A high Beta posterior mode after strong data

Question

You estimate a success probability p with a mild 	ext{Beta}(2, 2) prior, then observe a strong run of 20 successes in 25 trials. You want the single most probable value of p under the posterior, the mode or MAP estimate. What is the posterior mode of p?

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Intuition first. A strong run of successes should push the peak of the belief curve up toward a high value, with the mild prior only lightly holding it back. We update first, then read off the peak. The update. Beta is conjugate to success/failure data, so add successes to the first number and failures to the second: 	ext{Beta}(2,2)\ +\ 20 	ext{ successes and } 5 	ext{ failures}\ ightarrow\ 	ext{Beta}(2+20,\ 2+5) = 	ext{Beta}(22, 7). The mode formula. For 	ext{Beta}(a,b) with both numbers above 1: 	ext{mode} = \frac{a - 1}{a + b - 2}. Plug in. With a = 22 and b = 7: 	ext{mode} = \fr…

A high Beta posterior mode after strong data

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