Class Dirichlet
Inherits From: Distribution
Aliases:
- Class
tf.contrib.distributions.Dirichlet
- Class
tf.distributions.Dirichlet
Defined in tensorflow/python/ops/distributions/dirichlet.py
.
Dirichlet distribution.
The Dirichlet distribution is defined over the
(k-1)
-simplex using a positive,
length-k
vector concentration
(k > 1
). The Dirichlet is identically the
Beta distribution when k = 2
.
Mathematical Details
The Dirichlet is a distribution over the open (k-1)
-simplex, i.e.,
S^{k-1} = { (x_0, ..., x_{k-1}) in R^k : sum_j x_j = 1 and all_j x_j > 0 }.
The probability density function (pdf) is,
pdf(x; alpha) = prod_j x_j**(alpha_j - 1) / Z
Z = prod_j Gamma(alpha_j) / Gamma(sum_j alpha_j)
where:
x in S^{k-1}
, i.e., the(k-1)
-simplex,concentration = alpha = [alpha_0, ..., alpha_{k-1}]
,alpha_j > 0
,Z
is the normalization constant aka the multivariate beta function, and,Gamma
is the gamma function.
The concentration
represents mean total counts of class occurrence, i.e.,
concentration = alpha = mean * total_concentration
where mean
in S^{k-1}
and total_concentration
is a positive real number
representing a mean total count.
Distribution parameters are automatically broadcast in all functions; see examples for details.
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
Examples
import tensorflow_probability as tfp
tfd = tfp.distributions
# Create a single trivariate Dirichlet, with the 3rd class being three times
# more frequent than the first. I.e., batch_shape=[], event_shape=[3].
alpha = [1., 2, 3]
dist = tfd.Dirichlet(alpha)
dist.sample([4, 5]) # shape: [4, 5, 3]
# x has one sample, one batch, three classes:
x = [.2, .3, .5] # shape: [3]
dist.prob(x) # shape: []
# x has two samples from one batch:
x = [[.1, .4, .5],
[.2, .3, .5]]
dist.prob(x) # shape: [2]
# alpha will be broadcast to shape [5, 7, 3] to match x.
x = [[...]] # shape: [5, 7, 3]
dist.prob(x) # shape: [5, 7]
# Create batch_shape=[2], event_shape=[3]:
alpha = [[1., 2, 3],
[4, 5, 6]] # shape: [2, 3]
dist = tfd.Dirichlet(alpha)
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]
Compute the gradients of samples w.r.t. the parameters:
alpha = tf.constant([1.0, 2.0, 3.0])
dist = tfd.Dirichlet(alpha)
samples = dist.sample(5) # Shape [5, 3]
loss = tf.reduce_mean(tf.square(samples)) # Arbitrary loss function
# Unbiased stochastic gradients of the loss function
grads = tf.gradients(loss, alpha)
__init__
__init__(
concentration,
validate_args=False,
allow_nan_stats=True,
name='Dirichlet'
)
Initialize a batch of Dirichlet distributions. (deprecated)
Args:
concentration
: Positive floating-pointTensor
indicating mean number of class occurrences; aka "alpha". Impliesself.dtype
, andself.batch_shape
,self.event_shape
, i.e., ifconcentration.shape = [N1, N2, ..., Nm, k]
thenbatch_shape = [N1, N2, ..., Nm]
andevent_shape = [k]
.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.
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.
concentration
Concentration parameter; expected counts for that coordinate.
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.
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
.
total_concentration
Sum of last dim of concentration parameter.
validate_args
Python bool
indicating possibly expensive checks are enabled.
Methods
tf.distributions.Dirichlet.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
.
tf.distributions.Dirichlet.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
: Pythonstr
prepended to names of ops created by this function.
Returns:
cdf
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
tf.distributions.Dirichlet.copy
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 oftype(self)
initialized from the union of self.parameters and override_parameters_kwargs, i.e.,dict(self.parameters, **override_parameters_kwargs)
.
tf.distributions.Dirichlet.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
: Pythonstr
prepended to names of ops created by this function.
