Beta-Bernoulli conjugate update — derive.

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Answer

$Prior: p ~ Beta(α, β).$
$Data: n Bernoulli trials with k successes.$
$p\; \mid \;\mathrm{data}$
$Posterior mean = (α + k) / (α + β + n) \to smoothly interpolates between prior mean α/(α+β) and MLE k/n as n grows.$
$Uses: Thompson sampling for binary bandits, small-sample click-through rate estimation, MAP with a fair-coin prior Beta(1, 1) = Uniform.$

Check yourself — multiple choice

Random
Beta(α,β) prior + k successes in n trials → Beta(α+k, β+n-k) posterior; posterior mean interpolates prior and MLE
Not conjugate
Requires normality

Beta-Bernoulli: Beta(α+k, β+n-k); posterior mean = shrinkage of MLE toward prior.

#bayesian#estimation

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