Data's weight against a balanced Beta prior

Question

You estimate a success rate with a 	ext{Beta}(5, 5) prior, then observe 90 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: 5 and 5 add up to 10 pretend trials already seen. The real data adds 90 more. The posterior mean is a tug-of-war, and each side pulls in proportion to its share of the total trials. The rule. For a Beta prior, pretend trials = a + b and real trials = n, so: 	ext{data weight} = \frac{n}{(a+b) + n}. Plug in. Here a + b = 5 + 5 = 10 and n = 90: 	ext{data weight} = \frac{90}{10 + 90} = \frac{90}{100} = 0.9. What the number means. Ninety percent of the estimate comes from the real data and only ten percent from the prior. With this much dat…

Data's weight against a balanced Beta prior

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