Franco Taroni - Statistics and the Evaluation of Evidence for Forensic Scientists

This is minimised uniquely when . The negative of the Brier score is a strictly proper scoring rule with and (minimisation of a function corresponds to maximisation of the negative of the function).

The notion of exchangeability is illustrated with the following example of selection without replacement of items of a particular type, say, , from a small population. As an example of what might be, consider tablets in a consignment of drugs; the tablets may be either illicit ( ) or licit. The descriptor ‘small’ for the population size is used to indicate that removal of a member from the population, as in selection without replacement, effects the probability of possession of when the next member is selected for removal.

Denote the population size by . Of the items in the population, possess and do not and is not known. A sample of size is taken. The probability the first item selected from the population is of type is . If the first member selected from the population possesses , the probability the next member selected also possesses is . The population size is sufficiently small that cannot be approximated meaningfully by . Successive draws from the consignment are not independent in that knowledge of the outcome of one draw affects the probability of a particular outcome at the next draw.

Let be the number of members of the sample of size that possess . The probability distribution for is the hypergeometric distribution (Section 4.3.2 and Appendix A.2.5) and

This distribution does not depend on the order in which the members are drawn from the population, only on the number which possess and the number which do not. The property that the distribution is independent of the order is that of exchangeability.

As is not known, it is not possible to determine . However, it is possible given values for , and to make inferences about . A comparison of the frequentist and Bayesian approaches to this small consignment sampling problem is given in Section 4.3.2 and Aitken (1999).

Probabilities based on frequencies may be thought of as objective probabilities. They are considered objective in the sense that there is a well‐defined set of circumstances for the long‐run repetition of the trials, such that the corresponding probabilities are well‐defined and that one's personal or subjective views will not alter the value of the probabilities. Each person considering these circumstances will provide the same values for the probabilities. The frequency model relates to a relative frequency obtained in a long sequence of trials, assumed to be performed in an identical manner, physically independent of each other. Such a circumstance has certain difficulties. This point of view does not allow a statement of probability for any situation that does not happen to be embedded, at least conceptually, in a long sequence of events giving equally likely outcomes. However, note the following words of Lindley (2004):