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Models the distribution of finitely many importance-reweighted samples.
Inherits From: Distribution
tfp.experimental.distributions.ImportanceResample(
proposal_distribution,
target_log_prob_fn,
importance_sample_size,
sample_size=1,
stochastic_approximation_seed=None,
validate_args=False,
name=None
)
This wrapper adapts a proposal distribution towards a target density using
importance sampling.
Given a proposal q
, a target density p
(which may be unnormalized), and
an integer importance_sample_size
, it models the result of the following
sampling process:
- Draw
importance_sample_size
samplesz[k] ~ q
from the proposal. - Compute an importance weight
w[k] = p(z[k]) / q(z[k])
for each sample. - Return a sample
z[k*]
selected with probability proportional to the importance weights, i.e., withk* ~ Categorical(probs=w/sum(w))
.
In the limit where importance_sample_size -> inf
, the result z[k*]
of this
procedure would be distributed according to the target density p
. On the
other hand, if importance_sample_size == 1
, then the reweighting has no
effect and the result z[k*]
is simply a sample from q
. Finite values
of importance_sample_size
describe distributions that are intermediate
between p
and q
.
This distribution may also be understood as an explicit representation of the surrogate posterior that is implicitly assumed by importance-weighted variational objectives. [1, 2]
Examples
This distribution can be used directly for posterior inference via importance sampling:
tfd = tfp.distributions
tfed = tfp.experimental.distributions
def target_log_prob_fn(x):
prior = tfd.Normal(loc=0., scale=1.).log_prob(x)
# Multimodal likelihood.
likelihood = tf.reduce_logsumexp(
tfd.Normal(loc=x, scale=0.1).log_prob([-1., 1.]))
return prior + likelihood
# Use importance sampling to infer an approximate posterior.
approximate_posterior = tfed.ImportanceResample(
proposal_distribution=tfd.Normal(loc=0., scale=2.),
target_log_prob_fn=target_log_prob_fn,
importance_sample_size=100)
We can estimate posterior expectations directly using an importance-weighted sum of proposal samples:
# Directly compute expectations under the posterior via importance weights.
posterior_mean = approximate_posterior.self_normalized_expectation(
lambda x: x, importance_sample_size=1000)
posterior_variance = approximate_posterior.self_normalized_expectation(
lambda x: (x - posterior_mean)**2, importance_sample_size=1000)
Alternately, the same expectations can be estimated from explicit (unweighted)
samples. Note that sampling may be expensive because it performs resampling
internally. For example, to produce sample_size
samples requires first
proposing values of shape [sample_size, importance_sample_size]
([1000, 100]
in the code below) and then resampling down to [sample_size]
,
throwing most of the proposals away. For this reason you should prefer calling
self_normalized_expectation
over naive sampling to compute expectations.
posterior_samples = approximate_posterior.sample(1000)
posterior_mean_inefficient = tf.reduce_mean(posterior_samples)
posterior_variance_inefficient = tf.math.reduce_variance(posterior_samples)
# Calling `self_normalized_expectation` allows for a much lower `sample_size`
# because it uses the full set of `importance_sample_size` proposal samples to
# approximate the expectation at each of the `sample_size` Monte Carlo
# evaluations. This is formalized in Eq. 9 of [3].
posterior_mean_efficient = approximate_posterior.self_normalized_expectation(
lambda x: x, sample_size=10)
posterior_variance_efficient = (
approximate_posterior.self_normalized_expectation(
lambda x: (x - posterior_mean_efficient)**2, sample_size=10))
The posterior (log-)density cannot be computed directly, but may be
stochastically approximated. The prob
and log_prob
methods accept
arguments seed
and sample_size
to control the variance of the
approximation.
# Plot the posterior density.
from matplotlib import pylab as plt
xs = tf.linspace(-3., 3., 101)
probs = approximate_posterior.prob(xs, sample_size=10, seed=(42, 42))
plt.plot(xs, probs)
Connections to importance-weighted variational inference
Optimizing an importance-weighted variational bound provides a natural
approach to choose a proposal distribution for importance sampling.
