Why are Conjugate Priors Useful 🌱

Suppose our prior follows a beta distribution \(p(\theta) \sim Beta(\alpha,\beta)\) and we have a binomial likelihood \(p(x\mid\theta)\). \(p(\theta \mid x) = p(x \mid \theta) p(\theta)\) Since the beta is a conjugate prior for the binomial likelihood, we can show that \(p(\theta \mid x) \sim Beta(k+\alpha,n-k+\beta)\).

• We don’t need to actually calculate this \(p(\theta \mid x)\) with the computer since it just follows a known distribution and the MAP estimate for this distribution will be of the same form as the optimal estimate of \(p(\theta)\), just with the adjusted \(k+\alpha, \ n-k+\beta\) values.