Quasi-Probability: Why quasi-Monte-Carlo methods are statistically valid and how their errors can be estimated statistically

Abstract

The classical model of probability theory , due principally to Kolmogorov, defines probability as a totally-one measure on a sigma-algebra of subsets ( events ) of a given set (the sample space ), and random variables as real-valued functions on the sample space, such that the inverse images of all Borel sets are events. From this model, all the results of probability theory are derived. However, the assertion that any given concrete situation is subject to probability theory is a scientific hypothesis verifiable only experimentally , by appropriate sampling, and never totally certain. Furthermore, classical probability theory allows for the possibility of "outliers"—sampled values which are misleading. In particular, Kolmogorov's Strong Law of Large Numbers asserts that, if, as is usually the case, a random variable has a finite expectation (its integral over the sample space), then the average value of N independently sampled values of this function converges to the expectation with probability 1, as N tends to infinity. This implies that there may be sample sequences (belonging to a set of probability 0) for which this convergence does not occur; these are the "outliers". It is proposed to derive a large and important part of the classical probabilistic results, on the simple basis that sample sets of p -dimensional points are so constructed that distributions of samples of size N converge to given limit distributions, as N tends to infinity. It can then be shown that, for any Riemann-integrable random variable ( p -dimensional function) Ψ , the corresponding average value of a samples of size N of values of Ψ converges to the expectation (the Riemann integral) of Ψ , as N tends to infinity. This, and number of other useful results have already been proved, and further investigations are proceeding with much promise. By this device, the stochastic nature of some concrete situations is no longer, as heretofore, a likely scientific hypothesis , but rather a proven mathematical fact , and the problem of outliers is eliminated. This model may be referred to as "quasi-probability theory"; it is particularly appropriate for the large class of computations that are already referred to as "quasi-Monte-Carlo".