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.