Structured Statistical Families

This page covers distribution families whose observations are not scalar: records, tuples, sequences, sets, rankings, trees, graphs, vectors, matrices, and directional data. These are the families that make Mixle useful for heterogeneous data rather than only iid columns.

Combinators

Combinators turn smaller distributions into distributions over structured observations.

Family	Observation shape	Use when
CompositeDistribution / CompositeEstimator	tuple	fields are ordered by position.
RecordDistribution / RecordEstimator	named record	fields are dictionaries or schema-backed records.
DictRecordDistribution / DictRecordEstimator	dictionary record	dictionary shape should be explicit.
SequenceDistribution / SequenceEstimator	variable-length sequence	elements share a family and length may be modeled.
OptionalDistribution / OptionalEstimator	missing-or-present value	a field can be absent without dropping the whole record.
SelectDistribution / SelectEstimator	dispatch by type or field	different subfamilies apply to different observed cases.
ConditionalDistribution / ConditionalEstimator	conditional relation	a distribution depends on observed covariates.
TransformDistribution / TransformEstimator	deterministic transformed value	modeling scale differs from observation scale.
FiniteStochasticTransformDistribution	stochastic finite mapping	a latent finite source emits through a finite channel.
TruncatedDistribution / TruncatedEstimator	restricted support	legal or observed support is a subset of the base family.
CensoredDistribution / CensoredEstimator	censored observation	true value is partially hidden by thresholds or intervals.
SurvivalDistribution / SurvivalEstimator	event or censoring time	time-to-event data with censoring.
HurdleDistribution / HurdleEstimator	structural zero plus positive part	zero occurrence and positive magnitude are separate processes.
ZeroInflatedDistribution / ZeroInflatedEstimator	extra zeros	count data have more zeros than the base family explains.
WeightedDistribution / WeightedEstimator	weighted observation	examples carry frequency or importance weights.
IgnoredDistribution / IgnoredEstimator	ignored field	a field must pass through shape validation but not affect likelihood.
NullDistribution / NullEstimator	no information	neutral branch in a larger composition.

Combinators are how a heterogeneous row becomes one model:

from mixle.inference import optimize
from mixle.stats import CategoricalEstimator, GammaEstimator, RecordEstimator, field

estimator = RecordEstimator(
    (
        field("event", CategoricalEstimator()),
        field("duration", GammaEstimator()),
    )
)

model = optimize(records, estimator, out=None)

Multivariate And Matrix Families

Family	Observation	Use when
MultivariateGaussianDistribution / MultivariateGaussianEstimator	real vector	full-covariance Gaussian vector model.
DiagonalGaussianDistribution / DiagonalGaussianEstimator	real vector	independent Gaussian dimensions or high-dimensional baseline.
MultivariateStudentTDistribution / MultivariateStudentTEstimator	real vector	heavy-tailed vector residuals.
GaussianCopulaDistribution / GaussianCopulaEstimator	vector with copula dependence	separate marginal behavior from Gaussian dependence.
CompositionDistribution	compositional vector	parts constrained to a whole.
CategoricalMultinomialDistribution / MultinomialDistribution	finite-count vector	category counts from repeated trials.
IntegerMultinomialDistribution / IntegerMultinomialEstimator	integer-count vector	dense integer-coded multinomial counts.
DirichletMultinomialDistribution / DirichletMultinomialEstimator	count vector	over-dispersed multinomial counts.
MatrixNormalDistribution / MatrixNormalEstimator	matrix	row/column covariance structure.
WishartDistribution / WishartEstimator	positive-definite matrix	covariance-like random matrices.
InverseWishartDistribution / InverseWishartEstimator	positive-definite matrix	inverse covariance or Bayesian covariance prior.
LKJDistribution / LKJEstimator	correlation matrix	correlation priors or fitted correlation structure.

For full-covariance multivariate Gaussian fits, 0.6.2 improves the default numeric path in two ways. Weighted second moments are accumulated through a BLAS-backed matrix multiply instead of a naive tensor contraction, which keeps large mixture fits from spending most of their time in covariance assembly. The Cholesky path also has a minimal jitter fallback for nearly positive-definite float32 covariance estimates, so GPU or reduced-precision EM runs can recover from roundoff without changing the ordinary float64 fast path.

Directional Families

Directional observations live on circles, spheres, or orientation manifolds.

