tfp.experimental.substrates.jax.distributions.MultivariateStudentTLinearOperator

The [Multivariate Student's t-distribution](

Inherits From: Distribution

https://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: []

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<tr><th colspan="2"><h2 class="add-link">Args</h2></th></tr>

<tr>
<td>
`df`
</td>
<td>
A positive floating-point `Tensor`. Has shape `[B1, ..., Bb]` where `b
>= 0`.
</td>
</tr><tr>
<td>
`loc`
</td>
<td>
Floating-point `Tensor`. Has shape `[B1, ..., Bb, k]` where `k` is
the event size.
</td>
</tr><tr>
<td>
`scale`
</td>
<td>
Instance of `LinearOperator` with a floating `dtype` and shape
`[B1, ..., Bb, k, k]`.
</td>
</tr><tr>
<td>
`validate_args`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`allow_nan_stats`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`name`
</td>
<td>
The name to give Ops created by the initializer.
</td>
</tr>
</table>



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<tr><th colspan="2"><h2 class="add-link">Raises</h2></th></tr>

<tr>
<td>
`TypeError`
</td>
<td>
if not `scale.dtype.is_floating`.
</td>
</tr><tr>
<td>
`ValueError`
</td>
<td>
if not `scale.is_non_singular`.
</td>
</tr>
</table>





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<tr><th colspan="2"><h2 class="add-link">Attributes</h2></th></tr>

<tr>
<td>
`allow_nan_stats`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`batch_shape`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`df`
</td>
<td>
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`.
</td>
</tr><tr>
<td>
`dtype`
</td>
<td>
The `DType` of `Tensor`s handled by this `Distribution`.
</td>
</tr><tr>
<td>
`event_shape`
</td>
<td>
Shape of a single sample from a single batch as a `TensorShape`.

May be partially defined or unknown.
</td>
</tr><tr>
<td>
`loc`
</td>
<td>
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

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. scale 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

trainable_variables

validate_args Python bool indicating possibly expensive checks are enabled. variables

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.
**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

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.

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

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

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 (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 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 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.

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.

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.

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.
**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 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

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

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.

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.

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().

__getitem__

View source

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.stateless_normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.cholesky(cov)
loc = tf.random.stateless_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