contrib.distributions.DirichletMultinomial
tf.contrib.distributions.DirichletMultinomial
class tf.contrib.distributions.DirichletMultinomial
Defined in tensorflow/python/ops/distributions/dirichlet_multinomial.py
.
See the guide: Statistical Distributions (contrib) > Multivariate distributions
Dirichlet-Multinomial compound distribution.
The Dirichlet-Multinomial distribution is parameterized by a (batch of) length-k
concentration
vectors (k > 1
) and a total_count
number of trials, i.e., the number of trials per draw from the DirichletMultinomial. It is defined over a (batch of) length-k
vector counts
such that tf.reduce_sum(counts, -1) = total_count
. The Dirichlet-Multinomial is identically the Beta-Binomial distribution when k = 2
.
Mathematical Details
The Dirichlet-Multinomial is a distribution over k
-class counts, i.e., a length-k
vector of non-negative integer counts = n = [n_0, ..., n_{k-1}]
.
The probability mass function (pmf) is,
pmf(n; alpha, N) = Beta(alpha + n) / (prod_j n_j!) / Z Z = Beta(alpha) / N!
where:
-
concentration = alpha = [alpha_0, ..., alpha_{k-1}]
,alpha_j > 0
, -
total_count = N
,N
a positive integer, -
N!
isN
factorial, and, -
Beta(x) = prod_j Gamma(x_j) / Gamma(sum_j x_j)
is the multivariate beta function, and, -
Gamma
is the gamma function.
Dirichlet-Multinomial is a compound distribution, i.e., its samples are generated as follows.
- Choose class probabilities:
probs = [p_0,...,p_{k-1}] ~ Dir(concentration)
- Draw integers:
counts = [n_0,...,n_{k-1}] ~ Multinomial(total_count, probs)
The last concentration
dimension parametrizes a single Dirichlet-Multinomial distribution. When calling distribution functions (e.g., dist.prob(counts)
), concentration
, total_count
and counts
are broadcast to the same shape. The last dimension of of counts
corresponds single Dirichlet-Multinomial distributions.
Distribution parameters are automatically broadcast in all functions; see examples for details.
Examples
alpha = [1, 2, 3] n = 2 dist = DirichletMultinomial(n, alpha)
Creates a 3-class distribution, with the 3rd class is most likely to be drawn. The distribution functions can be evaluated on counts.
# counts same shape as alpha. counts = [0, 0, 2] dist.prob(counts) # Shape [] # alpha will be broadcast to [[1, 2, 3], [1, 2, 3]] to match counts. counts = [[1, 1, 0], [1, 0, 1]] dist.prob(counts) # Shape [2] # alpha will be broadcast to shape [5, 7, 3] to match counts. counts = [[...]] # Shape [5, 7, 3] dist.prob(counts) # Shape [5, 7]
Creates a 2-batch of 3-class distributions.
alpha = [[1, 2, 3], [4, 5, 6]] # Shape [2, 3] n = [3, 3] dist = DirichletMultinomial(n, alpha) # counts will be broadcast to [[2, 1, 0], [2, 1, 0]] to match alpha. counts = [2, 1, 0] dist.prob(counts) # Shape [2]
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 prior 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.
total_count
Number of trials used to construct a sample.
validate_args
Python bool
indicating possibly expensive checks are enabled.
Methods
__init__
__init__( total_count, concentration, validate_args=False, allow_nan_stats=True, name='DirichletMultinomial' )
Initialize a batch of DirichletMultinomial distributions.
Args:
-
total_count
: Non-negative floating point tensor, whose dtype is the same asconcentration
. The shape is broadcastable to[N1,..., Nm]
withm >= 0
. Defines this as a batch ofN1 x ... x Nm
different Dirichlet multinomial distributions. Its components should be equal to integer values. -
concentration
: Positive floating point tensor, whose dtype is the same asn
with shape broadcastable to[N1,..., Nm, k]
m >= 0
. Defines this as a batch ofN1 x ... x Nm
differentk
class Dirichlet multinomial distributions. -
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.
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.
Additional documentation from DirichletMultinomial
:
The covariance for each batch member is defined as the following:
Var(X_j) = n * alpha_j / alpha_0 * (1 - alpha_j / alpha_0) * (n + alpha_0) / (1 + alpha_0)
where concentration = alpha
and total_concentration = alpha_0 = sum_j alpha_j
.
The covariance between elements in a batch is defined as:
Cov(X_i, X_j) = -n * alpha_i * alpha_j / alpha_0 ** 2 * (n + alpha_0) / (1 + alpha_0)
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.
Additional documentation from DirichletMultinomial
:
For each batch of counts, value = [n_0, ..., n_{k-1}]
, P[value]
is the probability that after sampling self.total_count
draws from this Dirichlet-Multinomial distribution, the number of draws falling in class j
is n_j
. Since this definition is exchangeable; different sequences have the same counts so the probability includes a combinatorial coefficient.
Note:value
must be a non-negative tensor with dtypeself.dtype
, have no fractional components, and such thattf.reduce_sum(value, -1) = self.total_count
. Its shape must be broadcastable withself.concentration
andself.total_count
.
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.
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.
Additional documentation from DirichletMultinomial
:
For each batch of counts, value = [n_0, ..., n_{k-1}]
, P[value]
is the probability that after sampling self.total_count
draws from this Dirichlet-Multinomial distribution, the number of draws falling in class j
is n_j
. Since this definition is exchangeable; different sequences have the same counts so the probability includes a combinatorial coefficient.
Note:value
must be a non-negative tensor with dtypeself.dtype
, have no fractional components, and such thattf.reduce_sum(value, -1) = self.total_count
. Its shape must be broadcastable withself.concentration
andself.total_count
.
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
.
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/DirichletMultinomial