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

Probability is a fact about one's state of mind, not a fact about a phenomenon.

In summary, a person's assessment of their degree of belief (subjective probability) in the truth of a given statement or in the occurrence of an event (i) depends on information, (ii) may change as the information changes, and (iii) may differ from the assessment of others because different individuals may have different information or assessment criteria. In Savage's words (Savage 1954)

Probabilistic views hold that probability measures the confidence that a particular individual has in the truth of a particular proposition, for example, the probability that it will rain tomorrow. These views postulate that the individual concerned is in some way ‘reasonable’, but they do not deny the possibility that two reasonable individuals faced with the same information may have different degrees of confidence in the truth of the same proposition. (p. 3)

The only constraint in the assessment – as noted in Section 1.7.2– is that it must be coherent. Coherence may be understood through consideration of subjective probability in terms of betting, for example, on the outcome of a horse race. For the probabilities on winning for each horse in a race to be coherent, the sum of the probabilities over all the horses must be 1. This property characterises a ‘reasonable individual’. An example is presented in Section 1.7.6.

For a historical and philosophical discussion of subjective probabilities and a commentary on the work of de Finetti and Savage in the middle of the twentieth century, see Lindley (1980), Lad (1996), Taroni et al. (2001), Dawid (2004), Dawid and Galavotti (2009), Galavotti (2016, 2017), and Zynda (2016).

Savage, like de Finetti, viewed a personal probability as a numerical measure of the confidence a person has in the truth of a particular proposition. This opinion is viewed with scepticism today and was viewed with scepticism then (Savage 1967), as illustrated by Savage (1954).

I personally consider it more probable that a Republican president will be elected in 1996 than it will snow in Chicago sometime in the month of May, 1994. But even this late spring snow seems to me more probable than that Adolf Hitler is still alive. Many, after careful consideration, are convinced that such statements about probability to a person mean precisely nothing or, at any rate, that they mean nothing precisely. At the opposite extreme, others hold the meaning to be so self‐evident [ ]. (p. 27)7

The introduction of subjective probability through a betting scheme is straightforward. The concept is based on hypothetical bets (Scozzafava 1987):

The force of the argument does not depend on whether or not one actually intends to bet, yet a method of evaluating probabilities making one a sure loser if he had to gamble (whether or not he really will act so) would be suspicious and unreliable for any purposes whatsoever. (p. 685)

Consider a proposition that can only take one of two values, namely, ‘true’ and ‘false’. There is a lack of information on the actual value of and an operational system is needed for the quantification of the uncertainty about imparted by the lack of information. A value is regarded as an amount to be paid to bet on with the conditions that a unit amount will be paid if is true and nothing will be paid if is false. In other words, is the amount to be paid to obtain an amount equal to the value of , that is associating the value 1 with ‘true’ and the value 0 with ‘false’. This idea was expressed by de Finetti (1940) in the following terms.

The probability of event is, according to Mr NN, equal to 0.37, meaning that if the person was forced to accept bets for and against event , on the basis of the betting ratio which he can choose as he pleases, this person would choose . (p. 113) 8

Coherence, as briefly described in Section 1.7.2, is defined by the requirement that the choice of does not make the player a certain loser or a certain winner. Denote an event which is certain, sometimes known as a universal set , as and an event which is impossible, sometimes known as the empty set , as so that if and the two possible gains are

When or , there is no uncertainty in the outcome of the corresponding bet and so the coherence (in the absence of uncertainty) requires the respective gains to be zero. The values of the gains are therefore