example

``````@tfd.JointDistributionCoroutine
def model():
x = yield tfd.Normal(0, 1, name='x'),
y = yield tfd.Normal(0, 1, name='y'),
yield tfd.Normal(0, 1, name='z')
model.experimental_pin(z=1.).experimental_pin(y=.5).event_shape
# => StructTuple(x=[])
``````

Args
`*args` Positional arguments: a value structure or component values.
`**kwargs` Keyword arguments: a value structure or component values. 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 in addition to those pins already specified on `self`.

### `log_weight`

View source

Computes the log relative weight of the given sample.

This function computes the log-probability of the pinned parts at the given location, ignoring the probability of the unpinned parts.

``````The methods of `JointDistributionPinned` (`unnormalized_log_prob`,
`sample_weighted`, etc.) can be called by passing a single structure
of tensors, a sequence of tensor arguments, or using named args for each
part. For example:

```python
tfde = tfp.experimental.distributions

# 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 `__init__` styles are all permissible and produce
# `JointDistributionPinned` objects behaving identically.
PartialXY = collections.namedtuple('PartialXY', 'x,y')
PartialX = collections.namedtuple('PartialX', 'x')
OrderedDict = collections.OrderedDict
assert (tfde.JointDistributionPinned(jd, x=2.).pins ==
tfde.JointDistributionPinned(jd, x=2., z=None).pins ==
tfde.JointDistributionPinned(jd, dict(x=2.)).pins ==
tfde.JointDistributionPinned(jd, dict(x=2., y=None)).pins ==
tfde.JointDistributionPinned(jd, OrderedDict(x=2.)).pins ==
tfde.JointDistributionPinned(jd, OrderedDict(x=2., y=None)).pins ==
tfde.JointDistributionPinned(jd, PartialXY(x=2., y=None)).pins ==
tfde.JointDistributionPinned(jd, PartialX(x=2.)).pins ==
tfde.JointDistributionPinned(jd, None, None, 2.).pins ==
tfde.JointDistributionPinned(jd, [None, None, 2.]).pins)
# (Notice that the `pins` attribute is always resolved to a `dict`.)

pinned = tfde.JointDistributionPinned(jd, x=2.)
pinned.dtype
# ==> [tf.float32, tf.float32]
z, y = sample = pinned.sample_unpinned()

# The following calling styles are all permissable and produce the exactly
# the same output.
PartialZY = collections.namedtuple('PartialZY', 'z,y')
assert (pinned.log_weight(sample) ==
pinned.log_weight(z, y) ==
pinned.log_weight(z=z, y=y) ==
pinned.log_weight(PartialZY(z=z, y=y)))

# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
pinned.log_weight(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
pinned.log_weight(**sample)
```

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.

In general, return types of part-wise methods/properties are determined by
those of the underlying `JointDistribution`'s model type:

- `StructTuple` for `JointDistributionCoroutine`, and for
`JointDistributionNamed` with `namedtuple` model type.
- `collections.OrderedDict` for `JointDistributionNamed` with `OrderedDict`
model type.
- `dict` for `JointDistributionNamed` with `dict` model type.
- `tuple` or `list` for `JointDistributionSequential`.

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`). In the same way, JointDistributionPinned does
not accept unnamed pins for unordered `JointDistributionNamed` models.

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
pinned = tfde.JointDistributionPinned(
tfd.JointDistributionSequential(
[tfd.Exponential(1.), lambda s: tfd.Normal(0., s)]),
None, 1.2)
pinned.dtype  # => [tf.float32]
pinned.log_weight([4.])
# ==> Tensor with shape `[]`.
log_wt = pinned.log_weight(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
`pinned.log_weight(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_weights` log-weight of the given point, i.e. the log pinned evidence.

### `sample_and_log_weight`

View source

Draws unnormalized samples and their log-weights with ancestral sampling.

Since this object represents an unnormalized density, we are unable to directly sample the distribution. However, we can evaluate the relative density of different samples. This function returns the relative log-weight alongside the sample. This log-weight is the log-probability of the pinned parts at the sampled location (it differs from `unnormalized_log_prob` by the log-probability of the unpinned parts).

Args
`sample_shape` Shape prefix to use when sampling.
`seed` Optional seed for reproducible sampling.

Returns
`samples` unpinned parts drawn from the pinned distribution.
`log_weights` log-weight of the sample. (Log-probability of the pinned parts at the sampled location.)

### `sample_unpinned`

View source

Draws unnormalized samples using ancestral sampling.

Conceptually, this is comparable to calling `underlying.sample(value=pins)`, then stripping away the pinned parts.

Args
`sample_shape` Shape prefix to use when sampling.
`seed` Optional seed for reproducible sampling.

Returns
`samples` unpinned parts sampled from the underlying distribution.

### `unnormalized_log_prob`

View source

Computes the unnormalized log-probability.

