Statistics and the Evaluation of Evidence for Forensic Scientists. Franco Taroni
1.7.8. An example of a Dutch book is given to examine if a given person assigns subjective probabilities coherently. Consider a horse race with three horses,
The relationship between odds and probability is described briefly here with fuller details given in Chapter 2. An event with probability
Suppose the odds offered by the bookmaker are accepted by the person. Thus, their beliefs do not satisfy the additivity law of probability (condition 3). If any single bet is acceptable, they can all be accepted. This is equivalent to a bet on the certain event that one of
Judgements are required in all aspect of scientific investigation. The elicitation of probability distributions for uncertain quantities represents a challenging work for scientists and decision‐makers. O'Hagan (2019) recently wrote:
Subjective expert judgments play a part in all areas of scientific activity, and should be made with the care, rigour, and honesty that science demands. (p. 80)
A discussion can be found in Section 1.7.7.
1.7.7 Probabilities and Frequencies: The Role of Exchangeability
It is not uncommon for subjective (or personal) probabilities to be considered as a synonym for arbitrariness. This is not so; the use of subjectivism does not mean the use of acquired knowledge that is often available for consideration of relative frequencies is neglected. The main source of misunderstanding is concerned with the relationship between frequencies and beliefs. The two terms are, unfortunately, often regarded as equivalent since frequency data can be used to inform probabilities (Lindley 1991) but they are not equivalent. Dawid and Galavotti (2009, p. 100) quoted de Finetti's view:
every probability evaluation essentially depends on two components: (1) the objective component, consisting of the evidence of known data and facts; and (2) the subjective component, consisting of the opinion concerning unknown facts based on known evidence.
As emphasised more recently by D'Agostini (2016)
It is a matter of fact that relative frequency and probability are somehow connected within probability theory, without the need for identifying the two concepts. (p. 13)
It is reasonable to use relative frequencies to inform measures of belief and the relationship takes the form of a mathematical theorem, de Finetti's Representation theorem. According to the theorem, the convergence of one's personal probability towards the value of observed frequencies, as the number of observations increases, is a logical consequence of Bayes' theorem if a condition called exchangeability is satisfied by the degrees of belief prior to the observations (Dawid 2004).
As an illustration of the connection between frequency and probability, consider again an urn containing a certain number of balls, indistinguishable except by their colour, which is either white or black, and the number of balls of each colour being known. The extraction of a ball from this urn defines an experiment having two and only two possible outcomes that are generally denoted as success (say, the withdrawal of a white ball) or failure (say, the withdrawal of a black ball). Let
Suppose now the observer does not know the absolute value of balls present, nor the proportion that are of each colour. De Finetti (1931a) showed that every series of experiments having two and only two possible outcomes that can be taken as exchangeable (i.e. the probability assigned to the outcomes