Your posterior mean is a tug-of-war between prior and data

Question

You put a Beta(a, b) prior on a new strategy's win rate p, then observe s wins out of n trades. Show that the posterior mean is a weighted average of the prior mean and the observed win rate, and interpret the weights.

Accepted Answer

The conjugate update. For a Bernoulli likelihood, the Beta prior is conjugate: observing s wins and n - s losses updates the prior to p \mid 	ext{data} \sim 	ext{Beta}(a + s,\ b + n - s). The posterior mean is \dfrac{a + s}{a + b + n}. Rewrite as a blend. Split the numerator and denominator to expose two familiar pieces, the prior mean \mu0 = \frac{a}{a+b} and the data's win rate \hat p = \frac{s}{n}: \mathbb{E}[p \mid 	ext{data}] = \underbrace{\frac{a+b}{a+b+n}}{w{	ext{prior}}}\cdot\frac{a}{a+b} \;+\; \underbrace{\frac{n}{a+b+n}}{w{	ext{data}}}\cdot\frac{s}{n}. So the posterior mean is …

Your posterior mean is a tug-of-war between prior and data

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