``````The methods of `JointDistributionPinned` (`unnormalized_log_prob`,
`sample_weighted`, etc.) can be called by passing a single structure
of tensors, a sequence of tensor arguments, or using named args for each
part. For example:

```python
tfde = tfp.experimental.distributions

# 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 `__init__` styles are all permissible and produce
# `JointDistributionPinned` objects behaving identically.
PartialXY = collections.namedtuple('PartialXY', 'x,y')
PartialX = collections.namedtuple('PartialX', 'x')
OrderedDict = collections.OrderedDict
assert (tfde.JointDistributionPinned(jd, x=2.).pins ==
tfde.JointDistributionPinned(jd, x=2., z=None).pins ==
tfde.JointDistributionPinned(jd, dict(x=2.)).pins ==
tfde.JointDistributionPinned(jd, dict(x=2., y=None)).pins ==
tfde.JointDistributionPinned(jd, OrderedDict(x=2.)).pins ==
tfde.JointDistributionPinned(jd, OrderedDict(x=2., y=None)).pins ==
tfde.JointDistributionPinned(jd, PartialXY(x=2., y=None)).pins ==
tfde.JointDistributionPinned(jd, PartialX(x=2.)).pins ==
tfde.JointDistributionPinned(jd, None, None, 2.).pins ==
tfde.JointDistributionPinned(jd, [None, None, 2.]).pins)
# (Notice that the `pins` attribute is always resolved to a `dict`.)

pinned = tfde.JointDistributionPinned(jd, x=2.)
pinned.dtype
# ==> [tf.float32, tf.float32]
z, y = sample = pinned.sample_unpinned()

# The following calling styles are all permissable and produce the exactly
# the same output.
PartialZY = collections.namedtuple('PartialZY', 'z,y')
assert (pinned.unnormalized_log_prob(sample) ==
pinned.unnormalized_log_prob(z, y) ==
pinned.unnormalized_log_prob(z=z, y=y) ==
pinned.unnormalized_log_prob(PartialZY(z=z, y=y)))

# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
pinned.unnormalized_log_prob(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
pinned.unnormalized_log_prob(**sample)
```

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.

In general, return types of part-wise methods/properties are determined by
those of the underlying `JointDistribution`'s model type:

- `StructTuple` for `JointDistributionCoroutine`, and for
`JointDistributionNamed` with `namedtuple` model type.
- `collections.OrderedDict` for `JointDistributionNamed` with `OrderedDict`
model type.
- `dict` for `JointDistributionNamed` with `dict` model type.
- `tuple` or `list` for `JointDistributionSequential`.

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`). In the same way, JointDistributionPinned does
not accept unnamed pins for unordered `JointDistributionNamed` models.

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
pinned = tfde.JointDistributionPinned(
tfd.JointDistributionSequential(
[tfd.Exponential(1.), lambda s: tfd.Normal(0., s)]),
None, 1.2)
pinned.dtype  # => [tf.float32]
pinned.unnormalized_log_prob([4.])
# ==> Tensor with shape `[]`.
lp = pinned.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
`pinned.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
`unnormalized_log_prob` The joint log-probability of `*xs` or `**kwargs` with the pinned parts. It is unnormalized with respect to `*xs` or `**kwargs`.

