Applied Logic Series Volume 24, 2001, pp 137-159

The Logic of Bayesian Probability

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For the last eighty or so years it has been generally accepted that the theory of Bayesian probability is a theory of partial belief subject to rationality constraints. There is also a virtual consensus that both the measure of belief and the constraints to which it is subject can only be provided via utility theory. It is easy to see why this should be so. The underlying idea, accepted initially by both de Finetti and Ramsey in their seminal papers ([1964] and [1931] respectively, though the paper 1964, first published in 1937, built on earlier work), but going back at least as far as Bayes’ Memoir [1763], is that an agent’s degree of belief in or uncertainty about a proposition A can be assessed by their rate of substitution of a quantity of value for a conditional benefit [S if A is true, 0 if not]. The natural medium of value is, of course, money, but the obvious difficulties with sensitivity to loss and the consequent diminishing marginal value of money seem to lead, apparently inexorably, to the need to develop this idea within an explicit theory of utility. This was first done only in this century, by Ramsey [1931]; today it is customary to follow Savage [1954] and show that suitable axioms for preference determine a reflexive and transitive ordering ‘at least as probable as’ and thence, given a further assumption about how finely the state space can be partitioned, a unique probability function.