prior probability

There is a sizable scholarly literature discussing the so-called “Paradox of the Gatecrasher,” a simple thought experiment introduced many years ago by British philosopher L. Jonathan Cohen, an evidence problem designed to test the proper role of statistics in law. (For a survey of this legal literature, see footnote 8 of our latest paper “Visualizing Probabilistic Proof.”) Briefly, Wikipedia presents the gatecrasher problem as follows:

Statistical syllogisms may be used as legal evidence but it is usually believed that a legal decision should not be based solely on them. For example, in L. Jonathan Cohen‘s “gatecrasher paradox”, 499 tickets to a rodeo have been sold and 1000 people are observed in the stands. The rodeo operator sues a random attendee for non-payment of the entrance fee. The statistical syllogism:

1. 501 of the 1000 attendees have not paid
2. The defendant is an attendee
3. Therefore, on the balance of probabilities, the defendant has not paid

is a sound one, but it is felt to be unjust to burden a defendant with membership of a class, without evidence that bears directly on the defendant.

Notice, however, that Professor Cohen’s Gatecrasher Paradox is not really a paradox in the true sense of the word. In fact, it’s not even a mildly interesting problem, if you are a good Bayesian, that is. Why not? Because the statistical syllogism above only establishes a Bayesian prior, i.e. the initial probability that a randomly-selected attendee snuck in the rodeo without paying for his or her ticket. We still have to update our prior! Accordingly, let’s assume that the randomly-selected rodeo attendee in the example above is a man (let’s call him Mr X) and that our hypothetical Bayesian inquisitor is a woman (let’s call her Miss B). After finding her Bayesian prior, Miss B, our Bayesian puzzle-solver, would also ask Mr X to produce some scrap of evidence or proof that he paid for his rodeo ticket — a receipt, a ticket stub, a witness, etc. — and our Bayesian sleuth would then update her prior in a manner consistent with the evidence, including the lack of any evidence if that were the case.

Let’s solve the case of the gatecrasher