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Abstract:
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A new Bayes' Rule is derived from the author's remarkable symmetric utility rule for gambles (Bus. & Econ. Stat. Section Proceedings of ASA, 1976) -- an expected utility with probabilities^c + Renyi entropy (using same c), c = constant, where p^c means p raised to power c; the limiting case, c-->1, is the Expected Utility + Shannon Entropy Rule. The link between utility (U) and probability (P) is the recognition that U[x] = P[Success | x]. Substitute this into the utility-of-a-gamble rule, allow multiple kinds of Success, and normalize, to get a new marginal probability rule. Standard relationships of joint and marginal probabilities then yield an elegant new Bayes' Rule: P[xi | yj] = P[yj | xi]*(P[xi] / Sum[P[xk, {k, 1, n}]])^c / (P[yj] / Sum[P[yt, {t, 1, m}]] )^c , i=1,., n, j=1, ., m. Becomes usual Bayes' Rule as c-->1. Examples illustrate effects of c upon Bayesian inference.
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