contrib.distributions.StudentT
tf.contrib.distributions.StudentT
class tf.contrib.distributions.StudentT
Defined in tensorflow/python/ops/distributions/student_t.py
.
See the guide: Statistical Distributions (contrib) > Univariate (scalar) distributions
Student's t-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
.
Examples
Examples of initialization of one or a batch of distributions.
# Define a single scalar Student t distribution. single_dist = tf.distributions.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 = tf.distributions.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 = tf.distributions.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)
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
: Pythonbool
.
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.
df
Degrees of freedom in these Student's t distribution(s).
dtype
The DType
of Tensor
s 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.
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
.
Returns:
An instance of ReparameterizationType
.
scale
Scaling factors of these Student's t distribution(s).
validate_args
Python bool
indicating possibly expensive checks are enabled.
Methods
__init__
__init__( df, loc, scale, validate_args=False, allow_nan_stats=True, name='StudentT' )
Construct Student's t distributions.
The distributions have degree of freedom df
, mean loc
, and scale scale
.
The parameters df
, loc
, and scale
must be shaped in a way that supports broadcasting (e.g. df + loc + scale
is a valid operation).
Args:
-
df
: Floating-pointTensor
. The degrees of freedom of the distribution(s).df
must contain only positive values. -
loc
: Floating-pointTensor
. The mean(s) of the distribution(s). -
scale
: Floating-pointTensor
. The scaling factor(s) for the distribution(s). Note thatscale
is not technically the standard deviation of this distribution but has semantics more similar to standard deviation than variance. -
validate_args
: Pythonbool
, defaultFalse
. WhenTrue
distribution parameters are checked for validity despite possibly degrading runtime performance. WhenFalse
invalid inputs may silently render incorrect outputs. -
allow_nan_stats
: Pythonbool
, defaultTrue
. WhenTrue
, statistics (e.g., mean, mode, variance) use the value "NaN
" to indicate the result is undefined. WhenFalse
, an exception is raised if one or more of the statistic's batch members are undefined. -
name
: Pythonstr
name prefixed to Ops created by this class.
Raises:
-
TypeError
: if loc and scale are different dtypes.
batch_shape_tensor
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
cdf( value, name='cdf' )
Cumulative distribution function.
Given random variable X
, the cumulative distribution function cdf
is:
cdf(x) := P[X <= x]
Args:
-
value
:float
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
-
cdf
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
copy
copy(**override_parameters_kwargs)
Creates a deep copy of the distribution.
Note: the copy distribution may continue to depend on the original initialization arguments.
Args:
**override_parameters_kwargs: String/value dictionary of initialization arguments to override with new values.
Returns:
-
distribution
: A new instance oftype(self)
initialized from the union of self.parameters and override_parameters_kwargs, i.e.,dict(self.parameters, **override_parameters_kwargs)
.
covariance
covariance(name='covariance')
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
: The name to give this op.
Returns:
-
covariance
: Floating-pointTensor
with shape[B1, ..., Bn, k', k']
where the firstn
dimensions are batch coordinates andk' = reduce_prod(self.event_shape)
.
entropy
entropy(name='entropy')
Shannon entropy in nats.
event_shape_tensor
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
is_scalar_batch(name='is_scalar_batch')
Indicates that batch_shape == []
.
Args:
-
name
: The name to give this op.
Returns:
-
is_scalar_batch
:bool
scalarTensor
.
is_scalar_event
is_scalar_event(name='is_scalar_event')
Indicates that event_shape == []
.
Args:
-
name
: The name to give this op.
Returns:
-
is_scalar_event
:bool
scalarTensor
.
log_cdf
log_cdf( value, name='log_cdf' )
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
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
-
logcdf
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
log_prob
log_prob( value, name='log_prob' )
Log probability density/mass function.
Args:
-
value
:float
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
-
log_prob
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
log_survival_function
log_survival_function( value, name='log_survival_function' )
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
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
Tensor
of shape sample_shape(x) + self.batch_shape
with values of type self.dtype
.
mean
mean(name='mean')
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
mode(name='mode')
Mode.
param_shapes
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 tosample()
. -
name
: name to prepend ops with.
Returns:
dict
of parameter name to Tensor
shapes.
param_static_shapes
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 tosample()
.
Returns:
dict
of parameter name to TensorShape
.
Raises:
-
ValueError
: ifsample_shape
is aTensorShape
and is not fully defined.
prob
prob( value, name='prob' )
Probability density/mass function.
Args:
-
value
:float
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
-
prob
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
quantile
quantile( value, name='quantile' )
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
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
-
quantile
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
sample
sample( sample_shape=(), seed=None, name='sample' )
Generate samples of the specified shape.
Note that a call to sample()
without arguments will generate a single sample.
Args:
-
sample_shape
: 0D or 1Dint32
Tensor
. Shape of the generated samples. -
seed
: Python integer seed for RNG -
name
: name to give to the op.
Returns:
-
samples
: aTensor
with prepended dimensionssample_shape
.
stddev
stddev(name='stddev')
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
: The name to give this op.
Returns:
-
stddev
: Floating-pointTensor
with shape identical tobatch_shape + event_shape
, i.e., the same shape asself.mean()
.
survival_function
survival_function( value, name='survival_function' )
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
ordouble
Tensor
. -
name
: The name to give this op.
Returns:
Tensor
of shape sample_shape(x) + self.batch_shape
with values of type self.dtype
.
variance
variance(name='variance')
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
: The name to give this op.
Returns:
-
variance
: Floating-pointTensor
with shape identical tobatch_shape + event_shape
, i.e., the same shape asself.mean()
.
© 2017 The TensorFlow Authors. All rights reserved.
Licensed under the Creative Commons Attribution License 3.0.
Code samples licensed under the Apache 2.0 License.
https://www.tensorflow.org/api_docs/python/tf/contrib/distributions/StudentT