Probability the next prediction is correct

Question

A new forecasting algorithm has made 3 correct calls out of 4 so far. You start from a flat prior $\text{Beta}(1, 1)$ on its true success probability $p$ , meaning every value of $p$ looked equally plausible before any data.

Using Bayesian updating, what is the probability that the algorithm's next call is correct?

Accepted Answer

Intuition first. You are not asked for the true success rate but for the chance the very next call is right. The Bayesian answer averages the success probability over all the values p could still take, weighted by how believable each is now. That average turns out to equal the posterior mean, so you predict with the same number you would estimate. The update rule. Beta is conjugate to right/wrong data, so add correct calls to the first number and wrong calls to the second: 	ext{Beta}(a,b)\ +\ k 	ext{ correct and } (n-k) 	ext{ wrong}\ ightarrow\ 	ext{Beta}(a+k,\ b+n-k). Here a=1, b=1, wi…

Probability the next prediction is correct

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