Conjugate distribution

What is conjugate prior

In Bayesian statistics, if the posterior $p (θ | y)$ and the prior $p (θ)$ follow a same type of distribution (e.g. both follow the normal distribution), we will say this prior is conjugate for the likelihood $p (y | θ)$ .

Note

You can always say that a prior is a conjugate for a likelihood; but a posterior never has conjugates.

Why need conjugate prior

If the prior is conjugate for the likelihood, we can compute the posterior in closed form.

likelihood	conjugate prior
Bernoulli, binominal	Beta
Categorical distrbution	Dirichlet distribution (a multivariate generalization of the Beta distribution)
Poisson, exponential	Gamma
normal with known var	normal
normal with known mean	inverse-Gamma