Beta–binomial updating

With a Beta(a, b) prior for a success probability and s successes and f failures, the posterior is Beta(a + s, b + f) — the prior acts like a + b earlier observations.

\theta \sim \text{Beta}(a, b), \quad s \text{ successes}, f \text{ failures} \;\Rightarrow\; \theta \given \text{data} \sim \text{Beta}(a + s, \, b + f)

Conjugate update

$\theta$: unknown success probability
$a, b$: prior parameters (Beta(1, 1) is uniform); density proportional to $heta^{a-1}(1- heta)^{b-1}$
$s, f$: observed successes and failures

\E[\theta] = \frac{a}{a+b}, \qquad \text{mode} = \frac{a-1}{a+b-2} \; (a, b > 1), \qquad \Var(\theta) = \frac{ab}{(a+b)^2(a+b+1)}

Beta(a, b) summaries

$a, b$: parameters of the (posterior) Beta distribution

When to use

Estimating a rate (conversion, defect, response) with uncertainty, especially with small samples or when earlier evidence should count.
Sequential learning: today's posterior is tomorrow's prior.

Assumptions

Trials are independent with a common success probability (binomial likelihood).
The prior is chosen before seeing the data and stated openly; check sensitivity to it.

How to interpret

The posterior mean $(a + s)/(a + b + s + f)$ is a compromise between the prior mean and the observed proportion, weighted by $a + b$ versus $s + f$ . With lots of data the prior hardly matters.
The posterior is a full distribution: report its mean or median and a credible interval.

Common mistakes

Using a strong prior without justification, or hiding it.
Confusing the prior "pseudo-counts" with real data when reporting n.
Treating the posterior mean as the observed rate.

Example

12 successes in 40 trials. Uniform prior Beta(1, 1) → posterior Beta(13, 29): mean 0.310, mode 0.300, sd 0.070, 95% equal-tailed credible interval 0.181 to 0.455. A sceptical prior Beta(2, 8) (prior mean 0.2) → Beta(14, 36): mean 0.280, interval 0.166 to 0.411.