4th International Symposium on Imprecise Probabilities and Their Applications, Pittsburgh, Pennsylvania, 2005 Estimation of Chaotic Probabilities Leandro Chaves Rêgo School of Electrical and Computer Engin. Cornell University, Ithaca, NY, U.S.A. lcr26@cornell.edu Terrence L. Fine School of Electrical and Computer Engin. Cornell University, Ithaca, NY, U.S.A. tlfine@ece.cornell.edu Abstract A Chaotic Probability model is a usual set of proba- bility measures, M, the totality of which is endowed with an objective, frequentist interpretation as op- posed to being viewed as a statistical compound hy- pothesis or an imprecise behavioral subjective one. In the prior work of Fierens and Fine, given finite time series data, the estimation of the Chaotic Probability model is based on the analysis of a set of relative fre- quencies of events taken along a set of subsequences selected by a set of rules. Fierens and Fine proved the existence of families of causal subsequence selec- tion rules that can make M visible, but they did not provide a methodology for finding such family. This paper provides a universal methodology for finding a family of subsequences that can make M visible such that relative frequencies taken along such subse- quences are provably close enough to a measure in M with high probability. Keywords. Imprecise Probabilities, Foundations of Probability, Church Place Selection Rules, Probabilis- tic Reasoning, Complexity. 1 Introduction A large portion of this and the following section is drawn in detail from Fierens 2003 [3] and Fierens and Fine 2003 [2]. They are included in this work to give the reader the necessary background to understand it. 1.1 Scope of Chaotic Probabilities A Chaotic Probability model is a usual set of proba- bility measures, M, the totality of which is endowed with an objective, frequentist interpretation as op- posed to being viewed as a statistical compound hy- pothesis or an imprecise behavioral subjective one. This model was proposed by Fierens and Fine 2003, [2] [3]. In this setting, M is intended to model stable (although not stationary in the traditional stochastic sense) physical sources of long finite time series data that have highly irregular behavior and not to model states of belief or knowledge that are assuredly impre- cise. This work was in part inspired by the following quotation from Kolmogorov 1983 [6]: “In everyday language we call random those phenom- ena where we cannot find a regularity allowing us to predict precisely their results. Generally speaking, there is no ground to believe that random phenom- ena should possess any definite probability. Therefore, we should distinguish between randomness proper (as absence of any regularity) and stochastic randomness (which is the subject of probability theory).” The task of identifying real world data supporting the Chaotic Probability model is an important open re- search question. We believe the model may be useful for complex phenomena (e.g. weather forecast, finan- cial data) where the probability of events, interpreted as propensity to occur, varies in a chaotic way with different initial conditions of the random experiment. Due to the complexity of such phenomena, inferring one probability for each possible initial condition is infeasible and unrealistic. The Chaotic Probability model gives a coarse-grained picture of the phenom- ena, keeping track of the range of the possible proba- bilities of the events. 1.2 Previous Work and Overview There is some earlier literature that tries to develop a frequentist interpretation of imprecise probabilities (although most of the literature deals with a subjec- tive interpretation, see for example Walley 1991 [11]). For work on asymptotics or laws of large numbers for interval-valued probability models, see Fine et al. [5] [7] [9] [10]. Cozman and Chrisman 1997 [1] stud- ied the estimation of credal sets by analyzing limiting relative frequencies along a set of subsequences of a time series. However, as we said above, the focus of our Chaotic Probability model is the study of finite