Beta-Binomial Posterior Update

The Beta distribution is a conjugate prior for the Bernoulli likelihood.

Add the number of observed successes to the first parameter and failures to the second.

1. 1. Apply conjugacy

   For a $\text{Beta}(\alpha ,\beta )$ prior updated with $s$ successes and $f$ failures, the posterior is $\text{Beta}(\alpha +s,\beta +f)$.

2. 2. Plug in the observed data

   With $\alpha =2$, $\beta =3$, $s=7$, $f=3$, the posterior parameters follow directly.

   $$\text{Posterior}=\text{Beta}(2+7,3+3)=\text{Beta}(9,6)$$

Answer: Beta(9,6)