tfp.distributions.MultivariateStudentTLinearOperator

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The [Multivariate Student's t-distribution](

Inherits From: Distribution

tfp.distributions.MultivariateStudentTLinearOperator(
    df, loc, scale, validate_args=False, allow_nan_stats=True,
    name='MultivariateStudentTLinearOperator'
)

en.wikipedia.org/wiki/Multivariate_t-distribution) on R^k.

Mathematical Details

The probability density function (pdf) is,

pdf(x; df, loc, Sigma) = (1 + ||y||**2 / df)**(-0.5 (df + k)) / Z
where,
y = inv(Sigma) (x - loc)
Z = abs(det(Sigma)) sqrt(df pi)**k Gamma(0.5 df) / Gamma(0.5 (df + k))

where:

• df is a positive scalar.
• loc is a vector in R^k,
• Sigma is a positive definite shape' matrix inR^{k x k}, parameterized asscale @ scale.T` in this class,
• Z denotes the normalization constant, and,
• ||y||**2 denotes the squared Euclidean norm of y.

The Multivariate Student's t-distribution distribution is a member of the location-scale family, i.e., it can be constructed as,

X ~ MultivariateT(loc=0, scale=1)   # Identity scale, zero shift.
Y = scale @ X + loc

Examples

tfd = tfp.distributions

# Initialize a single 3-variate Student's t.
df = 3.
loc = [1., 2, 3]
scale = [[ 0.6,  0. ,  0. ],
         [ 0.2,  0.5,  0. ],
         [ 0.1, -0.3,  0.4]]
sigma = tf.matmul(scale, scale, adjoint_b=True)
# ==> [[ 0.36,  0.12,  0.06],
#      [ 0.12,  0.29, -0.13],
#      [ 0.06, -0.13,  0.26]]

mvt = tfd.MultivariateStudentTLinearOperator(
    df=df,
    loc=loc,
    scale=tf.linalg.LinearOperatorLowerTriangular(scale))

# Covariance is closely related to the sigma matrix (for df=3, it is 3x of the
# sigma matrix).

mvt.covariance().eval()
# ==> [[ 1.08,  0.36,  0.18],
#      [ 0.36,  0.87, -0.39],
#      [ 0.18, -0.39,  0.78]]

# Compute the pdf of an`R^3` observation; return a scalar.
mvt.prob([-1., 0, 1]).eval()  # shape: []

#### Args:


* <b>`df`</b>: A positive floating-point `Tensor`. Has shape `[B1, ..., Bb]` where `b
  >= 0`.
* <b>`loc`</b>: Floating-point `Tensor`. Has shape `[B1, ..., Bb, k]` where `k` is
  the event size.
* <b>`scale`</b>: Instance of `LinearOperator` with a floating `dtype` and shape
  `[B1, ..., Bb, k, k]`.
* <b>`validate_args`</b>: Python `bool`, default `False`. Whether to validate input
  with asserts. If `validate_args` is `False`, and the inputs are invalid,
  correct behavior is not guaranteed.
* <b>`allow_nan_stats`</b>: Python `bool`, default `True`. If `False`, raise an
  exception if a statistic (e.g. mean/variance/etc...) is undefined for
  any batch member If `True`, batch members with valid parameters leading
  to undefined statistics will return NaN for this statistic.
* <b>`name`</b>: The name to give Ops created by the initializer.


#### Attributes:

* <b>`allow_nan_stats`</b>:   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.

* <b>`batch_shape`</b>:   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.

* <b>`df`</b>:   The degrees of freedom of the distribution.

  This controls the degrees of freedom of the distribution. The tails of the
  distribution get more heavier the smaller `df` is. As `df` goes to
  infinitiy, the distribution approaches the Multivariate Normal with the same
  `loc` and `scale`.

* <b>`dtype`</b>:   The `DType` of `Tensor`s handled by this `Distribution`.
* <b>`event_shape`</b>:   Shape of a single sample from a single batch as a `TensorShape`.

