Limitations of Realistic Monte-Carlo Techniques in Estimating Interval Uncertainty

Because of the measurement errors, the result ỹ = f(x̃1, . . . , x̃n) of processing the measurement results x̃1, . . . , x̃n 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 = ỹ− y; however, this requires > n 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 technique for estimating ∆y – namely, Monte-Carlo simulation technique. A similar Monte-Carlo technique is also possible for the case of interval uncertainty, but the resulting simulation is not realistic: this technique uses Cauchy distributions which can result in arbitrarily small or arbitrarily large values, while we know that each measurement error ∆xi = x̃i − xi is located within the corresponding 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.