### `unnormalized_log_prob_parts`

View source

Computes the unnormalized log-probability of each part.

``````The methods of `JointDistributionPinned` (`unnormalized_log_prob`,
`sample_weighted`, etc.) can be called by passing a single structure
of tensors, a sequence of tensor arguments, or using named args for each
part. For example:

```python
tfde = tfp.experimental.distributions

# 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 `__init__` styles are all permissible and produce
# `JointDistributionPinned` objects behaving identically.
PartialXY = collections.namedtuple('PartialXY', 'x,y')
PartialX = collections.namedtuple('PartialX', 'x')
OrderedDict = collections.OrderedDict
assert (tfde.JointDistributionPinned(jd, x=2.).pins ==
tfde.JointDistributionPinned(jd, x=2., z=None).pins ==
tfde.JointDistributionPinned(jd, dict(x=2.)).pins ==
tfde.JointDistributionPinned(jd, dict(x=2., y=None)).pins ==
tfde.JointDistributionPinned(jd, OrderedDict(x=2.)).pins ==
tfde.JointDistributionPinned(jd, OrderedDict(x=2., y=None)).pins ==
tfde.JointDistributionPinned(jd, PartialXY(x=2., y=None)).pins ==
tfde.JointDistributionPinned(jd, PartialX(x=2.)).pins ==
tfde.JointDistributionPinned(jd, None, None, 2.).pins ==
tfde.JointDistributionPinned(jd, [None, None, 2.]).pins)
# (Notice that the `pins` attribute is always resolved to a `dict`.)

pinned = tfde.JointDistributionPinned(jd, x=2.)
pinned.dtype
# ==> [tf.float32, tf.float32]
z, y = sample = pinned.sample_unpinned()

# The following calling styles are all permissable and produce the exactly
# the same output.
PartialZY = collections.namedtuple('PartialZY', 'z,y')
assert (pinned.unnormalized_log_prob_parts(sample) ==
pinned.unnormalized_log_prob_parts(z, y) ==
pinned.unnormalized_log_prob_parts(z=z, y=y) ==
pinned.unnormalized_log_prob_parts(PartialZY(z=z, y=y)))

# These calling possibilities also imply that one can also use `*`
# expansion, if `sample` is a sequence:
pinned.unnormalized_log_prob_parts(*sample)
# and similarly, if `sample` is a map, one can use `**` expansion:
pinned.unnormalized_log_prob_parts(**sample)
```

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.

In general, return types of part-wise methods/properties are determined by
those of the underlying `JointDistribution`'s model type:

- `StructTuple` for `JointDistributionCoroutine`, and for
`JointDistributionNamed` with `namedtuple` model type.
- `collections.OrderedDict` for `JointDistributionNamed` with `OrderedDict`
model type.
- `dict` for `JointDistributionNamed` with `dict` model type.
- `tuple` or `list` for `JointDistributionSequential`.

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`). In the same way, JointDistributionPinned does
not accept unnamed pins for unordered `JointDistributionNamed` models.

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
pinned = tfde.JointDistributionPinned(
tfd.JointDistributionSequential(
[tfd.Exponential(1.), lambda s: tfd.Normal(0., s)]),
None, 1.2)
pinned.dtype  # => [tf.float32]
pinned.unnormalized_log_prob_parts([4.])
# ==> Tensor with shape `[]`.
lp_parts = pinned.unnormalized_log_prob_parts(4.)
# ==> Tensor with shape `[]`.
```

Notice that in the first call, `[4.]` is interpreted as a list of one
scalar while in the second call the input is a scalar. Hence both inputs
result in identical scalar outputs. If we wanted to pass an explicit
vector to the `Exponential` component---creating a vector-shaped batch
`pinned.unnormalized_log_prob_parts(np.array([4]))`.

Args:
*args: Positional arguments: a value structure or component values
(see above).
**kwargs: Keyword arguments: a value structure or component values
(see above). May also include `name`, specifying a Python string name
for ops generated by this method.
``````

Returns
`pinned` partial log-prob of each pinned part
`unpinned` partial log-prob of each unpinned part

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