  May be partially defined or unknown.

* <b>`loc`</b>:   The location parameter of the distribution.

  `loc` applies an elementwise shift to the distribution.

  ```none
  X ~ MultivariateT(loc=0, scale=1)   # Identity scale, zero shift.
  Y = scale @ X + loc
  ```

* <b>`name`</b>:   Name prepended to all ops created by this `Distribution`.
* <b>`name_scope`</b>:   Returns a `tf.name_scope` instance for this class.
* <b>`parameters`</b>:   Dictionary of parameters used to instantiate this `Distribution`.
* <b>`reparameterization_type`</b>:   Describes how samples from the distribution are reparameterized.

  Currently this is one of the static instances
  `tfd.FULLY_REPARAMETERIZED` or `tfd.NOT_REPARAMETERIZED`.

* <b>`scale`</b>:   The scale parameter of the distribution.

  `scale` applies an affine scale to the distribution.

>     X ~ MultivariateT(loc=0, scale=1)   # Identity scale, zero shift.
>     Y = scale @ X + loc

* <b>`submodules`</b>:   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
>     assert list(a.submodules) == [b, c]
>     assert list(b.submodules) == [c]
>     assert list(c.submodules) == []

* <b>`trainable_variables`</b>:   Sequence of trainable variables owned by this module and its submodules.

  Note: this method uses reflection to find variables on the current instance
  and submodules. For performance reasons you may wish to cache the result
  of calling this method if you don't expect the return value to change.

* <b>`validate_args`</b>:   Python `bool` indicating possibly expensive checks are enabled.
* <b>`variables`</b>:   Sequence of variables owned by this module and its submodules.

  Note: this method uses reflection to find variables on the current instance
  and submodules. For performance reasons you may wish to cache the result
  of calling this method if you don't expect the return value to change.


#### Raises:


* <b>`TypeError`</b>: if not `scale.dtype.is_floating`.
* <b>`ValueError`</b>: if not `scale.is_non_singular`.

## Methods

<h3 id="__getitem__"><code>__getitem__</code></h3>

<a target="_blank" href="https://github.com/tensorflow/probability/blob/v0.9.0/tensorflow_probability/python/distributions/distribution.py#L623-L650">View source</a>

```python
__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  # => [2]
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape  # => [4, 2, 3, 1]
mvn2.event_shape  # => [2]

Args:

• slices: slices from the [] operator

Returns:

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

__iter__

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.

Additional documentation from MultivariateStudentTLinearOperator:

The covariance for Multivariate Student's t equals

scale @ scale.T * df / (df - 2), when df > 2
infinity, when 1 < df <= 2
NaN, when df <= 1

If self.allow_nan_stats=False, then an exception will be raised rather than returning NaN.

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.

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 (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.

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_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.

mean

View source

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

Mean.

Additional documentation from MultivariateStudentTLinearOperator:

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

mode

View source

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

Mode.

param_shapes

View source

@classmethod
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

@classmethod
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 or tfp.util.SeedStream instance, for seeding PRNG.
• 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.

Additional documentation from MultivariateStudentTLinearOperator:

The standard deviation for Student's T equals

sqrt(diag(scale @ scale.T)) * 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.
• **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.

Additional documentation from MultivariateStudentTLinearOperator:

The variance for Student's T equals

diag(scale @ scale.T) * df / (df - 2), when df > 2
infinity, when 1 < df <= 2
NaN, when df <= 1

If self.allow_nan_stats=False, then an exception will be raised rather than returning NaN.

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().

with_name_scope

@classmethod
with_name_scope(
    cls, 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], 64]))
    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([8, 32]))
# ==> <tf.Tensor: ...>
mod.w
# ==> <tf.Variable ...'my_module/w:0'>

Args:

• method: The method to wrap.

Returns:

The original method wrapped such that it enters the module's name scope.