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tfp.experimental.substrates.jax.distributions.Horseshoe

Class Horseshoe

Horseshoe distribution.

Inherits From: Distribution

The so-called 'horseshoe' distribution is a Cauchy-Normal scale mixture, proposed as a sparsity-inducing prior for Bayesian regression.  It is symmetric around zero, has heavy (Cauchy-like) tails, so that large coefficients face relatively little shrinkage, but an infinitely tall spike at 0, which pushes small coefficients towards zero. It is parameterized by a positive scalar scale parameter: higher values yield a weaker sparsity-inducing effect.

Mathematical details

The Horseshoe distribution is centered at zero, with scale parameter $$\lambda$$. It is defined by:
$$X \sim \text {Horseshoe}(scale=\lambda) \, \equiv \, X \sim \text{Normal} (0, \, \lambda \cdot \sigma) \quad \text{where} \quad \sigma \sim \text{HalfCauchy} (0, \,1)$$

The probability density function,
$$\pi_\lambda(x) = \int_0^\infty \, \frac{1}{\sqrt{ 2\pi \lambda^2 t^2 }} \, \exp \left\{ -\frac{x^2}{2\lambda^2t^2} \right\} \, \frac{2}{\pi\left(1+t^2\right)} \mathrm{d} t$$

can be rewritten  as
$$\pi_\lambda(x) = \frac{1}{\sqrt{2 \pi^3 \lambda^2}} \, \exp \left\{ \frac{x^2}{2\lambda^2} \right\} \, E_1\left(\frac{x^2}{2\lambda^2}\right)$$

where E1(.) is the exponential integral function which can be approximated by elementary functions. 

Examples

Examples of initialization of one or a batch of distributions.

# Define a single scalar Horseshoe distribution.
dist = tfp.distributions.Horseshoe(scale=3.0)

# Evaluate the log_prob at 1, returning a scalar.
dist.log_prob(1.)

# Define a batch of two scalar valued Horseshoes.
# The first has scale 11.0, the second 22.0
dist = tfp.distributions.Horseshoe(scale=[11.0, 22.0])

# Evaluate the log_prob of the first distribution on 1.0, and the second on
# 1.5, returning a length two tensor.
dist.log_prob([1.0, 1.5])

# Evaluate the log_prob of both distributions at 2.0 and 2.5, returning a
# 2 x 2 tensor.
dist.log_prob([[2.0], [2.5]])

# Get 3 samples, returning a 3 x 2 tensor.
dist.sample()

 Carvalho, Polson, Scott. Handling Sparsity via the Horseshoe (2008).

 Barry, Parlange, Li. Approximation for the exponential integral (2000). Formula from Wikipedia.

__init__

__init__(
scale,
validate_args=False,
allow_nan_stats=True,
name='Horseshoe'
)

Construct a Horseshoe distribution with scale.

Args:

• scale: Floating point tensor; the scales of the distribution(s). Must contain only positive values.
• 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 (i.e., do not validate args).
• 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. Default value: True.
• name: Python str name prefixed to Ops created by this class. Default value: 'Horseshoe'.

Properties

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.

Returns:

• allow_nan_stats: Python bool.

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.

Returns:

• batch_shape: TensorShape, possibly unknown.

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.

Returns:

• event_shape: TensorShape, possibly unknown.

name

Name prepended to all ops created by this Distribution.

parameters

Dictionary of parameters used to instantiate this 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.

Returns:

An instance of ReparameterizationType.

scale

Distribution parameter for scale.

validate_args

Python bool indicating possibly expensive checks are enabled.

Methods

__getitem__

View source

__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.cholesky(cov)
loc = tf.random.normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape  # => [4, 5, 3]
mvn.event_shape  # => 
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape  # => [4, 2, 3, 1]
mvn2.event_shape  # => 

Args:

• slices: slices from the [] operator

Returns:

• dist: A new tfd.Distribution instance with sliced parameters.

View source

__iter__()

batch_shape_tensor

View source

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

View source

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

Cumulative distribution function.

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

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

Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.
• **kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

copy

View source

copy(**override_parameters_kwargs)

Creates a deep copy of the distribution.

Args:

• **override_parameters_kwargs: String/value dictionary of initialization arguments to override with new values.

Returns:

• distribution: A new instance of type(self) initialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

covariance

View source

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

Covariance.

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

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

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

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

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

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

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

Args:

• name: Python str prepended to names of ops created by this function.
• **kwargs: Named arguments forwarded to subclass implementation.

Returns:

• covariance: Floating-point Tensor with shape [B1, ..., Bn, k', k'] where the first n dimensions are batch coordinates and k' = reduce_prod(self.event_shape).

cross_entropy

View source

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.

Returns:

• cross_entropy: self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of (Shannon) cross entropy.

entropy

View source

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

Shannon entropy in nats.

event_shape_tensor

View source

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

View source

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

View source

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

View source

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.

Returns:

• kl_divergence: self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of the Kullback-Leibler divergence.

log_cdf

View source

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

Log cumulative distribution function.

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

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

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

Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.
• **kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

log_prob

View source

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

Log 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:

• 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(
value,
name='log_survival_function',
**kwargs
)

Log survival function.

Given random variable X, the survival function is defined:

log_survival_function(x) = Log[ P[X > x] ]
= Log[ 1 - P[X <= x] ]
= Log[ 1 - cdf(x) ]

Typically, different numerical approximations can be used for the log survival function, which are more accurate than 1 - cdf(x) when x >> 1.

Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.
• **kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

View source

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

Mean.

View source

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

Mode.

param_shapes

View source

param_shapes(
cls,
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

View source

param_static_shapes(
cls,
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

View source

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

Probability density/mass function.

Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.
• **kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

quantile

View source

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

Quantile function. Aka 'inverse cdf' or 'percent point function'.

Given random variable X and p in [0, 1], the quantile is:

quantile(p) := x such that P[X <= x] == p

Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.
• **kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

sample

View source

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.

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.
• **kwargs: Named arguments forwarded to subclass implementation.

Returns:

• samples: a Tensor with prepended dimensions sample_shape.

stddev

View source

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

Standard deviation.

Standard deviation is defined as,

stddev = E[(X - E[X])**2]**0.5

where X is the random variable associated with this distribution, E denotes expectation, and stddev.shape = batch_shape + event_shape.

Args:

• name: Python str prepended to names of ops created by this function.
• **kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

survival_function

View source

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

Survival function.

Given random variable X, the survival function is defined:

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

Args:

• value: float or double Tensor.
• name: Python str prepended to names of ops created by this function.
• **kwargs: Named arguments forwarded to subclass implementation.

Returns:

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

variance

View source

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

Variance.

Variance is defined as,

Var = E[(X - E[X])**2]

where X is the random variable associated with this distribution, E denotes expectation, and Var.shape = batch_shape + event_shape.

Args:

• name: Python str prepended to names of ops created by this function.
• **kwargs: Named arguments forwarded to subclass implementation.

Returns:

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