The most likely rate: a Beta posterior mode

Question

You estimate a success probability p with a 	ext{Beta}(2, 3) prior, then observe 10 successes in 14 trials. Instead of the posterior average, you want the single most probable value of p under the posterior, called the mode or MAP (maximum a posteriori) estimate. What is the posterior mode of p?

Accepted Answer

Intuition first. The posterior mean is the balance point of the belief curve, but the mode is its peak, the single value that the data and prior together make most plausible. For a Beta they usually differ slightly because the curve is lopsided. First we update, then we 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}(2,3)\ +\ 10 	ext{ successes and } 4 	ext{ failures}\ ightarrow\ 	ext{Beta}(2+10,\ 3+4) = 	ext{Beta}(12, 7). The mode formula. For a 	ext{Beta}(a,b) with both numbers abov…

The most likely rate: a Beta posterior mode

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