Learning from the Probability Assertions of Experts
Mario Di Bacco, Patrizio Frederic and Frank Lad
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Abstract
We develop a new outlook on the use of experts’ probabilities for inference, distinguishing the information content available to the experts from their probability assertions based on that information. Considered as functions of the data, the experts’ assessment functions provide statistics relevant to the event of interest. This allows us to specify a flexible combining function that represents a posterior probability of interest conditioned on all the information available to any of the experts; but it is computed as a function of their probability assertions. We work here in the restricted case of two experts, but the results are extendible in a variety of ways. Their probability assertions are shown to be almost sufficient for the direct specification of the desired posterior probability. A mixture distribution structure that allows integration in one dimension is required to yield the complete computation, accounting for the insufficiency. One sidelight of this development is a display of the moment structure of the logitnormal family of distributions. Another is a generalisation of the factorisation property of probabilities for the product of independent events, allowing a parametric characterisation of distributions which orders degrees of dependency. Three numerical examples portray an interesting array of combining functions. The coherent posterior probability for the event conditioned on the experts’ two probabilities does not specify an externally Bayesian operator on their probabilities. However, we identify a natural condition under which the contours of asserted probability pairs supporting identical inferences are the same as the contours specified by EB operators. Our discussion provides motivation for the differing function values on the contours. The unanimity and compromise properties of these functions are characterised numerically and geometrically. The results are quite promising for representing a vast array of attitudes toward experts, and for empirical studies.
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