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,...,x_n). How different?
Traditional engineering to error estimation in data processing assumes that we know the probabilities of different measurement error Dxi=Xi-xi.
In many practical situations, we only know the upper bound Di for this error; hence, after the measurement, the only information that we have about xi is that it belongs to the interval [xi]=[Xi-Di,Xi+Di]. In this case, it is important to find the range [y] of all possible values of y=f(x1,...,xn) when xi is in [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 Di, we have some partial information about the probabilities of different values of Dxi.