Need to Combine Interval and Probabilistic Uncertainty: What Needs to Be Computed, What Can Be Computed, What Can Be Feasibly Computed, and How Physics Can Help

In many practical situations, the quantity of interest is difficult to measure directly. In such situations, to estimate this quantity, we measure easier-to-measure quantities which are related to the desired one by a known relation, and we use the results of these measurement to estimate the desired quantity. How accurate is this estimate? Traditional engineering approach assumes that we know the probability distributions of measurement errors; however, in practice, we often only have partial information about these distributions. In some cases, we only know the upper bounds on the measurement errors; in such cases, the only thing we know about the actual value of each measured quantity is that it is somewhere in the corresponding interval. Interval computation estimates the range of possible values of the desired quantity under such interval uncertainty. In other situations, in addition to the intervals, we also have partial information about the probabilities. In this paper, we describe how to solve this problem in the linearized case, what is computable and what is feasibly computable in the general case, and, somewhat surprisingly, how physics ideas -- that initial conditions are not abnormal, that every theory is only approximate -- can help with the corresponding computations