Conjugate update and MAP

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Problem

A Bernoulli success probability $p$ has prior $\operatorname{Beta}(2,3)$ . After observing $7$ successes and $3$ failures, find the posterior distribution and its MAP.

Reveal answer or reference solution

The Bernoulli likelihood contributes $p^7(1-p)^3$ . Multiplying by the prior kernel $p^{2-1}(1-p)^{3-1}$ gives posterior

p\mid\text{data}\sim\operatorname{Beta}(9,6)

Since both parameters exceed $1$ , the beta mode is

\frac{9-1}{9+6-2}=\frac8{13}

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Conjugate update and MAP

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