Conjugate Prior

https://en.wikipedia.org/wiki/Conjugate_prior

To understand conjugate prior, we must first understand Bayesian Inference.

If, given a likelihood function $p(x\mid \theta )$, the posterior distribution $p(\theta \mid x)$ is in the same probability distribution family as the prior probability distribution $p(\theta )$, the prior and posterior are then called conjugate distributions with respect to that likelihood function.

The prior is called a conjugate prior for the likelihood function $p(x\mid \theta )$.