tfp.distributions.RelaxedBernoulli

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RelaxedBernoulli distribution with temperature and logits parameters.

Inherits From: AutoCompositeTensorDistribution, Distribution, AutoCompositeTensor

tfp.distributions.RelaxedBernoulli(
    temperature,
    logits=None,
    probs=None,
    validate_args=False,
    allow_nan_stats=True,
    name='RelaxedBernoulli'
)

The RelaxedBernoulli is a distribution over the unit interval (0,1), which continuously approximates a Bernoulli. The degree of approximation is controlled by a temperature: as the temperature goes to 0 the RelaxedBernoulli becomes discrete with a distribution described by the logits or probs parameters, as the temperature goes to infinity the RelaxedBernoulli becomes the constant distribution that is identically 0.5.

The RelaxedBernoulli distribution is a reparameterized continuous distribution that is the binary special case of the RelaxedOneHotCategorical distribution (Maddison et al., 2016; Jang et al., 2016). For details on the binary special case see the appendix of Maddison et al. (2016) where it is referred to as BinConcrete. If you use this distribution, please cite both papers.

Some care needs to be taken for loss functions that depend on the log-probability of RelaxedBernoullis, because computing log-probabilities of the RelaxedBernoulli can suffer from underflow issues. In many case loss functions such as these are invariant under invertible transformations of the random variables. The KL divergence, found in the variational autoencoder loss, is an example. Because RelaxedBernoullis are sampled by a Logistic random variable followed by a tf.sigmoid op, one solution is to treat the Logistic as the random variable and tf.sigmoid as downstream. The KL divergences of two Logistics, which are always followed by a tf.sigmoid op, is equivalent to evaluating KL divergences of RelaxedBernoulli samples. See Maddison et al., 2016 for more details where this distribution is called the BinConcrete.

An alternative approach is to evaluate Bernoulli log probability or KL directly on relaxed samples, as done in Jang et al., 2016. In this case, guarantees on the loss are usually violated. For instance, using a Bernoulli KL in a relaxed ELBO is no longer a lower bound on the log marginal probability of the observation. Thus care and early stopping are important.

Examples

Creates three continuous distributions, which approximate 3 Bernoullis with probabilities (0.1, 0.5, 0.4). Samples from these distributions will be in the unit interval (0,1).

temperature = 0.5
p = [0.1, 0.5, 0.4]
dist = RelaxedBernoulli(temperature, probs=p)

Creates three continuous distributions, which approximate 3 Bernoullis with logits (-2, 2, 0). Samples from these distributions will be in the unit interval (0,1).

temperature = 0.5
logits = [-2, 2, 0]
dist = RelaxedBernoulli(temperature, logits=logits)

Creates three continuous distributions, whose sigmoid approximate 3 Bernoullis with logits (-2, 2, 0).

temperature = 0.5
logits = [-2, 2, 0]
dist = Logistic(logits/temperature, 1./temperature)
samples = dist.sample()
sigmoid_samples = tf.sigmoid(samples)
# sigmoid_samples has the same distribution as samples from
# RelaxedBernoulli(temperature, logits=logits)

Creates three continuous distributions, which approximate 3 Bernoullis with logits (-2, 2, 0). Samples from these distributions will be in the unit interval (0,1). Because the temperature is very low, samples from these distributions are almost discrete, usually taking values very close to 0 or 1.

temperature = 1e-5
logits = [-2, 2, 0]
dist = RelaxedBernoulli(temperature, logits=logits)

Creates three continuous distributions, which approximate 3 Bernoullis with logits (-2, 2, 0). Samples from these distributions will be in the unit interval (0,1). Because the temperature is very high, samples from these distributions are usually close to the (0.5, 0.5, 0.5) vector.

temperature = 100
logits = [-2, 2, 0]
dist = RelaxedBernoulli(temperature, logits=logits)

Chris J. Maddison, Andriy Mnih, and Yee Whye Teh. The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables. 2016.

Eric Jang, Shixiang Gu, and Ben Poole. Categorical Reparameterization with Gumbel-Softmax. 2016.

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.

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.

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.

logits_parameter

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logits_parameter(
    name=None
)

Logits computed from non-None input arg (probs or logits).

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.

probs_parameter

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probs_parameter(
    name=None
)

Probs computed from non-None input arg (probs or logits).

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.

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__

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__iter__()