Probabilities, intervals, what next? extension of interval computations to situations with partial information about probabilities

Abstract. In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easier-to-measure quantities x1,..., xn which are related to y by a known relation y = f(x1,..., xn). Measurements are never 100 % accurate; hence, the measured values �xi are different from xi, and the resulting estimate �y = f(�x1,..., �xn) is different from the desired value y = f(x1,..., xn). How different? Traditional engineering to error estimation in data processing assumes that we know the probabilities of different def measurement error ∆xi = �xi − xi. In many practical situations, we only know the upper bound ∆i for this error; hence, after the measurement, the only information that we have about xi is that it belongs def to the interval xi = [�xi − ∆i, �xi + ∆i]. In this case, it is important to find the range y of all possible values of y = f(x1,..., xn) when xi ∈ xi. We start with a brief overview of the corresponding interval computation problems. We then discuss what to do when, in addition to the upper bounds ∆i, we have some partial information about the probabilities of different values of ∆xi.