Probability as a Measure of Information Added |
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Authors: | Peter Milne |
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Affiliation: | 1. University of Stirling, Stirling, United Kingdom
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Abstract: | Some propositions add more information to bodies of propositions than do others. We start with intuitive considerations on
qualitative comparisons of information added. Central to these are considerations bearing on conjunctions and on negations. We find that we can discern two distinct,
incompatible, notions of information added. From the comparative notions we pass to quantitative measurement of information
added. In this we borrow heavily from the literature on quantitative representations of qualitative, comparative conditional
probability. We look at two ways to obtain a quantitative conception of information added. One, the most direct, mirrors Bernard
Koopman’s construction of conditional probability: by making a strong structural assumption, it leads to a measure that is,
transparently, some function of a function P which is, formally, an assignment of conditional probability (in fact, a Popper function). P reverses the information added order and mislocates the natural zero of the scale so some transformation of this scale is
needed but the derivation of P falls out so readily that no particular transformation suggests itself. The Cox–Good–Aczél method assumes the existence of
a quantitative measure matching the qualitative relation, and builds on the structural constraints to obtain a measure of
information that can be rescaled as, formally, an assignment of conditional probability. A classical result of Cantor’s, subsequently
strengthened by Debreu, goes some way towards justifying the assumption of the existence of a quantitative scale. What the
two approaches give us is a pointer towards a novel interpretation of probability as a rescaling of a measure of information
added. |
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