A gentle prior meets a tiny sample

Question

A coin has unknown head probability p, and your prior is p \sim 	ext{Beta}(2, 2), a gentle bump centered at 	frac12. You flip it 4 times and see 3 heads. What is the posterior distribution and its mean, and how does it compare to the MLE?

Accepted Answer

Beta is conjugate to the binomial, so updating is "add heads to the first parameter, tails to the second": p \mid 	ext{data} \sim 	ext{Beta}(a + k,\; b + n - k) = 	ext{Beta}(2 + 3,\; 2 + 1) = 	ext{Beta}(5, 3). The posterior mean is \mathbb{E}[p \mid 	ext{data}] = \frac{5}{5 + 3} = \frac{5}{8} = 0.625. The MLE is k/n = 3/4 = 0.75. The posterior mean sits between the prior mean 	frac12 and the sample frequency 0.75, pulled toward the prior. The shrinkage is a precision-weighted blend. The prior acts like a + b = 4 pseudo-observations, and the data adds n = 4 real ones, so the data's weigh…

A gentle prior meets a tiny sample

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