Specialty › Bayesian

Proportion (Beta-Binomial)

A Beta-Binomial conjugate model gives the Bayesian posterior for a single proportion, with a credible interval.

What is Proportion (Beta-Binomial)?

The Beta distribution is the conjugate prior for the binomial likelihood: starting from Beta(α, β) and observing s successes in n trials, the posterior is exactly Beta(α + s, β + n − s). No MCMC, no approximation — the posterior is closed-form.

Common priors: Beta(1, 1) = uniform (every proportion equally likely a priori — the 'no information' default); Beta(0.5, 0.5) = Jeffreys (invariant to reparametrisation, common in modern Bayesian practice); Beta(α, β) with α + β small and α/(α + β) = expected value when you have prior beliefs but limited confidence.

vs frequentist binomial test: the Bayesian CrI (equal-tail) is interpretable as 'the true π is in this range with probability 0.95', not the awkward 'in repeated sampling the CI would cover the truth 95% of the time'. With a uniform prior the CrI matches the Bayesian Wilson CI numerically.

When should I use Proportion (Beta-Binomial)?

Estimating a single proportion when you want a probability statement about the parameter.
When you have genuine prior information (historical data, expert elicitation) you'd like to fold in.
Reporting in fields that prefer Bayesian framings (decision analysis, machine learning).

What data does it need?

Binary column + prior Beta(α, β).

What does it report?

Posterior Beta parameters, mean, SD, equal-tail 95% credible interval.

What does it assume?

Independent Bernoulli trials.
Prior is a genuine reflection of beliefs or a deliberate 'uninformative' choice.

Formula

posterior π ~ Beta(α + s, β + n − s)

How do I interpret the result?

With a uniform prior and modest n, the CrI is essentially the Bayesian analogue of the Wilson interval — very close numerically, very different interpretively.

References

Gelman et al. (2013). Bayesian Data Analysis, 3rd ed. — Ch. 2.