Probability and paternity testing.

A probability can be viewed as an estimate of a variable that is sometimes 1 and sometimes 0. To have validity, the probability must equal the expected value of that variable. To have utility, the average squared deviation of the probability from the value of that variable should be small. It is shown that probabilities of paternity calculated by the use of Bayes' theorem under appropriate assumptions are valid, but they can vary in utility. In particular, a recently proposed probability of paternity has less utility than the usual one based on the paternity index. Using an arbitrary prior probability in the calculation cannot lead to a valid probability unless, by chance, the chosen prior probability happens to be appropriate. Appropriate assumptions regarding both the prior probability and gene or genotypic frequencies can be estimated from prior experience.