When a strong Beta prior dominates the data

Question

You estimate a success rate with a strong $\text{Beta}(15, 5)$ prior, then observe only 5 new success-or-failure trials. The posterior mean is a weighted average of the prior mean and the observed rate.

What fraction of the posterior-mean estimate is contributed by the observed data (rather than the prior)?

Accepted Answer

Intuition first. A Beta prior counts as pretend trials: 15 and 5 add up to 20 pretend trials already seen, a heavy prior. Your real data adds only 5 trials. The posterior mean is a tug-of-war, and with so many pretend trials the prior should win most of the pull. The rule. Pretend trials = a + b and real trials = n, so: 	ext{data weight} = \frac{n}{(a+b) + n}. Plug in. Here a + b = 15 + 5 = 20 and n = 5: 	ext{data weight} = \frac{5}{20 + 5} = \frac{5}{25} = 0.2. What the number means. Only twenty percent of the estimate comes from the 5 real trials; the strong prior supplies the other eight…

When a strong Beta prior dominates the data

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