We say that $\pi(\theta)$ is a conjugate prior for $f(\theta \ |\ x)$ if the posterior, $\pi(\theta \ | \ x) \propto \pi(\theta) \ f(\theta \ | \ x )$, is from the same model family as the prior.
Common conjugate families include
- [[beta-binomial conjugate]]
- [[normal-normal conjugate]]
- [[inverse gamma-normal conjugate]]
- [[gamma-Poisson conjugate]]
- [[gamma-exponential conjugate]]