Importance-weighted bounds are available directly in TFP via the
importance_sample_size
argument to tfp.vi.monte_carlo_variational_loss
and tfp.vi.fit_surrogate_posterior
. For example, we might improve on the
example above by replacing the fixed proposal distribution with a learned
proposal:
proposal_distribution = tfp.experimental.util.make_trainable(tfd.Normal)
importance_sample_size = 100
importance_weighted_losses = tfp.vi.fit_surrogate_posterior(
target_log_prob_fn,
surrogate_posterior=proposal_distribution,
optimizer=tf.optimizers.Adam(0.1),
num_steps=200,
importance_sample_size=importance_sample_size)
approximate_posterior = tfed.ImportanceResample(
proposal_distribution=proposal_distribution,
target_log_prob_fn=target_log_prob_fn,
importance_sample_size=importance_sample_size)
Note that although the importance-resampled approximate_posterior
serves
ultimately as the surrogate posterior, only the bare proposal distribution
is passed as the surrogate_posterior
argument to fit_surrogate_posterior
.
This is because the importance_sample_size
argument tells
fit_surrogate_posterior
to compute an importance-weighted bound directly
from the proposal distribution. Mathematically, it would be equivalent to omit
the importance_sample_size
argument and instead pass an ImportanceResample
distribution as the surrogate posterior:
equivalent_but_less_efficient_losses = tfp.vi.fit_surrogate_posterior(
target_log_prob_fn,
surrogate_posterior=tfed.ImportanceResample(
proposal_distribution=proposal_distribution,
target_log_prob_fn=target_log_prob_fn,
importance_sample_size=importance_sample_size),
optimizer=tf.optimizers.Adam(0.1),
num_steps=200)
but this approach is not recommended, because it performs redundant
evaluations of the target_log_prob_fn
compared to the direct bound shown
above.
References
[1] Yuri Burda, Roger Grosse, and Ruslan Salakhutdinov. Importance Weighted Autoencoders. In International Conference on Learning Representations, 2016. https://arxiv.org/abs/1509.00519 [2] Chris Cremer, Quaid Morris, and David Duvenaud. Reinterpreting Importance-Weighted Autoencoders. In International Conference on Learning Representations, Workshop track, 2017. https://arxiv.org/abs/1704.02916 [3] Justin Domke, Daniel Sheldon. Importance Weighting and Variational Inference. In Neural Information Processing Systems (NIPS), 2018. https://arxiv.org/abs/1808.09034
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_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
|
The list or structure of lists of active shard axis names. |
importance_sample_size
|
|
name
|
Name prepended to all ops created by this Distribution .
|
name_scope
|
Returns a tf.name_scope instance for this class.
|
non_trainable_variables
|
Sequence of non-trainable variables owned by this module and its submodules. |
parameters
|
Dictionary of parameters used to instantiate this Distribution .
|
proposal_distribution
|
|
reparameterization_type
|
Describes how samples from the distribution are reparameterized.
Currently this is one of the static instances
|
sample_size
|
|
stochastic_approximation_seed
|
|
submodules
|
Sequence of all sub-modules.
Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on).
|
target_log_prob_fn
|
|
trainable_variables
|
Sequence of trainable variables owned by this module and its submodules. |
validate_args
|
Python bool indicating possibly expensive checks are enabled.
|
variables
|
Sequence of variables owned by this module and its submodules. |
Methods
batch_shape_tensor
batch_shape_tensor(
name='batch_shape_tensor'
)
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
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
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
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
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
.
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
entropy(
name='entropy', **kwargs
)
Shannon entropy in nats.
event_shape_tensor
event_shape_tensor(
name='event_shape_tensor'
)
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
experimental_default_event_space_bijector(
*args, **kwargs
)
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
@classmethod
experimental_fit( value, sample_ndims=1, validate_args=False, **init_kwargs )
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
experimental_local_measure(
value, backward_compat=False, **kwargs
)
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_sample_and_log_prob
experimental_sample_and_log_prob(
sample_shape=(), seed=None, name='sample_and_log_prob', **kwargs
)
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.
Additional documentation from ImportanceResample
:
Re-use proposal samples for lower variance in the log-prob estimate.
