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Half-Student's t distribution.

Inherits From: Distribution, AutoCompositeTensor

The half-Student's t distribution has three parameters: degree of freedom df, location loc, and scale scale. It represents the right half of the two symmetric halves in a Student's t distribution.

Mathematical Details

The probability density function (pdf) for the half-Student's t distribution is given by

pdf(x; df, loc, scale) = (1 + y**2 / df)**(-0.5 (df + 1)) / Z,
y = (x - loc) / scale
Z = 2 * scale * sqrt(df * pi) * gamma(0.5 * df) / gamma(0.5 * (df + 1))


  • df is a positive scalar in R,
  • loc is a scalar in R,
  • scale is a positive scalar in R,
  • Z is the normalization constant, and
  • Gamma is the gamma function.

The support of the distribution is given by the interval [loc, infinity).

Samples of this distribution are reparameterized (pathwise differentiable). The derivatives are computed using the approach described in the paper

Michael Figurnov, Shakir Mohamed, Andriy Mnih. Implicit Reparameterization Gradients, 2018


import tensorflow_probability as tfp
tfd = tfp.distributions

# Define a single scalar Student t distribution.
single_dist = tfd.HalfStudentT(df=3, loc=0, scale=1)

# Evaluate the pdf at 1, returning a scalar Tensor.

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

# Evaluate the pdf of the first distribution at 1.5, and the second on 2.5,
# returning a length two tensor.
multi_dist.prob([1.5, 2.5])

# Get 3 samples, returning a 3 x 2 tensor.

Arguments are broadcast when possible.

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

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

Compute the gradients of samples w.r.t. the parameters via implicit reparameterization through the gamma:

df = tf.constant(2.0)
loc = tf.constant(2.0)
scale = tf.constant(11.0)
dist = tfd.HalfStudentT(df=df, loc=loc, scale=scale)
with tf.GradientTape() as tape:, loc, scale))
  loss = tf.reduce_mean(dist.sample(5))
  # Unbiased stochastic gradients of the loss function
  grads = tape.gradient(loss, (df, loc, scale))

df Floating-point Tensor. The degrees of freedom of the distribution(s). df must contain only positive values.
loc Floating-point Tensor; the location(s) of the distribution(s).
scale Floating-point Tensor; the scale(s) 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: 'HalfStudentT'.

TypeError if loc and scale have different dtype.

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 Distribution parameter for the degrees of freedom.
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.

experimental_shard_axis_names The list or structure of lists of active shard axis names.
loc Distribution parameter for the location.
name Name prepended to all ops created by this Distribution.
name_scope Returns a tf.name_scope instance for this class.
non_trainable_variables Sequence of non-trainable variables owned by this module and its submodules.

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.

scale Distribution parameter for the scale.
submodules Sequence of all sub-modules.

Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on).

a = tf.Module()
b = tf.Module()
c = tf.Module()
a.b = b
b.c = c
list(a.submodules) == [b, c]
list(b.submodules) == [c]
list(c.submodules) == []

trainable_variables Sequence of trainable variables owned by this module and its submodules.

validate_args Python bool indicating possibly expensive checks are enabled.
variables Sequence of variables owned by this module and its submodules.



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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.

name name to give to the op

batch_shape Tensor.


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Cumulative distribution function.

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

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

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

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


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Creates a deep copy of the distribution.

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

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).


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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.

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