Beta posterior mode below one-half

Question

You estimate a success probability p with a 	ext{Beta}(4, 2) prior (leaning optimistic, centered around two-thirds), then observe just 3 successes in 9 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 prior expected a high success rate, but the data came in low. The mode, the peak of the combined belief curve, should land between the two, somewhere below one-half. 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}(4,2)\ +\ 3 	ext{ successes and } 6 	ext{ failures}\ ightarrow\ 	ext{Beta}(4+3,\ 2+6) = 	ext{Beta}(7, 8). 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 = 7 and…

Beta posterior mode below one-half

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