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

Statistics and the Evaluation of Evidence for Forensic Scientists

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 [ images ]. (p. 27)7

1.7.6 The Quantification of Probability Through a Betting Scheme

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 images that can only take one of two values, namely, ‘true’ and ‘false’. There is a lack of information on the actual value of images and an operational system is needed for the quantification of the uncertainty about images imparted by the lack of information. A value images is regarded as an amount to be paid to bet on images with the conditions that a unit amount will be paid if images is true and nothing will be paid if images is false. In other words, images is the amount to be paid to obtain an amount equal to the value of images , 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 images is, according to Mr NN, equal to 0.37, meaning that if the person was forced to accept bets for and against event images , on the basis of the betting ratio images which he can choose as he pleases, this person would choose images . (p. 113)⁸

Coherence, as briefly described in Section 1.7.2, is defined by the requirement that the choice of images 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 images and an event which is impossible, sometimes known as the empty set, as images so that if images and images the two possible gains are

When images or images , 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

This happens when images for images and images for images . Therefore if the subjective probability of images , that represents our degree of belief on images , is defined as an amount images , which makes a personal bet on the event or proposition images coherent, then the probability images satisfies two conditions.

1 (1) ;

2 (2) .

Consider the case of images possible bets on events images that partition images ; i.e. images are mutually exclusive and exhaustive (Scozzafava 1987, p. 686). Let images , be the amount paid for a coherent bet on images . These bets can be regarded as a single bet on images with amount images . Another condition may be specified from the requirement of coherence, namely

1 (3) .

These conditions are the axioms of probability. Further details are given by de Finetti (1931b) and in Скачать книгу