Beta
Inherits From: Distribution
tf.contrib.distributions.Beta
tf.distributions.Beta
Defined in tensorflow/python/ops/distributions/beta.py
.
See the guide: Statistical Distributions (contrib) > Univariate (scalar) distributions
Beta distribution.
The Beta distribution is defined over the (0, 1)
interval using parameters concentration1
(aka "alpha") and concentration0
(aka "beta").
The probability density function (pdf) is,
pdf(x; alpha, beta) = x**(alpha - 1) (1 - x)**(beta - 1) / Z Z = Gamma(alpha) Gamma(beta) / Gamma(alpha + beta)
where:
concentration1 = alpha
,concentration0 = beta
,Z
is the normalization constant, and,Gamma
is the gamma function.The concentration parameters represent mean total counts of a 1
or a 0
, i.e.,
concentration1 = alpha = mean * total_concentration concentration0 = beta = (1. - mean) * total_concentration
where mean
in (0, 1)
and total_concentration
is a positive real number representing a mean total_count = concentration1 + concentration0
.
Distribution parameters are automatically broadcast in all functions; see examples for details.
# Create a batch of three Beta distributions. alpha = [1, 2, 3] beta = [1, 2, 3] dist = Beta(alpha, beta) dist.sample([4, 5]) # Shape [4, 5, 3] # `x` has three batch entries, each with two samples. x = [[.1, .4, .5], [.2, .3, .5]] # Calculate the probability of each pair of samples under the corresponding # distribution in `dist`. dist.prob(x) # Shape [2, 3]
# Create batch_shape=[2, 3] via parameter broadcast: alpha = [[1.], [2]] # Shape [2, 1] beta = [3., 4, 5] # Shape [3] dist = Beta(alpha, beta) # alpha broadcast as: [[1., 1, 1,], # [2, 2, 2]] # beta broadcast as: [[3., 4, 5], # [3, 4, 5]] # batch_Shape [2, 3] dist.sample([4, 5]) # Shape [4, 5, 2, 3] x = [.2, .3, .5] # x will be broadcast as [[.2, .3, .5], # [.2, .3, .5]], # thus matching batch_shape [2, 3]. dist.prob(x) # Shape [2, 3]
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.
allow_nan_stats
: Python bool
.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.
batch_shape
: TensorShape
, possibly unknown.concentration0
Concentration parameter associated with a 0
outcome.
concentration1
Concentration parameter associated with a 1
outcome.
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.
event_shape
: TensorShape
, possibly unknown.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
.
An instance of ReparameterizationType
.
total_concentration
Sum of concentration parameters.
validate_args
Python bool
indicating possibly expensive checks are enabled.
__init__
__init__( concentration1=None, concentration0=None, validate_args=False, allow_nan_stats=True, name='Beta' )
Initialize a batch of Beta distributions.
concentration1
: Positive floating-point Tensor
indicating mean number of successes; aka "alpha". Implies self.dtype
and self.batch_shape
, i.e., concentration1.shape = [N1, N2, ..., Nm] = self.batch_shape
.concentration0
: Positive floating-point Tensor
indicating mean number of failures; aka "beta". Otherwise has same semantics as concentration1
.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.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.
name
: name to give to the opbatch_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]
Additional documentation from Beta
:
Note:x
must have dtypeself.dtype
and be in[0, 1].
It must have a shape compatible withself.batch_shape()
.
value
: float
or double
Tensor
.name
: Python str
prepended to names of ops created by this function.cdf
: a Tensor
of shape sample_shape(x) + self.batch_shape
with values of type self.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.
**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)
.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.
name
: Python str
prepended to names of ops created by this function.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
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)
, (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
.
other
: tf.distributions.Distribution
instance.name
: Python str
prepended to names of ops created by this function.cross_entropy
: self.dtype
Tensor
with shape [B1, ..., Bn]
representing n
different calculations of (Shanon) cross entropy.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
.
name
: name to give to the opevent_shape
: Tensor
.is_scalar_batch
is_scalar_batch(name='is_scalar_batch')
Indicates that batch_shape == []
.
name
: Python str
prepended to names of ops created by this function.is_scalar_batch
: bool
scalar Tensor
.is_scalar_event
is_scalar_event(name='is_scalar_event')
Indicates that event_shape == []
.
name
: Python str
prepended to names of ops created by this function.is_scalar_event
: bool
scalar Tensor
.kl_divergence
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 (Shanon) cross entropy, and H[.]
denotes (Shanon) entropy.
other
: tf.distributions.Distribution
instance.name
: Python str
prepended to names of ops created by this function.kl_divergence
: self.dtype
Tensor
with shape [B1, ..., Bn]
representing n
different calculations of the Kullback-Leibler divergence.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
.
Additional documentation from Beta
:
Note:x
must have dtypeself.dtype
and be in[0, 1].
It must have a shape compatible withself.batch_shape()
.
value
: float
or double
Tensor
.name
: Python str
prepended to names of ops created by this function.logcdf
: a Tensor
of shape sample_shape(x) + self.batch_shape
with values of type self.dtype
.log_prob
log_prob( value, name='log_prob' )
Log probability density/mass function.
Additional documentation from Beta
:
Note:x
must have dtypeself.dtype
and be in[0, 1].
It must have a shape compatible withself.batch_shape()
.
value
: float
or double
Tensor
.name
: Python str
prepended to names of ops created by this function.log_prob
: a Tensor
of shape sample_shape(x) + self.batch_shape
with values of type self.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
.
value
: float
or double
Tensor
.name
: Python str
prepended to names of ops created by this function.Tensor
of shape sample_shape(x) + self.batch_shape
with values of type self.dtype
.
mean
mean(name='mean')
Mean.
mode
mode(name='mode')
Mode.
Additional documentation from Beta
:
Note: The mode is undefined whenconcentration1 <= 1
orconcentration0 <= 1
. Ifself.allow_nan_stats
isTrue
,NaN
is used for undefined modes. Ifself.allow_nan_stats
isFalse
an exception is raised when one or more modes are undefined.
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
.
sample_shape
: Tensor
or python list/tuple. Desired shape of a call to sample()
.name
: name to prepend ops with.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.
sample_shape
: TensorShape
or python list/tuple. Desired shape of a call to sample()
.dict
of parameter name to TensorShape
.
ValueError
: if sample_shape
is a TensorShape
and is not fully defined.prob
prob( value, name='prob' )
Probability density/mass function.
Additional documentation from Beta
:
Note:x
must have dtypeself.dtype
and be in[0, 1].
It must have a shape compatible withself.batch_shape()
.
value
: float
or double
Tensor
.name
: Python str
prepended to names of ops created by this function.prob
: a Tensor
of shape sample_shape(x) + self.batch_shape
with values of type self.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
value
: float
or double
Tensor
.name
: Python str
prepended to names of ops created by this function.quantile
: a Tensor
of shape sample_shape(x) + self.batch_shape
with values of type self.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.
sample_shape
: 0D or 1D int32
Tensor
. Shape of the generated samples.seed
: Python integer seed for RNGname
: name to give to the op.samples
: a Tensor
with prepended dimensions sample_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
.
name
: Python str
prepended to names of ops created by this function.stddev
: Floating-point Tensor
with shape identical to batch_shape + event_shape
, i.e., the same shape as self.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).
value
: float
or double
Tensor
.name
: Python str
prepended to names of ops created by this function.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
.
name
: Python str
prepended to names of ops created by this function.variance
: Floating-point Tensor
with shape identical to batch_shape + event_shape
, i.e., the same shape as self.mean()
.
© 2018 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/distributions/Beta