On the definition of objective probabilities by empirical similarity

We suggest to define objective probabilities by similarity-weighted empirical frequencies, where more similar cases get a higher weight in the computation of frequencies. This formula is justified intuitively and axiomatically, but raises the question, which similarity function should be used? We propose to estimate the similarity function from the data, and thus obtain objective probabilities. We compare this definition to others, and attempt to delineate the scope of situations in which objective probabilities can be used.

[1]  Judea Pearl,et al.  Fusion, Propagation, and Structuring in Belief Networks , 1986, Artif. Intell..

[2]  D. Ellsberg,et al.  Risk, ambiguity, and decision , 2001 .

[3]  J. L. Hodges,et al.  Discriminatory Analysis - Nonparametric Discrimination: Consistency Properties , 1989 .

[4]  I. Gilboa,et al.  Inductive Inference: An Axiomatic Approach , 2001 .

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

[6]  Illtyd Trethowan.  Causality , 1938 .

[7]  D. Schmeidler,et al.  A More Robust Definition of Subjective Probability , 1992 .

[8]  D. McFadden.  Conditional logit analysis of qualitative choice behavior , 1972 .

[9]  D. Ellsberg.  Decision, probability, and utility: Risk, ambiguity, and the Savage axioms , 1961 .

[10]  H. Akaike.  An approximation to the density function , 1954 .

[11]  C. D. Kemp,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[12]  G. Shafer.  Savage revisited , 1990 .

[13]  Itzhak Gilboa,et al.  Empirical Similarity , 2004, The Review of Economics and Statistics.

[14]  F. J. Anscombe,et al.  A Definition of Subjective Probability , 1963 .

[15]  David W. Scott,et al.  Multivariate Density Estimation: Theory, Practice, and Visualization , 1992, Wiley Series in Probability and Statistics.

[16]  Itzhak Gilboa,et al.  A theory of case-based decisions , 2001 .

[17]  D. W. Scott,et al.  Multivariate Density Estimation, Theory, Practice and Visualization , 1992 .

[18]  L. J. Savage,et al.  The Foundations of Statistics , 1955 .

[19]  Itzhak Gilboa,et al.  A similarity-based approach to prediction , 2011 .

[20]  I. Gilboa,et al.  Case-Based Decision Theory , 1995 .

[21]  Itzhak Gilboa,et al.  Rationality of Belief , 2004 .

[22]  M. Rostek.  Quantile Maximization in Decision Theory , 2009 .

[23]  Dov Samet,et al.  Probabilities as Similarity-Weighted Frequencies , 2004 .

[24]  Itzhak Gilboa,et al.  Axiomatization of an Exponential Similarity Function , 2004, Math. Soc. Sci..