By Franco Taroni; Wiley InterScience (Online service); et al
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Additional resources for Data analysis in forensic science : a Bayesian decision perspective
Ii) for any A, there exists a value α ∗ such that you will be indifferent between the ‘degenerate’ gamble (A, A; α ∗ , 1 − α ∗ ) and the gamble (C , B ; α ∗ , 1 − α ∗ ). Note that, in practice, this method to measure utility is subject to measurement errors (like any measure of empirical quantities), especially when the value of α ∗ is very near to either 0 or 1. Methods to improve the measurement can be found in the technical literature (Keeney and Raiffa 1976). Formal proofs of the expected utility theorem were given for the ﬁrst time by Ramsey (1931) and von Neumann and Morgenstern (1953).
An illustrative example for this is provided by courts that, typically, seek to reduce their uncertainty about a defendant’s true connection with a criminal act (Lindley 2006). Often, part of this effort is thought to be achieved on the basis of evidence as offered by forensic scientists. g. deciding if a suspect should be found guilty for the offence for which he has been charged) and taking such assessment seriously reﬂects the intention of promoting accurate decision making (Fienberg and Schervish 1986; Kaplan 1968; Kaye 1988; Redmayne 2001; Robertson and Vignaux 1993).
That the ordering is complete means, as already explained in von Neumann’s and Morgenstern’s quotation, that for any two gambles f and g, you are able to tell either which one of the two you prefer or that you are indifferent between them. That the ordering is transitive means that if you prefer gamble f to gamble g and prefer gamble g to gamble h, then you prefer also f to h; and if you 28 Scientiﬁc Reasoning and Decision Making are indifferent between f and g and between g and h, then you are also indifferent between f and h.