Family	Support	Use when
VonMisesDistribution / VonMisesEstimator	circle	circular angles.
WrappedNormalDistribution / WrappedNormalEstimator	circle	wrapped Gaussian-like angular data.
WrappedCauchyDistribution / WrappedCauchyEstimator	circle	heavier-tailed circular data.
VonMisesFisherDistribution / VonMisesFisherEstimator	sphere	directional vectors around a mean direction.
WatsonDistribution / WatsonEstimator	axial sphere	orientations where sign is not meaningful.
KentDistribution / KentEstimator	sphere	directional data with elliptical concentration.
BinghamDistribution / BinghamEstimator	antipodal directional support	axial orientation and shape.
ProjectedNormalDistribution / ProjectedNormalEstimator	sphere/circle	directions induced by projecting Gaussian vectors.

Sequences And Markov Families

The sequence package covers explicit Markov structure over observed states:

• MarkovChainDistribution and MarkovChainEstimator;

• IntegerMarkovChainDistribution and IntegerMarkovChainEstimator;

• MarkovTransform and SparseMarkovTransform families.

Use these when the observation is an observed-state sequence. Use HMMs and Latent Structure when the state path is hidden.

Sets

Set families model unordered finite collections:

• BernoulliSetDistribution and BernoulliSetEstimator;

• IntegerBernoulliSetDistribution and IntegerBernoulliSetEstimator;

• IntegerBernoulliEditDistribution and IntegerBernoulliEditEstimator;

• IntegerStepBernoulliEditDistribution and IntegerStepBernoulliEditEstimator.

Use set families when membership matters but order does not. Use edit families when the likelihood is naturally expressed as set changes from a reference.

Rankings And Pairwise Preferences

Ranking families model permutations, partial orderings, or pairwise outcomes:

• MallowsDistribution and MallowsEstimator;

• GeneralizedMallowsDistribution and GeneralizedMallowsEstimator;

• GeneralizedMallowsModelDistribution and GeneralizedMallowsModelEstimator;

• PlackettLuceDistribution and PlackettLuceEstimator;

• SpearmanRankingDistribution and SpearmanRankingEstimator;

• BradleyTerryDistribution and BradleyTerryEstimator;

• ThurstoneDistribution and ThurstoneEstimator;

• ThurstoneMostellerDistribution and ThurstoneMostellerEstimator;

• DavidsonDistribution and DavidsonEstimator;

• RaoKupperDistribution and RaoKupperEstimator;

• LowRankPermutationDistribution and LowRankPermutationEstimator;

• EwensDistribution and EwensEstimator;

• MatchingDistribution and MatchingEstimator.

Use these for preference data, rankings, pairwise comparison, matching, or ordering uncertainty. For consensus analysis outside a distribution family, see Analysis Utilities.

Trees And Graphs

Tree and graph families score structured graph-valued observations:

• ChowLiuTreeDistribution and ChowLiuTreeEstimator;

• IntegerChowLiuTreeDistribution and IntegerChowLiuTreeEstimator;

• SpanningTreeDistribution and SpanningTreeEstimator;

• ErdosRenyiGraphDistribution and ErdosRenyiGraphEstimator;

• StochasticBlockGraphDistribution and StochasticBlockGraphEstimator;

• RandomDotProductGraphDistribution and RandomDotProductGraphEstimator;

• KnowledgeGraphDistribution and KnowledgeGraphEstimator;

• KnowledgeGraphEnsemble and fit_knowledge_graph_ensemble;

• temporal graph grammar families including TemporalGraphGrammarDistribution and labeled, homophily, churning, latent, attributed, and latent-churning variants.

Use graph distributions when the graph itself is the observation. Use Relations when a graph algorithm is the decision layer after a model has produced edge or node scores.

Workflow

The safest workflow for structured families is:

1. Start with the simplest shape-preserving family.

2. Fit locally on a representative sample.

3. Inspect field-level or component-level likelihoods.

4. Add latent structure only when held-out scoring supports it.

5. Add enumeration, relation, or production constraints only after the probabilistic model is behaving sensibly.

API Reference

Generated reference pages:

• mixle.stats.combinator package;

• mixle.stats.multivariate package;

• mixle.stats.matrix package;

• mixle.stats.directional package;

• mixle.stats.sequences package;

• mixle.stats.sets package;

• mixle.stats.rankings package;

• mixle.stats.trees package;

• mixle.stats.graphs package.