Why are Conjugate Priors Useful 🌱

Suppose our prior follows a beta distribution $p\left(\theta \right)\sim Beta(\alpha ,\beta )$ and we have a binomial likelihood $p(x\mid \theta )$. $p(\theta \mid x)=p(x\mid \theta )p\left(\theta \right)$ 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 $p\left(\theta \right)$, just with the adjusted $k+\alpha ,n-k+\beta$ values.