Some asymptotic results for the combination of evidence problem

AbstractThe combination of evidence problem is treated here as the construction of a posterior possibility function (or probability function, as a special case) describing an unknown state parameter vector of interest. This function exhibits the appropriate components contributing to knowledge of the parameter, including conditions or inference rules, relating the parameter with observable characteristics or attributes, and errors or confidences of observed or reported data. Multivalued logic operators - in particular, disjunction, conjunction, and implication operators, where needed – are used to connect these components and structure the posterior function. Typically, these operators are well-defined for only a finite number of arguments. Yet, often in the problem at hand, a number of observable attributes represent probabilistic concepts in the form of probability density functions. This occur, for example, for attributes representing ordinary numerical measurements- as opposed to those attributes representing linguistic-based information, where non-probabilistic possibility functions are used. Thus the problem of discretization of probabilistic attributes arises, where p.d.f.'s are truncated and discretized to probability functions. As the discretization process becomes finer and finer, intuitively the posterior function should better and better represent the information available. Hence, the basic question that arises is: what is the limiting behavior of the resulting posterior functions when the level of discretization becomes infinitely fine, and, in effect, the entire p.d.f.'s are used?It is shown in this paper that under mild analytic conditions placed upon the relevant functions and operators involved, nontrivial limits in the above sense do exist and involve monotone transforms of statistical expectations of functions of random variable corresponding to the p.d.f.'s for the probabilistic attributes