tf.contrib.distributions.RelaxedBernoulli

RelaxedBernoulli distribution with temperature and logits parameters.

Inherits From: TransformedDistribution

tf.contrib.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.

Args
temperature	An 0-D Tensor, representing the temperature of a set of RelaxedBernoulli distributions. The temperature should be positive.
logits	An N-D Tensor representing the log-odds of a positive event. Each entry in the Tensor parametrizes an independent RelaxedBernoulli distribution where the probability of an event is sigmoid(logits). Only one of logits or probs should be passed in.
probs	An N-D Tensor representing the probability of a positive event. Each entry in the Tensor parameterizes an independent Bernoulli distribution. Only one of logits or probs should be passed in.
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.
allow_nan_stats	Python bool, default True. When True, statistics (e.g., mean, mode, variance) use the value "NaN" to indicate the result is undefined. When False, an exception is raised if one or more of the statistic's batch members are undefined.
name	Python str name prefixed to Ops created by this class.

Raises
ValueError	If both probs and logits are passed, or if neither.

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.
bijector	Function transforming x => y.
distribution	Base distribution, p(x).
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.
logits	Log-odds of 1.
name	Name prepended to all ops created by this Distribution.
parameters	Dictionary of parameters used to instantiate this Distribution.
probs	Probability of 1.
reparameterization_type	Describes how samples from the distribution are reparameterized. Currently this is one of the static instances distributions.FULLY_REPARAMETERIZED or distributions.NOT_REPARAMETERIZED.
temperature	Distribution parameter for the location.
validate_args	Python bool indicating possibly expensive checks are enabled.

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.

Args
name	name to give to the op

Returns
batch_shape	Tensor.

cdf

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cdf(
    value, name='cdf'
)

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.

Returns
cdf	a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

copy

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copy(
    **override_parameters_kwargs
)

Creates a deep copy of the distribution.

Note: the copy distribution may continue to depend on the original initialization arguments.

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

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covariance(
    name='covariance'
)

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.

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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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), (Shanon) 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 (Shanon) cross entropy.

entropy

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entropy(
    name='entropy'
)

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.

Args
name	name to give to the op

Returns
event_shape	Tensor.

is_scalar_batch

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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

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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

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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 (Shanon) cross entropy, and H[.] denotes (Shanon) 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

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log_cdf(
    value, name='log_cdf'
)

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.

Returns
logcdf	a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_prob

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log_prob(
    value, name='log_prob'
)

Log probability density/mass function.

Args
value	float or double Tensor.
name	Python str prepended to names of ops created by this function.

Returns
log_prob	a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_survival_function

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log_survival_function(
    value, name='log_survival_function'
)

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.

Returns
Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

mean

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mean(
    name='mean'
)

Mean.

mode

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mode(
    name='mode'
)

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().

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

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@classmethod
param_static_shapes(
    sample_shape
)

param_shapes with static (i.e. TensorShape) shapes.

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.

prob

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prob(
    value, name='prob'
)

Probability density/mass function.

Args
value	float or double Tensor.
name	Python str prepended to names of ops created by this function.

Returns
prob	a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

quantile

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quantile(
    value, name='quantile'
)

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.

Returns
quantile	a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

sample

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sample(
    sample_shape=(), seed=None, name='sample'
)

Generate samples of the specified shape.

Note that a call to sample() without arguments will generate a single sample.

Args
sample_shape	0D or 1D int32 Tensor. Shape of the generated samples.
seed	Python integer seed for RNG
name	name to give to the op.

Returns
samples	a Tensor with prepended dimensions sample_shape.

stddev

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stddev(
    name='stddev'
)

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.

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(
    value, name='survival_function'
)

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.

Returns
Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

variance

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variance(
    name='variance'
)

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.

Returns
variance	Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().