Note: this method reuses the same proposal samples `z[k]` for both sampling
and approximate `log_prob` evaluation. Thus, calling `sample_and_log_prob`
is *not* equivalent to calling `sample` followed by `log_prob`, which would
use two independent sets of proposal samples and in general return a
different stochastic approximation to the log-density of the sampled points.
In particular, `log_prob` returns a stochastic lower bound (which becomes
tighter as `sample_size` increases) on the log-density of the
importance-resampled distribution , while this method returns a
single-sample stochastic *upper* bound. This guarantees that plugging an
`ImportanceResample` surrogate posterior into a variational evidence lower
bound (ELBO) preserves a valid lower bound---in fact, the IWAE bound [1]---
which would otherwise not be the case for `log_prob` with finite values of
`sample_size`. (This said, explicitly computing an IWAE bound via
<a href="../../../tfp/vi/monte_carlo_variational_loss"><code>tfp.vi.monte_carlo_variational_loss</code></a> is more efficient and stable than
this implicit construction using an `ImportanceResample` surrogate, and so
should be the preferred approach in general.)
#### Mathematical details
The `log_prob` estimate computed in this method is given by
```
surrogate_log_prob(x) = target_log_prob_fn(x) - log(mean(weights(z)))
```
where
`weights(z)[k] = exp(target_log_prob_fn(z[k]) - proposal.log_prob(z[k]))`
are the importance weights of the proposal samples `z[k]` from which `x` was
selected. Since we know that we selected `x` from among these
proposal samples, we may conclude that these samples are more likely
to lead to us selecting `x` than would be the case for 'typical' proposal
samples in the absence of such knowledge. The implied estimate of
`prob(x)` is therefore biased upwards.
The motivation for this estimate is that plugging it into the ELBO recovers
the IWAE objective:
```
ELBO = target_log_prob(x) - surrogate_log_prob(x)
(for x ~ surrogate)
= target_log_prob(x) - (target_log_prob(x) - log(mean(weights(z))))
(for z[k] ~ proposal)
= log(mean(weights(z)))
= IWAE
```
Because the IWAE objective lower-bounds the *true* ELBO
of the importance-resampled distribution (i.e., the ELBO that we would
compute using
`surrogate_log_prob(x) = ImportanceResample.log_prob(x, sample_size=inf)`;
see section 5.3 of Cremer et al. [2]),
it follows that the quantity `surrogate_log_prob(x)` estimated here is an
upper bound on the *true* log_prob of the importance-resampled distribution.
kwargs
:
importance_sample_size
: optional integerTensor
number of proposals used to define the distribution. IfNone
, defaults toself.importance_sample_size
.
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
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
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 .
|
kl_divergence
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.
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
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
log_prob(
value, name='log_prob', **kwargs
)
Log probability density/mass function.
Additional documentation from ImportanceResample
:
The density of an importance-resampled distribution is
not generally available in closed form. This method follows algorithm (2) of
Cremer et al. [2] to compute an unbiased estimate of prob(x)
, which
corresponds by Jensen's inequality to a stochastic lower
bound on log_prob(x)
. The estimation variance decreases, and the corresponding
bound tightens, as sample_size
increases; an infinitely large sample_size
would recover the true (log-)density.
kwargs
:
importance_sample_size
: optional integerTensor
number of proposals used to define the distribution. If not specified, defaults toself.importance_sample_size
.sample_size
: intTensor
number of samples used to reduce variance in the estimated density for a givenimportance_sample_size
. IfNone
, defaults toself.sample_size
.seed
: PRNG seed; seetfp.random.sanitize_seed
for details.
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 | |
---|---|
log_prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
log_survival_function
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
mean(
name='mean', **kwargs
)
Mean.
mode
mode(
name='mode', **kwargs
)
Mode.
param_shapes
@classmethod
param_shapes( sample_shape, name='DistributionParamShapes' )
Shapes of parameters given the desired shape of a call to sample()
. (deprecated)
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
@classmethod
param_static_shapes( sample_shape )
param_shapes with static (i.e. TensorShape
) shapes. (deprecated)
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
@classmethod
parameter_properties( dtype=tf.float32, num_classes=None )
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.ParameterPropertiesdict mapping constructor argument names to ParameterProperties`
instances.
|
Raises | |
---|---|
NotImplementedError
|
if the distribution class does not implement
_parameter_properties .
|
prob
prob(
value, name='prob', **kwargs
)
Probability density/mass function.