Returns:
covariance
: Floating-pointTensor
with shape[B1, ..., Bn, k', k']
where the firstn
dimensions are batch coordinates andk' = reduce_prod(self.event_shape)
.
tf.distributions.Dirichlet.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
.
Args:
other
:tfp.distributions.Distribution
instance.name
: Pythonstr
prepended to names of ops created by this function.
Returns:
cross_entropy
:self.dtype
Tensor
with shape[B1, ..., Bn]
representingn
different calculations of (Shanon) cross entropy.
tf.distributions.Dirichlet.entropy
entropy(name='entropy')
Shannon entropy in nats.
tf.distributions.Dirichlet.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
.
tf.distributions.Dirichlet.is_scalar_batch
is_scalar_batch(name='is_scalar_batch')
Indicates that batch_shape == []
.
Args:
name
: Pythonstr
prepended to names of ops created by this function.
Returns:
is_scalar_batch
:bool
scalarTensor
.
tf.distributions.Dirichlet.is_scalar_event
is_scalar_event(name='is_scalar_event')
Indicates that event_shape == []
.
Args:
name
: Pythonstr
prepended to names of ops created by this function.
Returns:
is_scalar_event
:bool
scalarTensor
.
tf.distributions.Dirichlet.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.
Args:
other
:tfp.distributions.Distribution
instance.name
: Pythonstr
prepended to names of ops created by this function.
Returns:
kl_divergence
:self.dtype
Tensor
with shape[B1, ..., Bn]
representingn
different calculations of the Kullback-Leibler divergence.
tf.distributions.Dirichlet.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
: Pythonstr
prepended to names of ops created by this function.
Returns:
logcdf
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
tf.distributions.Dirichlet.log_prob
log_prob(
value,
name='log_prob'
)
Log probability density/mass function.
Additional documentation from Dirichlet
:
Args:
value
:float
ordouble
Tensor
.name
: Pythonstr
prepended to names of ops created by this function.
Returns:
log_prob
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
tf.distributions.Dirichlet.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
: Pythonstr
prepended to names of ops created by this function.
Returns:
Tensor
of shape sample_shape(x) + self.batch_shape
with values of type
self.dtype
.
tf.distributions.Dirichlet.mean
mean(name='mean')
Mean.
tf.distributions.Dirichlet.mode
mode(name='mode')
Mode.
Additional documentation from Dirichlet
:
tf.distributions.Dirichlet.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.
tf.distributions.Dirichlet.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.
tf.distributions.Dirichlet.prob
prob(
value,
name='prob'
)
Probability density/mass function.
Additional documentation from Dirichlet
:
Args:
value
:float
ordouble
Tensor
.name
: Pythonstr
prepended to names of ops created by this function.
Returns:
prob
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
tf.distributions.Dirichlet.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
: Pythonstr
prepended to names of ops created by this function.
Returns:
quantile
: aTensor
of shapesample_shape(x) + self.batch_shape
with values of typeself.dtype
.
tf.distributions.Dirichlet.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 RNGname
: name to give to the op.
Returns:
samples
: aTensor
with prepended dimensionssample_shape
.
tf.distributions.Dirichlet.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
: Pythonstr
prepended to names of ops created by this function.
Returns:
stddev
: Floating-pointTensor
with shape identical tobatch_shape + event_shape
, i.e., the same shape asself.mean()
.
tf.distributions.Dirichlet.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
: Pythonstr
prepended to names of ops created by this function.
Returns:
Tensor
of shape sample_shape(x) + self.batch_shape
with values of type
self.dtype
.
tf.distributions.Dirichlet.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
.
Args:
name
: Pythonstr
prepended to names of ops created by this function.
Returns:
variance
: Floating-pointTensor
with shape identical tobatch_shape + event_shape
, i.e., the same shape asself.mean()
.