tfp.experimental.distributions.ImportanceResample

Models the distribution of finitely many importance-reweighted samples.

Inherits From: Distribution

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:

  1. Draw importance_sample_size samples z[k] ~ q from the proposal.
  2. Compute an importance weight w[k] = p(z[k]) / q(z[k]) for each sample.
  3. Return a sample z[k*] selected with probability proportional to the importance weights, i.e., with k* ~ 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

proposal_distribution Instance of tfd.Distribution used to generate proposals. This may be a joint distribution.
target_log_prob_fn Python callable representation of a (potentially unnormalized) target log-density. This should accept samples from the proposal, i.e., lp = target_log_prob_fn(proposal_distribution.sample()).
importance_sample_size integer Tensor number of proposals used in the distribution of a single sample. Larger values better approximate the target distribution, at the cost of increased computation and memory usage.
sample_size integer Tensor number of Monte Carlo samples used to reduce variance in stochastic methods such as log_prob, prob, and self_normalized_expectation. Note that increasing importance_sample_size leads to a more accurate approximation of the target distribution (reducing bias and variance), while increasing sample_size improves the precision of estimates under the intermediate distribution corresponding to a particular finite importance_sample_size (i.e., it reduces variance only and does not affect the sampling distribution). If unsure, it's generally safe to leave sample_size at its default value of 1 and focus on increasing importance_sample_size instead. Default value: 1.
stochastic_approximation_seed optional PRNG key used in stochastic approximations for methods such as log_prob, prob, and self_normalized_expectation. This seed does not affect sampling. Default value: None.
validate_args Python bool. Whether to validate input with asserts. If validate_args is False, and the inputs are invalid, correct behavior is not guaranteed. Default value: False.
name Python str name for this distribution. If None, defaults to 'importance_resample'. Default value: None.

allow_nan_stats Python bool describing behavior when a stat is undefined.

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

batch_shape Shape of a single sample from a single event index as a TensorShape.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

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

May be partially defined or unknown.

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 tfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.

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).

a = tf.Module()
b = tf.Module()
c = tf.Module()
a.b = b
b.c = c
list(a.submodules) == [b, c]
True
list(b.submodules) == [c]
True
list(c.submodules) == []
True

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

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

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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

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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

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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_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.

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 integer Tensor number of proposals used to define the distribution. If None, defaults to self.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 Tensors, 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.

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.

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.

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 integer Tensor number of proposals used to define the distribution. If not specified, defaults to self.importance_sample_size.
  • sample_size: int Tensor number of samples used to reduce variance in the estimated density for a given importance_sample_size. If None, defaults to self.sample_size.
  • seed: PRNG seed; see tfp.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

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.

mean

View source

Mean.

mode

View source

Mode.

param_shapes

View source

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

View source

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

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.ParameterPropertiesdict mapping constructor argument names toParameterProperties` instances.

Raises
NotImplementedError if the distribution class does not implement _parameter_properties.

prob

View source

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

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 ImportanceResample:

kwargs:
  • importance_sample_size: optional integer Tensor number of proposals used to define the distribution. If None, defaults to self.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

View source

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

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

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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

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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

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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

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, &#x27;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.Variables and tf.Tensors 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 &#x27;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__

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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__

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