Probably Not, 2nd Edition

12Estimation Problems

The Number of Locomotives Problem

An important use of Bayes Theorem is for estimating parameters in a situation where there is limited information. The locomotive problem is an example of this estimation category of calculations. We are told that a railroad company numbers its locomotives consecutively from 1 to N, where N is the total number of locomotives owned by the company (and presumably they're all out running on tracks somewhere). We see a locomotive with number 60 painted on it. We are asked to estimate how many locomotives this company has.

We have – at this time – very little information to help us create a prior. Therefore, let us start with the simplest prior possible. If N is the total number of locomotives owned by the company, then our prior is a uniform distribution from 1 to N. We'll try to gain some insight into the implications of our choice of N as we go along.

The likelihoods are straightforward. First, if the company has less than 60 locomotives, there is 0 probability that we saw locomotive number 60. If the company has 60 locomotives, then the probability that the one we saw was number 60 is exactly 1/60. If they have 61 locomotives, then the probability that we saw number 60 is 1/61, etc. In general, if the company has N locomotives, then the probability that we saw number 60 (or any particular locomotive) is 1/N.

Since we are starting with a uniform prior distribution, the posterior curve and the likelihood curve look exactly ...