## Bayesian billiards

This simple thought-experiment appears in the first draft of our latest paper:

Imagine a large square table [like the one pictured above] so level that any billiard ball thrown upon the table has an equal chance of landing on any part of the table. Also, imagine yourself blindfolded so that you cannot see the surface of the table. Now, imagine yourself asking a friend to throw or toss a cue ball onto the table. Your task in this thought experiment is to guess where the cue ball had landed … How would you make this guess without looking at the table (and without asking your friend to tell you where the cue ball is located)?

One possible solution, the Bayesian solution, is to ask your friend to throw a second billiard ball onto the table and then asking him to tell you whether the second ball landed to the right or left of the cue ball. If the second ball landed to the left of the cue ball, for example, then the cue ball is more likely to sit toward the right half of the table. In order to improve your guess, you should ask your friend to repeat this procedure many times, throwing n number of balls onto the table and reporting the relative location (i.e. left or right) of each subsequent ball in relation to the cue ball. Eventually, given a sufficiently large n, you should be able to narrow down the actual location of the cue ball …

Notice, though, that for Bayes’ solution-method to work the way Bayes himself intended, your friend must by willing to accurately and truthfully report the relative location of each subsequent ball thrown onto the table. That is, there is an implicit assumption in Bayes’ original thought-experiment–that the observer or information sender is always telling the truth … and it is this assumption that we wish to challenge here.

Our paper is titled “Bayesian manipulation of legal outcomes.” Can you see a direct analogy between Bayes’ thought-experiment and litigation generally? (Image of the square pool table courtesy of math.nyu.)