Let's say you're playing ping pong. For each point you win or lose, how would you update your probability of winning the next point?

Question

Let's say I start out believing that my probability of winning the next point is in the interval $[0.25, 0.5]$ with 50% confidence. If I win the next point, what is an intuitive or "good" way to update my interval? (I still want an interval with 50% confidence but I want the interval to be as tight as possible.) I was thinking we could square root the endpoints of the interval and get $[0.5, 0.7]$. And for each point you win, you keep square rooting the endpoints. For each point you lose, perhaps square the endpoints?

Is there a more intuitive way to do this? I think this may be an open ended question..

Answer 1

First: The exact answer to the update is dependent on the precise prior distribution you are working with. Your problem constrains that only to the extent that $$F(5.0) - F(2.5) = .5$$ There are many possible actual priors that have that property; for example, a Gaussian centered on $0.375$ with a $\sigma$ of about $0.34$ (with suitable cutoffs at 0 and 1), or a uniform distribution on $(0,0.75)$. The updates you want depend heavily on the precise prior. For example, if the prior is that your chances of winning is uniform on $(0, 0,75)$, then no sequence of wins will cause you to say that $p > 0.75$.

Second, the choice of confidence interval has some subtleties having to do with experimental results that are very unlikely given your prior. You would like your confidence interval to place half the "outside" probability on either side of the interval. But, for example, if your prior is that uniform distribution, and you do get ten straight wins, this might become problematic. See work by Cousins and Feldman for a discussion of that problem.

If you are willing to say that your prior is that $p$ is Gaussian distributed with mean $0.375$ and $\sigma= 0.34$, then the math for updating the estimate of the distribution is given by the Kalman Filter technique. That is the practical best you will do, although it does spill over into the non-physical region of $p$ outside of $(0,1)$ if your data is wildly unlikely (say 50 straight wins) given your hypothesis.