Decision Making with Linear Constraints on Probabilities

Techniques for decision making with knowledge of linear constraints on condition probabilities are examined. These constraints arise naturally in many situations: upper and lower condition probabilities are known; an ordering among the probabilities is determined; marginal probabilities or bounds on such probabilities are known, e.g., data are available in the form of a probabilistic database (Cavallo and Pittarelli, 1987a); etc. Standard situations of decision making under risk and uncertainty may also be characterized by linear constraints. Each of these types of information may be represented by a convex polyhedron of numerically determinate condition probabilities. A uniform approach to decision making under risk, uncertainty, and partial uncertainty based on a generalized version of a criterion of Hurwicz is proposed, Methods for processing marginal probabilities to improve decision making using any of the criteria discussed are presented.

Maximum Uncertainty Procedures for Interval-Valued Probability Distributions

Measures of uncertainty and divergence are introduced for interval-valued probability distributions and are shown to have desirable mathematical properties. A maximum uncertainty inference procedure for marginal interval distributions is presented. A technique for reconstruction of interval distributions from projections is developed based on this inference procedure

Decisions with Limited Observations over a Finite Product Space: the Klir Effect

Probability estimation by maximum entropy reconstruction of an initial relative frequency estimate from its projection onto a hypergraph model of the approximate conditional independence relations exhibited by it is investigated. The results of this study suggest that use of this estimation technique may improve the quality of decisions that must be made on the basis of limited observations over a decomposable finite product space.

Some Problems for Convex Bayesians

We discuss problems for convex Bayesian decision making and uncertainty representation. These include the inability to accommodate various natural and useful constraints and the possibility of an analog of the classical Dutch Book being made against an agent behaving in accordance with convex Bayesian prescriptions. A more general set-based Bayesianism may be as tractable and would avoid the difficulties we raise.

Anytime Decision Making with Imprecise Probabilities

This paper examines methods of decision making that are able to accommodate limitations on both the form in which uncertainty pertaining to a decision problem can be realistically represented and the amount of computing time available before a decision must be made. The methods are anytime algorithms in the sense of Boddy and Dean 1991. Techniques are presented for use with Frisch and Haddawy's [1992] anytime deduction system, with an anytime adaptation of Nilsson's [1986] probabilistic logic, and with a probabilistic database model.