# Scoring rules, entropy, and imprecise probabilities

Published

Journal Article

Suppose that a risk-averse expected utility maximizer with a precise probability distribution p bets opti- mally against a risk neutral opponent (or equiva- lently invests in an incomplete market for contingent claims) whose beliefs (or prices) are described by a convex set Q of probability distributions. This utility- maximization problem is the dual of the problem of .nding the distribution q in Q that minimizes a gen- eralized divergence (relative entropy) with respect to p. A special case is that of logarithmic utility, in which the corresponding divergence is the Kullback- Leibler divergence, but we present a closed-form so- lution for the entire family of linear-risk-tolerance (a.k.a. HARA) utility functions and show that this corresponds to a particular parametric family of gen- eralized divergences, which is derived from an entropy measure originally proposed by Arimoto and which is also related to a generalization of pseudospherical scoring rule originally proposed by I.J. Good. A vari- Ant of this decision problem, in which the decision maker has quasilinear utility for consumption over two periods, leads to the family of power divergences, which is related to a generalization of the power fam- ily of scoring rules. Copyright © 2007 by SIPTA.

### Duke Authors

### Cited Authors

- Nau, R; Jose, VR; Winkler, R

### Published Date

- December 1, 2007

### Published In

- Isipta 2007 Proceedings of the 5th International Symposium on Imprecise Probability: Theories and Applications

### Start / End Page

- 307 - 315

### Citation Source

- Scopus