Conjugate Priors

When the posterior is in the same family as the prior, the prior is called conjugate to the likelihood. Conjugate analysis gives closed-form posteriors and turns Bayesian updating into mechanical parameter arithmetic.

The most common conjugate pairs:

Beta is conjugate to Bernoulli/Binomial.Gamma is conjugate to Poisson and Exponential rates.Normal-Inverse-Gamma is conjugate to Normal with unknown mean and variance.Dirichlet is conjugate to Multinomial.

Why bother? Closed-form updates are fast and sanity-checkable, no MCMC needed. Even when your final model uses flexible priors and numerical sampling, a conjugate baseline gives you intuition for what the data are saying and a check on the more complex code.

Conjugate models are especially handy in production systems where Bayesian updates need to happen on every event — A/B test dashboards, click-through rate estimators, multi-armed bandits.