Data's weight against a Gamma prior's pretend days

Question

You estimate a daily event rate with a 	ext{Gamma}(\alpha, \beta) prior whose rate parameter is \beta = 2, which acts like 2 pretend days of observation. You then observe the process for 8 real days. The posterior mean is a weighted average of the prior mean and the observed daily average. What fraction of the posterior-mean estimate is contributed by the observed data (rather than the prior)?

Accepted Answer

Intuition first. For a Gamma prior on a Poisson rate, the rate parameter \beta acts like a number of pretend days already watched. Here that is 2 pretend days. The real data adds 8 more days. The posterior mean is a tug-of-war between prior and data, weighted by each side's share of the total days. The rule. Pretend days = \beta and real days = n, so: 	ext{data weight} = \frac{n}{\beta + n}. Plug in. Here \beta = 2 and n = 8: 	ext{data weight} = \frac{8}{2 + 8} = \frac{8}{10} = 0.8. What the number means. Eighty percent of the rate estimate comes from the 8 real days and twenty percent from…

Data's weight against a Gamma prior's pretend days

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