How much weight does the data get?

Question

You estimate a click-through rate with a $\text{Beta}(3, 7)$ prior, then observe 30 new click-or-no-click 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 behaves like pretend past trials: the two numbers 3 and 7 add up to 10 pretend trials already "seen." Your real data adds 30 more trials. The posterior mean is just a tug-of-war between prior and data, and each side's pull is simply its share of the total trials. The rule. For a Beta prior, the pretend trial count is a + b, and the real count is n. The data's weight is its share of the grand total: 	ext{data weight} = \frac{n}{(a+b) + n}. In words: real trials divided by (pretend trials plus real trials). Plug in. Here a + b = 3 + 7 = 10 pretend trials and n = 3…

How much weight does the data get?

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