Quick Reference

A cheat sheet for conjugate model pairs to help you quickly find the right functions for your use case.

How to Use This Table

Find your likelihood (the distribution of your data)
Look up the corresponding model function from conjugate.models
Check the Functions column for:
Helper functions to extract sufficient statistics from raw data
Predictive functions for posterior predictive distributions
See examples for practical usage patterns

Note: Not all models have closed-form predictive distributions. When available, predictive functions follow the pattern <model>_predictive.

Coverage

This reference includes 26 conjugate model pairs with:

✅ Model functions for all 26 pairs (posterior updates)
✅ Helper functions for 25 pairs (raw data → sufficient statistics)
✅ Predictive functions for 18 pairs (posterior predictive distributions)

Predictive functions are available for the most commonly used models, particularly those based on Beta, Gamma, and Normal priors.

Note: For models without closed-form predictive distributions, see Unsupported Distributions for a guide on generating posterior predictive samples using sampling techniques.

Discrete Likelihoods

Likelihood	Prior / Posterior	Model	Functions	Examples
Binomial	Beta	`binomial_beta`	Helper: `binomial_beta_inputs` Predictive: `binomial_beta_predictive`	Binomial, Bandit, Bayesian Update, Vectorized
Bernoulli	Beta	`bernoulli_beta`	Helper: `bernoulli_beta_inputs` Predictive: `bernoulli_beta_predictive`	Raw Data Workflow
Negative Binomial	Beta	`negative_binomial_beta`	Helper: `negative_binomial_beta_inputs` Predictive: `negative_binomial_beta_predictive`	-
Geometric	Beta	`geometric_beta`	Helper: `geometric_beta_inputs` Predictive: `geometric_beta_predictive`	-
Hypergeometric	BetaBinomial	`hypergeometric_beta_binomial`	Helper: `hypergeometric_beta_binomial_inputs` Predictive: -	-
Categorical	Dirichlet	`categorical_dirichlet`	Helper: `categorical_dirichlet_inputs` Predictive: `categorical_dirichlet_predictive`	-
Multinomial	Dirichlet	`multinomial_dirichlet`	Helper: `multinomial_dirichlet_inputs` Predictive: `multinomial_dirichlet_predictive`	Raw Data Workflow
Poisson	Gamma	`poisson_gamma`	Helper: `poisson_gamma_inputs` Predictive: `poisson_gamma_predictive`	Bootstrap, Limit, Raw Data Workflow

Continuous Likelihoods

Likelihood	Prior / Posterior	Model	Functions	Examples
Exponential	Gamma	`exponential_gamma`	Helper: - Predictive: `exponential_gamma_predictive`	Thompson, Raw Data Workflow
Gamma (known shape)	Gamma	`gamma_known_shape`	Helper: `gamma_known_shape_inputs` Predictive: `gamma_known_shape_predictive`	-
Gamma (known rate)	GammaKnownRateProportional	`gamma_known_rate`	Helper: `gamma_known_rate_inputs` Predictive: -	-
Gamma	GammaProportional	`gamma`	Helper: `gamma_inputs` Predictive: -	-
Inverse Gamma (known rate)	Gamma	`inverse_gamma_known_rate`	Helper: `inverse_gamma_known_rate_inputs` Predictive: -	-
Normal (known variance)	Normal	`normal_known_variance`	Helper: `normal_known_variance_inputs` Predictive: `normal_known_variance_predictive`	Shortest Path
Normal (known precision)	Normal	`normal_known_precision`	Helper: `normal_known_precision_inputs` Predictive: `normal_known_precision_predictive`	-
Normal (known mean)	InverseGamma	`normal_known_mean`	Helper: `normal_known_mean_inputs` Predictive: `normal_known_mean_predictive`	-
Normal	NormalInverseGamma	`normal`	Helper: `normal_inputs` Predictive: `normal_predictive`	-
Linear Regression	NormalInverseGamma	`linear_regression`	Helper: `linear_regression_inputs` Predictive: `linear_regression_predictive`	Linear Regression
Log Normal	NormalInverseGamma	`log_normal`	Helper: `log_normal_inputs` Predictive: -	-
Uniform	Pareto	`uniform_pareto`	Helper: `uniform_pareto_inputs` Predictive: -	-
Pareto	Gamma	`pareto_gamma`	Helper: `pareto_gamma_inputs` Predictive: -	Unsupported Distributions
Beta	BetaProportional	`beta`	Helper: `beta_inputs` Predictive: -	Raw Data Workflow
Von Mises (known κ)	VonMisesKnownConcentration	`von_mises_known_concentration`	Helper: `von_mises_known_concentration_inputs` Predictive: -	-
Von Mises (known μ)	VonMisesKnownDirectionProportional	`von_mises_known_direction`	Helper: `von_mises_known_direction_inputs` Predictive: -	-
Weibull (known shape)	InverseGamma	`weibull_inverse_gamma_known_shape`	Helper: `weibull_inverse_gamma_known_shape_inputs` Predictive: -	-

Multivariate

Likelihood	Prior / Posterior	Model	Functions	Examples
Multivariate Normal (known cov)	MultivariateNormal	`multivariate_normal_known_covariance`	Helper: `multivariate_normal_known_covariance_inputs` Predictive: `multivariate_normal_known_covariance_predictive`	-
Multivariate Normal (known precision)	MultivariateNormal	`multivariate_normal_known_precision`	Helper: `multivariate_normal_known_precision_inputs` Predictive: `multivariate_normal_known_precision_predictive`	-
Multivariate Normal (known mean)	InverseWishart	`multivariate_normal_known_mean`	Helper: `multivariate_normal_known_mean_inputs` Predictive: -	-
Multivariate Normal	NormalInverseWishart	`multivariate_normal`	Helper: `multivariate_normal_inputs` Predictive: `multivariate_normal_predictive`	-