An Additive Theory of Bayesian Evidence Accrual
LOS ALAMOS NATIONAL LAB NM ANALYSIS AND ASSESSMENT DIV
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We derive a theory of data fusion based on an additive approach to Bayesian evidence combination and accrual. Although the additive method can be stated in terms of simple formulae of probability, it is surprisingly rich. It is robust against errors in data, and analysis and numerical simulations indicate that estimated probabilities of hypotheses converge to the expected value of a multiplicative Bayesian update as evidence that is mostly but not necessarily entirely correct is accrued. We summarize the method and principal results in the first part of the paper. The method relies on a representation theorem for expected values of uncertain probabilities that is an extension of a theorem of deFinettis deFinetti, 1937. The extension states that the expected value of a function of uncertain probabilities can be represented as a weighted sum of exchangeable random variables. We use the extended theorem to show that the additive method approximates the expected value of the ordinary Bayesian posterior, and they are equal in the limit. In the second part of the paper, we sketch proofs of our theorems, derive the additive rule and contrast the additive approach with others, especially multiplicative Bayesian updating on one hand and various consensus-based rules on the other. We show that the additive approach is much less sensitive to anomalous data than is Bayesian updating. The additive method, while similar in spirit to consensus approaches, is not ad hoc.
- Statistics and Probability