Combining interval and probabilistic uncertainty: What is computable?

Abstract

In many practical problems, we need to process measurement results. For example, we need such data processing to predict future values of physical quantities. In these computations, it is important to take into account that measurement results are never absolutely exact, that there is always measurement uncertainty, because of which the measurement results are, in general, somewhat different from the actual (unknown) values of the corresponding quantities. In some cases, all we know about measurement uncertainty is an upper bound; in this case, we have an interval uncertainty, meaning that all we know about the actual value is that is belongs to a certain interval. In other cases, we have some information-usually partial-about the corresponding probability distribution. New data processing challenges appear all the time; in many of these cases, it is important to come up with appropriate algorithms for taking uncertainty into account. Before we concentrate our efforts on designing such algorithms, it is important to make sure that such an algorithm is possible in the first place, i.e., that the corresponding problem is algorithmically computable. In this paper, we analyze the computability of such uncertainty-related problems. It turns out that in a naive (straightforward) formulation, many such problems are not computable, but they become computable if we reformulate them in appropriate practice-related terms.