mixle.stats.univariate.discrete.beta_binomial module

Beta-binomial distribution – an overdispersed binomial (Beta-binomial compound).

If the success probability of a binomial is itself Beta(a, b) distributed and integrated out, the count k in n trials follows the beta-binomial:

P(k; n, a, b) = C(n, k) B(k + a, n - k + b) / B(a, b), k = 0, …, n,

with mean n a/(a+b) and a variance inflated over the binomial by the intra-class correlation rho = 1/(a+b+1). It is the standard model for overdispersed bounded counts (clustered trials, batch defect rates). The number of trials n is a fixed, known parameter; a and b are estimated by moments – rho from the dispersion and the mean fixing a/(a+b).

class BetaBinomialDistribution(n, a, b, name=None, keys=None)[source]

Bases: SequenceEncodableProbabilityDistribution

Beta-binomial distribution over {0, ..., n} with shape parameters a, b > 0.

Parameters:
density(x)[source]

Return the probability mass at a single count x.

Parameters:

x (int)

Return type:

float

log_density(x)[source]

Return the log probability mass at x (-inf outside {0, ..., n}).

Parameters:

x (int)

Return type:

float

seq_log_density(x)[source]

Return vectorized log-mass for an array of counts.

Parameters:

x (ndarray)

Return type:

ndarray

sampler(seed=None)[source]

Return a sampler for drawing counts from this distribution.

Parameters:

seed (int | None)

Return type:

BetaBinomialSampler

estimator(pseudo_count=None)[source]

Return a method-of-moments estimator for a, b at the fixed number of trials n.

Parameters:

pseudo_count (float | None)

Return type:

BetaBinomialEstimator

dist_to_encoder()[source]

Return the data encoder used by this distribution for vectorized methods.

Return type:

BetaBinomialDataEncoder

class BetaBinomialSampler(dist, seed=None)[source]

Bases: DistributionSampler

Draw counts as p ~ Beta(a, b) then k ~ Binomial(n, p).

Parameters:

  • dist (BetaBinomialDistribution)

  • seed (int | None)

sample(size=None)[source]

Draw one count or an array of iid counts.

Parameters:

size (int | None)

Return type:

int | ndarray

class BetaBinomialAccumulator(name=None, keys=None)[source]

Bases: SequenceEncodableStatisticAccumulator

Accumulate weighted first and second moments for beta-binomial estimation.

Parameters:

  • name (str | None)

  • keys (str | None)

update(x, weight, estimate)[source]

Accumulate weighted first and second moments for one count.

Parameters:

  • x (int)

  • weight (float)

  • estimate (BetaBinomialDistribution | None)

Return type:

None

initialize(x, weight, rng)[source]

Initialize statistics from one count.

Parameters:
Return type:

None

seq_update(x, weights, estimate)[source]

Accumulate weighted moments from encoded counts.

Parameters:

  • x (ndarray)

  • weights (ndarray)

  • estimate (BetaBinomialDistribution | None)

Return type:

None

seq_initialize(x, weights, rng)[source]

Initialize statistics from encoded counts.

Parameters:
Return type:

None

combine(suff_stat)[source]

Merge another beta-binomial sufficient-statistic tuple.

Parameters:

suff_stat (tuple[float, float, float])

Return type:

BetaBinomialAccumulator

value()[source]

Return accumulated sum, second moment sum, and count.

Return type:

tuple[float, float, float]

from_value(x)[source]

Replace accumulator contents from a sufficient-statistic tuple.

Parameters:

x (tuple[float, float, float])

Return type:

BetaBinomialAccumulator

key_merge(stats_dict)[source]

Merge keyed statistics into stats_dict when keys are configured.

Parameters:

stats_dict (dict[str, Any])

Return type:

None

key_replace(stats_dict)[source]

Replace this accumulator from keyed statistics when available.

Parameters:

stats_dict (dict[str, Any])

Return type:

None

acc_to_encoder()[source]

Return the encoder used by this accumulator.

Return type:

BetaBinomialDataEncoder

class BetaBinomialAccumulatorFactory(name=None, keys=None)[source]

Bases: StatisticAccumulatorFactory

Factory for BetaBinomialAccumulator.

Parameters:

  • name (str | None)

  • keys (str | None)

make()[source]

Create a fresh beta-binomial accumulator.

Return type:

BetaBinomialAccumulator

class BetaBinomialEstimator(n, min_conc=1.0e-6, max_conc=1.0e8, name=None, keys=None)[source]

Bases: ParameterEstimator

Method-of-moments estimator for the beta-binomial shape parameters.

Parameters:
accumulator_factory()[source]

Return an accumulator factory for beta-binomial moments.

Return type:

BetaBinomialAccumulatorFactory

estimate(nobs, suff_stat)[source]

Estimate beta-binomial shape parameters from weighted moments.

Parameters:
Return type:

BetaBinomialDistribution

class BetaBinomialDataEncoder[source]

Bases: DataSequenceEncoder

Encode beta-binomial counts as a float array.

seq_encode(x)[source]

Encode counts as a floating-point array.

Parameters:

x (Sequence[int])

Return type:

ndarray