TensorFlow

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 tf.compat / v1 / v1.distributions.ReparameterizationType


Student's t-distribution.

Inherits From: Distribution

This distribution has parameters: degree of freedom df, location loc, and scale.

Mathematical details

The probability density function (pdf) is,

pdf(x; df, mu, sigma) = (1 + y**2 / df)**(-0.5 (df + 1)) / Z
where,
y = (x - mu) / sigma
Z = abs(sigma) sqrt(df pi) Gamma(0.5 df) / Gamma(0.5 (df + 1))

where:

  • loc = mu,
  • scale = sigma, and,
  • Z is the normalization constant, and,
  • Gamma is the gamma function.

The StudentT distribution is a member of the location-scale family, i.e., it can be constructed as,

X ~ StudentT(df, loc=0, scale=1)
Y = loc + scale * X

Notice that scale has semantics more similar to standard deviation than variance. However it is not actually the std. deviation; the Student's t-distribution std. dev. is scale sqrt(df / (df - 2)) when df > 2.

Samples of this distribution are reparameterized (pathwise differentiable). The derivatives are computed using the approach described in (Figurnov et al., 2018).

Examples

Examples of initialization of one or a batch of distributions.

import tensorflow_probability as tfp
tfd = tfp.distributions

# Define a single scalar Student t distribution.
single_dist = tfd.StudentT(df=3)

# Evaluate the pdf at 1, returning a scalar Tensor.
single_dist.prob(1.)

# Define a batch of two scalar valued Student t's.
# The first has degrees of freedom 2, mean 1, and scale 11.
# The second 3, 2 and 22.
multi_dist = tfd.StudentT(df=[2, 3], loc=[1, 2.], scale=[11, 22.])

# Evaluate the pdf of the first distribution on 0, and the second on 1.5,
# returning a length two tensor.
multi_dist.prob([0, 1.5])

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

Arguments are broadcast when possible.

# Define a batch of two Student's t distributions.
# Both have df 2 and mean 1, but different scales.
dist = tfd.StudentT(df=2, loc=1, scale=[11, 22.])

# Evaluate the pdf of both distributions on the same point, 3.0,
# returning a length 2 tensor.
dist.prob(3.0)

Compute the gradients of samples w.r.t. the parameters:

df = tf.constant(2.0)
loc = tf.constant(2.0)
scale = tf.constant(11.0)
dist = tfd.StudentT(df=df, loc=loc, scale=scale)
samples = dist.sample(5)  # Shape [5]
loss = tf.reduce_mean(tf.square(samples))  # Arbitrary loss function
# Unbiased stochastic gradients of the loss function
grads = tf.gradients(loss, [df, loc, scale])

References:

Implicit Reparameterization Gradients: Figurnov et al., 2018 (pdf)

df Floating-point Tensor. The degrees of freedom of the distribution(s). df must contain only positive values.
loc Floating-point Tensor. The mean(s) of the distribution(s).
scale Floating-point Tensor. The scaling factor(s) for the distribution(s). Note that scale is not technically the standard deviation of this distribution but has semantics more similar to standard deviation than variance.
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.

TypeError if loc and scale are different dtypes.

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.

df Degrees of freedom in these Student's t distribution(s).
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.

loc Locations of these Student's t distribution(s).
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 distributions.FULLY_REPARAMETERIZED or distributions.NOT_REPARAMETERIZED.

scale Scaling factors of these Student's t distribution(s).
validate_args Python bool indicating possibly expensive checks are enabled.

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.

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

copy

View source

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.

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), (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

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.

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 (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

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.

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.

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

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.

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

mean

View source

Mean.

Additional documentation from StudentT:

The mean of Student's T equals loc if df > 1, otherwise it is NaN. If self.allow_nan_stats=True, then an exception will be raised rather than returning NaN.

mode

View source

Mode.

param_shapes

View source

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_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

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

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.

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.

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

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.

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.

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

View source

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.

Additional documentation from StudentT:

The variance for Student's T equals

df / (df - 2), when df > 2
infinity, when 1 < df <= 2
NaN, when df <= 1

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().

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