When the MAP estimate equals the plain frequency

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You estimate a success probability p with a flat 	ext{Beta}(1, 1) prior, then observe 7 successes in 10 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. The mode is the peak of the belief curve. With a flat prior that treats every value equally, the peak should line up with what the raw data alone suggests. 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}(1,1)\ +\ 7 	ext{ successes and } 3 	ext{ failures}\ ightarrow\ 	ext{Beta}(1+7,\ 1+3) = 	ext{Beta}(8, 4). 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 = 8 and b = 4: 	ext{…

When the MAP estimate equals the plain frequency

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