Limitations of Realistic Monte-Carlo Techniques

Because of the measurement errors, the result Y = f(X1, ..., Xn) of processing the measurement results X1, ..., Xn is, in general, different from the value y = f(x1, ..., xn) that we would obtain if we knew the exact values x1, ..., xn of all the inputs. In the linearized case, we can use numerical differentiation to estimate the resulting difference Y -- y; however, this requires \u3en calls to an algorithm computing f, and for complex algorithms and large $n$ this can take too long. In situations when for each input xi, we know the probability distribution of the measurement error, we can use a faster Monte-Carlo simulation technique to estimate Y -- y. A similar Monte-Carlo technique is also possible for the case of interval uncertainty, but the resulting simulation is not realistic: while we know that each measurement error Xi -- xi is located within the corresponding interval, the algorithm requires that we use Cauchy distributions which can result in values outside this interval. In this paper, we prove that this non-realistic character of interval Monte-Carlo simulations is inevitable: namely, that no realistic Monte-Carlo simulation can provide a correct bound for Y -- y