ALGEBRA AND OPTIMIZATION FOR REASONING WITH IMPRECISE PROBABILITIES

Purpose. One of the central issues in statistics and decision analysis is how to handle probabilities and utilities whose precise values are not known. Here I discuss a method for modeling and computing with imprecise probabilities: Algebraic Inference with Constraints on Expressions of Probability (ALICE-P).

Method. You know from high-school geometry that mathematical models imply algebraic relationships. For example, the model of a right triangle implies a specific relationship between the lengths a and b of the sides and the length c of the hypotenuse; this is given by the familiar theorem of Pythagoras. Probability models are the same way. For example, the model of two correlated random variables A and B implies a specific relationship between the prior probability Pr(A), the conditional probability Pr(B|A), and the posterior probability Pr(A|B); this is given by the familiar theorem of Bayes.

In the ALICE-P method the analysis of probability models is separated into an initial symbolic probabilistic inference phase in which the algebraic relationships implied by the model structure are computed, followed by a numeric optimization phase in which constraints on the parameters used to specify the basic probabilities and utilities in the model are propagated into constraints on the posterior probabilities or expected utilities of interest. Additionally, the ALICE-P method allows the user to specify a varying amount of information about each probability measure in an incremental way, from a perfectly vacuous distribution through a precise density function through a specific value in the underlying sample space. Posterior probabilities are computed both in symbolic form as equations and in numeric form as strict interval bounds.

Results and Conclusion. The method of Algebraic Inference with Constrains on Expressions of Probability is a powerful technique which generalizes classical probability theory to handle imprecise probability measures; it is to classical probability theory as elementary algebra is to arithmetic. Unlike alternatives such as Dempster-Shafer theory, this work is perfectly compatible with classical probability theory and grounded in mathematical axioms. Furthermore, ALICE-P does not rely on assumptions of linearity which are incorrect for general probability models; instead the full nonlinear optimization is performed.