Bayesian Statistics: An Introduction, 4th Edition

7.3 Informative stopping rules

7.3.1 An example on capture and recapture of fish

A stopping rule s is said to be informative if its distribution depends on θ in such a way that it conveys information about θ in addition to that available from the values of . The point of this section is to give a non-trivial example of an informative stopping rule; the example is due to Roberts (1967).

Consider a capture–recapture situation for a population of fish in a lake. The total number N of fish is unknown and is the parameter of interest (i.e. it is the θ of the problem). It is known that R of the fish have been captured tagged and released, and we shall write S for the number of untagged fish. Because S=N–R and R is known, we can treat S as the unknown parameter instead of N, and it is convenient to do so. A random sample of n fish is then drawn (without replacement) from the lake. The sample yields r tagged fish and S=N–R untagged ones.

Assume that there is an unknown probability of catching each fish independently of each other. Then the stopping rule is given by the binomial distribution as

so that is a nuisance parameter such that . Note that this stopping rule is informative, because ...