Vladik Kreinovich, Gennady N. Solopchenko, Scott A. Ferson, Lev Ginzburg, Richard Alo
PROBABILITIES, INTERVALS, WHAT NEXT? EXTENSION OF INTERVAL COMPUTATIONS TO SITUATIONS WITH PARTIAL INFORMATION ABOUT PROBABILITIES
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,m are different from xi, and the resulting estimate ym = f(x1,m,..., xn,m) 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 measurement error Δxi (def) = xi,m - xi.
In many practical situations, we only know the upper bound Δi for this error; hence, after the measurement, the only information that we have about xi is that it belongs to the interval xi (def) = [xi,m - Δi, xi,m + Δi]. In this case, it is important to find the range y of all possible values of y = f(x1,..., xn) when xi ∈ xn.
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 Δi, we have some partial information about the probabilities of different values of Δxi.
