Beta Posterior Mean Calculator
Update a Beta prior with observed successes and failures to get the Bayesian posterior mean, mode and interval for a coin or conversion rate.
How it works
Posterior = Beta(α + successes, β + failures) · Mean = α′ ÷ (α′ + β′) · Mode = (α′ − 1) ÷ (α′ + β′ − 2)
The Beta distribution is the conjugate prior for a binomial proportion, so Bayesian updating is just addition: add your observed successes to α and failures to β. The posterior mean, α′ ÷ (α′ + β′), blends the prior with the data — with a uniform Beta(1,1) prior it becomes (successes + 1) ÷ (total + 2), Laplace's rule of succession, which sensibly pulls extreme observed rates toward 50% when data is scarce.
Worked example
Uniform prior Beta(1,1), then observe 7 heads and 3 tails. Posterior is Beta(8, 4). Mean = 8 ÷ 12 = 66.7%, notably gentler than the raw 7/10 = 70% because the prior still carries weight. As you flip more coins the posterior mean converges on the true rate.
Sources: Beta distribution — Wikipedia · Conjugate prior — Wikipedia
Frequently asked questions
Why use a posterior mean instead of the raw success rate?
The raw rate overreacts to small samples — 1 head in 1 flip reads as 100%. The posterior mean folds in a prior belief, so with a uniform prior 1/1 becomes 2/3 (66.7%), a far more sensible estimate until more data arrives.
What prior should I choose?
Beta(1,1) is uniform and assumes nothing. Beta(2,2) mildly favours 50% (good for a coin you expect to be fair). To encode a prior belief of roughly r% success, pick α and β with α ÷ (α+β) ≈ r, using larger values for stronger belief.
Is this credible interval the same as a confidence interval?
No. A credible interval is a genuine probability statement — there is about a 95% probability the true rate lies inside it, given the prior and data. A frequentist confidence interval makes a subtler claim about long-run coverage of the method.
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This calculator is for educational purposes. Double-check important results before acting on them.