Univariate Families

This page is the practical catalog for scalar distributions in mixle.stats. The generated API reference lists every class and method; this page explains which family to choose, what support it assumes, and how scalar families compose into larger Mixle models.

The naming pattern is consistent:

• FooDistribution is a fitted probability model.

• FooEstimator is the object passed to optimize to fit that family.

• FooEnumerator exists only for families with finite or countable support where exact support traversal is implemented.

Most scalar families are re-exported from mixle.stats:

from mixle.inference import optimize
from mixle.stats import GammaEstimator, GaussianEstimator, PoissonEstimator

duration = optimize(durations, GammaEstimator(), out=None)
residual = optimize(residuals, GaussianEstimator(), out=None)
counts = optimize(event_counts, PoissonEstimator(), out=None)

Continuous Families

Family	Support	Use when
GaussianDistribution / GaussianEstimator	real line	symmetric residuals, measurement noise, simple continuous baselines.
StudentTDistribution / StudentTEstimator	real line	residuals have heavier tails than a Gaussian.
SkewNormalDistribution / SkewNormalEstimator	real line	residuals are continuous but asymmetric.
LaplaceDistribution / LaplaceEstimator	real line	sharper center and heavier tails than Gaussian.
LogisticDistribution / LogisticEstimator	real line	symmetric real-valued data with logistic tails.
UniformDistribution / UniformEstimator	bounded interval	only a finite range is known or a flat baseline is needed.
BetaDistribution / BetaEstimator	[0, 1]	proportions, probabilities, or bounded rates.
GammaDistribution / GammaEstimator	positive real	durations, magnitudes, waiting times, positive skew.
ExponentialDistribution / ExponentialEstimator	non-negative real	memoryless waiting-time baseline.
WeibullDistribution / WeibullEstimator	non-negative real	failure times and hazards that rise or fall with age.
LogGaussianDistribution / LogGaussianEstimator	positive real	multiplicative noise or log-normal-like magnitudes.
InverseGammaDistribution / InverseGammaEstimator	positive real	variance-like positive quantities and Bayesian scale components.
InverseGaussianDistribution / InverseGaussianEstimator	positive real	first-passage-time-like positive data.
HalfNormalDistribution / HalfNormalEstimator	non-negative real	magnitudes of zero-centered Gaussian errors.
RayleighDistribution / RayleighEstimator	non-negative real	radial magnitudes from two Gaussian components.
RicianDistribution / RicianEstimator	non-negative real	magnitude with nonzero signal plus Gaussian noise.
NakagamiDistribution / NakagamiEstimator	non-negative real	flexible fading or positive magnitude data.
ParetoDistribution / ParetoEstimator	tail above a threshold	heavy-tailed size, wealth, severity, or frequency data.
GeneralizedParetoDistribution / GeneralizedParetoEstimator	threshold exceedances	peaks-over-threshold extreme-value modeling.
GeneralizedExtremeValueDistribution / GeneralizedExtremeValueEstimator	real line with shape-dependent tail	block maxima or minima.
GumbelDistribution / GumbelEstimator	real line	light-tailed extreme-value baseline.
GeneralizedGaussianDistribution / GeneralizedGaussianEstimator	real line	symmetric residuals with tunable tail shape.
ExponentiallyModifiedGaussianDistribution / ExponentiallyModifiedGaussianEstimator	real line with right skew	Gaussian noise plus exponential delay.
TweedieDistribution / TweedieEstimator	power-variance support	compound-like continuous/count mass patterns.

Discrete Families

Family	Support	Use when
CategoricalDistribution / CategoricalEstimator	arbitrary labels	labels, tokens, classes, states, finite outcomes.
IntegerCategoricalDistribution / IntegerCategoricalEstimator	integer labels	dense integer-valued categories where integer encoders matter.
BernoulliDistribution / BernoulliEstimator	{0, 1}	binary events.
BinomialDistribution / BinomialEstimator	bounded counts	successes out of a fixed number of trials.
BetaBinomialDistribution / BetaBinomialEstimator	bounded counts	over-dispersed binomial counts.
PoissonDistribution / PoissonEstimator	non-negative integers	count data with mean close to variance.
NegativeBinomialDistribution / NegativeBinomialEstimator	non-negative integers	over-dispersed count data.
GeometricDistribution / GeometricEstimator	positive or non-negative waiting count	trials until success.
LogSeriesDistribution / LogSeriesEstimator	positive integers	rare-species or heavily right-skewed counts.
SkellamDistribution / SkellamEstimator	all integers	difference of two Poisson counts.
PointMassDistribution / PointMassEstimator	one value	deterministic fields, constants, or degenerate baselines.
IntegerUniformSpikeDistribution / IntegerUniformSpikeEstimator	integer support with spike behavior	integer-valued data with a prominent preferred value.

Choosing Between Similar Families

For real-valued residuals:

• start with GaussianEstimator;

• use StudentTEstimator when a few large residuals should not dominate;

• use SkewNormalEstimator when positive and negative deviations behave differently;

• use LaplaceEstimator when absolute-error-like behavior is more natural than squared-error-like behavior.

For positive durations:

• start with GammaEstimator;

• use ExponentialEstimator as a simple memoryless baseline;

• use WeibullEstimator when the hazard changes with age;

• use LogGaussianEstimator when multiplicative effects dominate.

For counts:

• start with PoissonEstimator;

• use NegativeBinomialEstimator when variance exceeds the mean;

• use BetaBinomialEstimator for bounded over-dispersed counts;

• use SkellamEstimator when the observation is a difference of counts.

For extremes:

• use GeneralizedExtremeValueEstimator for block maxima;

• use GeneralizedParetoEstimator for threshold exceedances;

• use Analysis Utilities for tail diagnostics before committing to a tail family.

Composition Example

Scalar families are often leaves in a larger record model:

from mixle.inference import optimize
from mixle.stats import (
    CategoricalEstimator,
    CompositeEstimator,
    GammaEstimator,
    NegativeBinomialEstimator,
)

rows = [
    ("login", 0.8, 2),
    ("purchase", 4.2, 5),
    ("login", 1.1, 1),
]

estimator = CompositeEstimator(
    (
        CategoricalEstimator(),       # event type
        GammaEstimator(),             # latency
        NegativeBinomialEstimator(),  # repeated attempts
    )
)

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

The same scalar leaf can appear inside a mixture, HMM emission, survival wrapper, transform, record field, or process model.

Enumeration

Finite and countable discrete families may expose enumerators. Use Enumeration and Ranking for top-k and support traversal. Use Operations when a continuous family must be quantized into finite support for downstream enumeration.

API Reference

The generated scalar-family modules live under:

• mixle.stats.univariate.continuous package;

• mixle.stats.univariate.discrete package.