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tfp.substrates.numpy.distributions.Empirical

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Empirical distribution.

Inherits From: Distribution

The Empirical distribution is parameterized by a [batch] multiset of samples. It describes the empirical measure (observations) of a variable.

Mathematical Details

The probability mass function (pmf) and cumulative distribution function (cdf) are

pmf(k; s1, ..., sn) = sum_i I(k)^{k == si} / n
I(k)^{k == si} == 1, if k == si, else 0.
cdf(k; s1, ..., sn) = sum_i I(k)^{k >= si} / n
I(k)^{k >= si} == 1, if k >= si, else 0.

Examples


# Initialize a empirical distribution with 4 scalar samples.
dist = Empirical(samples=[0., 1., 1., 2.])
dist.cdf(1.)
==> 0.75
dist.prob([0., 1.])
==> [0.25, 0.5] # samples will be broadcast to
                  [[0., 1., 1., 2.], [0., 1., 1., 2.]] to match event.

# Initialize a empirical distribution with a [2] batch of scalar samples.
dist = Empirical(samples=[[0., 1.], [1., 2.]])
dist.cdf([0., 2.])
==> [0.5, 1.]
dist.prob(0.)
==> [0.5, 0] # event will be broadcast to [0., 0.] to match samples.

# Initialize a empirical distribution with 4 vector-like samples.
dist = Empirical(samples=[[0., 0.], [0., 1.], [0., 1.], [1., 2.]],
                 event_ndims=1)
dist.cdf([0., 1.])
==> 0.75
dist.prob([[0., 1.], [1., 2.]])
==> [0.5, 0.25] # samples will be broadcast to shape [2, 4, 2] to match event.

# Initialize a empirical distribution with a [2] batch of vector samples.
dist = Empirical(samples=[[[0., 0.], [0., 1.]], [[0., 1.], [1., 2.]]],
                 event_ndims=1)
dist.cdf([[0., 0.], [0., 1.]])
==> [0.5, 0.5]
dist.prob([0., 1.])
==> [0.5, 1.] # event will be broadcast to shape [[0., 1.], [0., 1.]]
                to match samples.

samples Numeric Tensor of shape [B1, ..., Bk, S, E1, ..., En], k, n >= 0. Samples or batches of samples on which the distribution is based. The first k dimensions index into a batch of independent distributions. Length of S dimension determines number of samples in each multiset. The last n dimension represents samples for each distribution. n is specified by argument event_ndims.
event_ndims Python int32, default 0. number of dimensions for each event. When 0 this distribution has scalar samples. When 1 this distribution has vector-like samples.
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.

ValueError if the rank of samples is statically known and less than event_ndims + 1.

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.

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.

experimental_shard_axis_names The list or structure of lists of active shard axis names.
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 tfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.

samples Distribution parameter.
trainable_variables

validate_args Python bool indicating possibly expensive checks are enabled.
variables

Methods

batch_shape_tensor

View source

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

View source

Cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

cdf(x) := P[X <= x]

Args
value float or double Tensor.
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

Returns
cdf a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

compute_num_samples

View source

Compute and return the number of values in self.samples.

Returns
num_samples int32 Tensor containing the number of entries in self.samples. If self.samples has shape [..., S, E1, ..., Ee] where the E's are event dims, this method returns a Tensor whose values is S.

copy

View source

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 of type(self) initialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

covariance

View source

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 Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

Returns
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

View source

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), (Shannon) 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 Python str prepended to names of ops created by this function.

Returns
cross_entropy self.dtype Tensor with shape [B1, ..., Bn] representing