tfp.experimental.distributions.ImportanceResample

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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:

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

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

Methods

batch_shape_tensor

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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.

cdf

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

Cumulative distribution function.

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

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

copy

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copy(
    **override_parameters_kwargs
)

Creates a deep copy of the distribution.

covariance

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

Covariance.

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

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

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

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

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

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

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

cross_entropy

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cross_entropy(
    other, name='cross_entropy'
)

Computes the (Shannon) cross entropy.

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

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

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

entropy

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

Shannon entropy in nats.

event_shape_tensor

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event_shape_tensor(
    name='event_shape_tensor'
)

Shape of a single sample from a single batch as a 1-D int32 Tensor.

experimental_default_event_space_bijector

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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.

experimental_fit

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@classmethod
experimental_fit(
    value, sample_ndims=1, validate_args=False, **init_kwargs
)

Instantiates a distribution that maximizes the likelihood of x.

experimental_local_measure

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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.

experimental_sample_and_log_prob

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

is_scalar_batch

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is_scalar_batch(
    name='is_scalar_batch'
)

Indicates that batch_shape == [].

is_scalar_event

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is_scalar_event(
    name='is_scalar_event'
)

Indicates that event_shape == [].

kl_divergence

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kl_divergence(
    other, name='kl_divergence'
)

Computes the Kullback--Leibler divergence.

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

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

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

log_cdf

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

Log cumulative distribution function.

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

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

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

log_prob

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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 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.

log_survival_function

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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.

mean

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

Mean.

mode

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

Mode.

param_shapes

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@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.

param_static_shapes

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@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.

parameter_properties

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@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.

prob

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

Probability density/mass function.

quantile

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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

sample

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

self_normalized_expectation

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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.

stddev

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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.

survival_function

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

Survival function.

Given random variable X, the survival function is defined:

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

unnormalized_log_prob

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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.

variance

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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.

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.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 'my_module/Variable:0' shape=(2, 3) dtype=float32,
numpy=..., dtype=float32)>

__getitem__

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__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]

__iter__

View source

__iter__()