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 | |
---|---|
prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .
|
quantile
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 .
|
sample
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.
Additional documentation from ImportanceResample
:
kwargs
:
importance_sample_size
: optional integerTensor
number of proposals used to define the distribution. IfNone
, defaults toself.importance_sample_size
.
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 .
|
self_normalized_expectation
self_normalized_expectation(
fn,
importance_sample_size=None,
sample_size=None,
seed=None,
name='self_normalized_expectation'
)
Approximates the expectation of fn(x).
This function applies self-normalized importance sampling with the given
proposal distribution to approximate expectations under the target
distribution. By using all of the importance_sample_size
proposal
samples to approximate the expectation, this will in general give
lower-variance estimates than those obtained by explicit sampling
(tf.reduce_sum(fn(self.sample(sample_size)), axis=0)
), since the latter
returns only one point from each set of importance_sample_size
proposals.
Concretely, this function draws importance_sample_size
samples
x[1], x[2], ...
from
self.proposal_distribution
, computes their importance weights
w[k] = target_log_prob_fn(x[k]) / proposal_distribution.log_prob(x[k])
,
and returns the weighted sum
sum(w[k]/sum(w) * fn(x[k]) for k in range(importance_sample_size))
. If
sample_size > 1
is specified, the previous procedure is performed
multiple times and the results averaged to reduce variance.
Args | |
---|---|
fn
|
Python callable that takes samples from self.proposal_distribution
and returns a (structure of) Tensor value(s). This may represent a
prediction derived from a posterior sample, or even a simple statistic;
for example, the expectation of fn = lambda x: x is the posterior
mean.
|
importance_sample_size
|
int Tensor number of samples used to define the
distribution under which the expectation is taken. If None , defaults
to self.importance_sample_size .
Default value: None .
|
sample_size
|
int Tensor number of samples used to reduce variance in the
expectation for a given importance_sample_size . If None , defaults
to self.sample_size .
Default value: None .
|
seed
|
PRNG seed; see tfp.random.sanitize_seed for details. If None ,
defaults to self.stochastic_approximation_seed .
Default value: None .
|
name
|
Python string name for ops created by this function.
Default value: self_normalized_expectation .
|
Returns | |
---|---|
expected_value
|
(structure of) Tensor value(s) estimate of the
expectation of fn(x) under the target distribution.
|
stddev
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
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 .
|
unnormalized_log_prob
unnormalized_log_prob(
value, name='unnormalized_log_prob', **kwargs
)
Potentially unnormalized log probability density/mass function.
This function is similar to log_prob
, but does not require that the
return value be normalized. (Normalization here refers to the total
integral of probability being one, as it should be by definition for any
probability distribution.) This is useful, for example, for distributions
where the normalization constant is difficult or expensive to compute. By
default, this simply calls log_prob
.
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 | |
---|---|
unnormalized_log_prob
|
a Tensor of shape
sample_shape(x) + self.batch_shape with values of type self.dtype .
|
variance
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() .
|
with_name_scope
@classmethod
with_name_scope( method )
Decorator to automatically enter the module name scope.
class MyModule(tf.Module):
@tf.Module.with_name_scope
def __call__(self, x):
if not hasattr(self, 'w'):
self.w = tf.Variable(tf.random.normal([x.shape[1], 3]))
return tf.matmul(x, self.w)
Using the above module would produce tf.Variable
s and tf.Tensor
s whose
names included the module name:
mod = MyModule()
mod(tf.ones([1, 2]))
<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>
mod.w
<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,
numpy=..., dtype=float32)>
Args | |
---|---|
method
|
The method to wrap. |
Returns | |
---|---|
The original method wrapped such that it enters the module's name scope. |
__getitem__
__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.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.linalg.cholesky(cov)
loc = tf.random.normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape # => [4, 5, 3]
mvn.event_shape # => [2]
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape # => [4, 2, 3, 1]
mvn2.event_shape # => [2]
Args | |
---|---|
slices
|
slices from the [] operator |
Returns | |
---|---|
dist
|
A new tfd.Distribution instance with sliced parameters.
|
__iter__
__iter__()