How the distributions are updated
When you choose a beta distribution like the following as the prior distribution for the parameter \(p\) , which is the success probability of Bernoulli trials.
$$ \mbox{Beta}(\alpha, \beta) $$
Let \(n\) the total trials and \(k\) the success trials. Then the poterior distribution is also a beta distribution, and the \(\alpha\) and \(\beta\) are updated like this.
$$ \mbox{Beta}(\alpha + k, \beta + n - k) $$
The Bayesian estimation of binomial distribution is that simple.
