Why Some Families of Probability Distributions Are Practically Efficient: A Symmetry-Based Explanation

Out of many possible families of probability distributions, some families turned out to be most efficient in practical situations. Why these particular families and not others? To explain this empirical success, we formulate the general problem of selecting a distribution with the largest possible utility under appropriate constraints. We then show that if we select the utility functional and the constraints which are invariant under natural symmetries – shift and scaling corresponding to changing the starting point and the measuring unit for describing the corresponding quantity x. then the resulting optimal families of probability distributions indeed include most of the empirically successful families. Thus, we get a symmetry-based explanation for their empirical success. Vladik Kreinovich Department of Computer Science, University of Texas at El Paso, 500 W. University, El Paso, Texas 79968, USA, e-mail: [email protected] Olga Kosheleva University of Texas at El Paso, 500 W. University, El Paso, TX 79968, USA, e-mail: [email protected] Hung T. Nguyen Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003, USA, and Faculty of Economics, Chiang Mai University, Chiang Mai, Thailand, e-mail: [email protected] Songsak Sriboonchitta Faculty of Economics, Chiang Mai University, Chiang Mai, Thailand, e-mail: [email protected]