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Dirichlet distribution.
Inherits From: Distribution
tf.compat.v1.distributions.Dirichlet(
concentration, validate_args=False, allow_nan_stats=True, name='Dirichlet'
)
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.
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.
Warning: Some components of the samples can be zero due to finite precision.
This happens more often when some of the concentrations are very small.
Make sure to round the samples to np.finfo(dtype).tiny before computing the
density.
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
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)
concentration: Positive floating-point Tensor indicating mean number
of class occurrences; aka "alpha". Implies self.dtype, and
self.batch_shape, self.event_shape, i.e., if
concentration.shape = [N1, N2, ..., Nm, k] then
batch_shape = [N1, N2, ..., Nm] and
event_shape = [k].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.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.
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.
concentration: Concentration parameter; expected counts for that coordinate.
dtype: The DType of Tensors handled by this Distribution.
event_shape: Shape of a single sample from a single batch as a TensorShape.
May be partially defined or 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.
total_concentration: Sum of last dim of concentration parameter.
validate_args: Python bool indicating possibly expensive checks are enabled.
batch_shape_tensorbatch_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.cdfcdf(
value, name='cdf'
)
Cumulative distribution function.
Given random variable X, the cumulative distribution function cdf is:
cdf(x) := P[X <= x]
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.copycopy(
**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).covariancecovariance(
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_entropycross_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: tfp.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.entropyentropy(
name='entropy'
)
Shannon entropy in nats.
event_shape_tensorevent_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_batchis_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_eventis_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_divergencekl_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: tfp.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_cdflog_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.
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_problog_prob(
value, name='log_prob'
)
Log probability density/mass function.
Additional documentation from Dirichlet:
Note: value must be a non-negative tensor with
dtype self.dtype and be in the (self.event_shape() - 1)-simplex, i.e.,
tf.reduce_sum(value, -1) = 1. It must have a shape compatible with
self.batch_shape() + self.event_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_functionlog_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.
meanmean(
name='mean'
)
Mean.
modemode(
name='mode'
)
Mode.
Additional documentation from Dirichlet:
Note: The mode is undefined when any concentration <= 1. If
self.allow_nan_stats is True, NaN is used for undefined modes. If
self.allow_nan_stats is False an exception is raised when one or more
modes are undefined.
param_shapes@classmethod
param_shapes(
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@classmethod
param_static_shapes(
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.probprob(
value, name='prob'
)
Probability density/mass function.
Additional documentation from Dirichlet:
Note: value must be a non-negative tensor with
dtype self.dtype and be in the (self.event_shape() - 1)-simplex, i.e.,
tf.reduce_sum(value, -1) = 1. It must have a shape compatible with
self.batch_shape() + self.event_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.quantilequantile(
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.samplesample(
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.stddevstddev(
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_functionsurvival_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.
variancevariance(
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().