ML Prep

Conjugate update and MAP

Problem

A Bernoulli success probability pp has prior Beta⁡(2,3)\operatorname{Beta}(2,3). After observing 77 successes and 33 failures, find the posterior distribution and its MAP.

Reveal answer or reference solution

The Bernoulli likelihood contributes p7(1−p)3p^7(1-p)^3. Multiplying by the prior kernel p2−1(1−p)3−1p^{2-1}(1-p)^{3-1} gives posterior

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

Since both parameters exceed 11, the beta mode is

9−19+6−2=813\frac{9-1}{9+6-2}=\frac8{13}
Outcome

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