A strong prior versus forty flips

Question

A coin has unknown head probability p, and your prior is p \sim 	ext{Beta}(10, 10), a confident belief centered at 	frac12 and worth 20 pseudo-flips. You then flip the coin 40 times and see 30 heads. What is the posterior mean, and how does it compare to the MLE?

Accepted Answer

Update by adding heads and tails to the Beta parameters: p \mid 	ext{data} \sim 	ext{Beta}(10 + 30,\; 10 + 10) = 	ext{Beta}(40, 20). Posterior mean: \mathbb{E}[p \mid 	ext{data}] = \frac{40}{40 + 20} = \frac{40}{60} = \frac23 \approx 0.667. The MLE is 30/40 = 0.75. Even with 40 observations, the posterior mean is dragged noticeably back toward the prior mean 	frac12, landing at 0.667 rather than 0.75. The reason is the prior's weight. It is worth a + b = 20 pseudo-flips, and the data is n = 40 real flips, so the data's weight is \frac{40}{20 + 40} = \frac23: \mathbb{E}[p \mid 	ext{data}…

A strong prior versus forty flips

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