Scoring Rules, Generalized Entropy, and Utility Maximization

Information measures arise in many disciplines, including forecasting (where scoring rules are used to provide incentives for probability estimation), signal processing (where information gain is measured in physical units of relative entropy), decision analysis (where new information can lead to improved decisions), and finance (where investors optimize portfolios based on their private information and risk preferences). In this paper, we generalize the two most commonly used parametric families of scoring rules and demonstrate their relation to well-known generalized entropies and utility functions, shedding new light on the characteristics of alternative scoring rules as well as duality relationships between utility maximization and entropy minimization. In particular, we show that weighted forms of the pseudospherical and power scoring rules correspond exactly to measures of relative entropy (divergence) with convenient properties, and they also correspond exactly to the solutions of expected utility ...

[1]  D. A. Sprott,et al.  Foundations of Statistical Inference. , 1972 .

[2]  A. Hendrickson,et al.  Proper Scores for Probability Forecasters , 1971 .

[3]  R. L. Winkler.  The Quantification of Judgment: Some Methodological Suggestions , 1967 .

[4]  K. Pearson.  On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is Such that it Can be Reasonably Supposed to have Arisen from Random Sampling , 1900 .

[5]  Suguru Arimoto,et al.  Information-Theoretical Considerations on Estimation Problems , 1971, Inf. Control..

[6]  B. D. Finetti.  La prévision : ses lois logiques, ses sources subjectives , 1937 .

[7]  Wojciech Slomczynski,et al.  Utility maximizing entropy and the second law of thermodynamics , 2004, math/0410115.

[8]  D. Friedman.  Effective Scoring Rules for Probabilistic Forecasts , 1983 .

[9]  Prem K. Goel,et al.  Information Measures and Bayesian Hierarchical Models , 1983 .

[10]  A. Raftery,et al.  Strictly Proper Scoring Rules, Prediction, and Estimation , 2007 .

[11]  B. H. Lavenda,et al.  Qualms concerning Tsallis's condition of pseudo-additivity as a definition of non-extensivity , 2003 .

[12]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[13]  Victor Richmond R. Jose,et al.  A Characterization for the Spherical Scoring Rule , 2009 .

[14]  R. Nau.  Should Scoring Rules be Effective , 1985 .

[15]  Ludger Rüschendorf,et al.  Minimax and minimal distance martingale measures and their relationship to portfolio optimization , 2001, Finance Stochastics.

[16]  Karl Pearson F.R.S..  X. On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling , 2009 .

[17]  Jan Havrda,et al.  Quantification method of classification processes. Concept of structural a-entropy , 1967, Kybernetika.

[18]  A. H. Murphy,et al.  Scoring rules and the evaluation of probabilities , 1996 .

[19]  Contents , 2003 .

[20]  H. Kyburg,et al.  Studies in Subjective Probability (2nd ed). , 1981 .

[21]  M. Frittelli.  The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets , 2000 .

[22]  L. Bregman.  The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[23]  L. J. Savage.  Elicitation of Personal Probabilities and Expectations , 1971 .

[24]  Dick E. Boekee,et al.  The R-Norm Information Measure , 1980, Inf. Control..

[25]  Palaniappan Kannappan,et al.  A Directed-Divergence Function of Type β , 1972, Inf. Control..

[26]  A. Dawid.  The geometry of proper scoring rules , 2007 .

[27]  J. Eric Bickel,et al.  Some Comparisons among Quadratic, Spherical, and Logarithmic Scoring Rules , 2007, Decis. Anal..

[28]  J McCarthy,et al.  MEASURES OF THE VALUE OF INFORMATION. , 1956, Proceedings of the National Academy of Sciences of the United States of America.

[29]  R. Cooke.  Experts in Uncertainty: Opinion and Subjective Probability in Science , 1991 .

[30]  A. Dawid,et al.  Game theory, maximum entropy, minimum discrepancy and robust Bayesian decision theory , 2004, math/0410076.

[31]  G. Brier.  VERIFICATION OF FORECASTS EXPRESSED IN TERMS OF PROBABILITY , 1950 .

[32]  F. Delbaen,et al.  Exponential Hedging and Entropic Penalties , 2002 .

[33]  R. Selten.  Axiomatic Characterization of the Quadratic Scoring Rule , 1998 .

[34]  Nicole El Karoui,et al.  Pricing Via Utility Maximization and Entropy , 2000 .

[35]  D. Haussler,et al.  MUTUAL INFORMATION, METRIC ENTROPY AND CUMULATIVE RELATIVE ENTROPY RISK , 1997 .

[36]  R. L. Winkler.  Scoring Rules and the Evaluation of Probability Assessors , 1969 .

[37]  David Lindley.  Scoring rules and the inevitability of probability , 1982 .

[38]  R. L. Winkler.  Evaluating probabilities: asymmetric scoring rules , 1994 .