Probabilistic arithmetic

Abstract

This thesis develops the idea of probabilistic arithmetic. The aim is to replace arithmetic operations on numbers with arithmetic operations on random variables. Specifically, we are interested in numerical methods of calculating convolutions of probability distributions. The long-term goal is to be able to handle random problems (such as the determination of the distribution of the roots of random algebraic equations) using algorithms which have been developed for the deterministic case. To this end, in this thesis we survey a number of previously proposed methods for calculating convolutions and representing probability distributions and examine their defects. We develop some new results for some of these methods (the Laguerre transform and the histogram method), but ultimately find them unsuitable. We find that the details on how the ordinary convolution equations are calculated are secondary to the difficulties arising due to dependencies.When random variables appear repeatedly in an expression it is not possible to determine the distribution of the overall expression by pairwise application of the convolution relations. We propose a method for partially overcoming this problem in the form of dependency bounds. These are bounds on the distribution of a function of random variables when only the marginal distributions of the variables are known. They are based on the Frkchet bounds for joint distribution functions.We develop efficient numerical methods for calculating these dependency bounds and show how they can be extended in a number of ways. Furthermore we show how they are related to the "extension principle" of fuzzy set theory which allows the calculation of functions of fuzzy variables. We thus provide a probabilistic interpretation of fuzzy variables. We also study the limiting behaviour of the dependency bounds. This shows the usefulness of interval arithmetic in some situations. The limiting result also provides a general law of large numbers for fuzzy variables. Interrelationships with a number of other ideas are also discussed.A number of potentially fruitful areas for future research are identified and the possible applications of probabilistic arithmetic, which include management of numeric uncertainty in artificial intelligence systems and the study of random systems, are discussed. Whilst the solution of random algebraic equations is still a long way off, the notion of dependency bounds developed in this thesis would appear to be of independent interest. The bounds are useful for determining robustness of independence assumptions: one can determine the range of possible results when nothing is known about the joint dependence structure of a set of random variables.

Bayesian Test of Significance for Conditional Independence: The Multinomial Model

“…The same result concerning the product of non-negative random variables can be expressed by the Mellin convolution of the probability density functions, as demonstrated by Kaplan and Lin [10], Springer [11] and Williamson [12].…”

“…The Mellin convolution W 1 ⊗ W 2 gives the distribution function of the product of two independent random variables, with distribution functions W 1 and W 2 ; see Kaplan and Lin [13] and Williamson [12]. Furthermore, the commutative and associative properties follow immediately for the Mellin convolution,…”

Loss Distribution Approach for Operational Risk

“…Let us consider two operational risks. Using results of Frank, Nelsen and Schweizer [1987], Williamson [1989] shows that the dependency bounds of G when the dependence function is larger than a given copula C − are…”

Possibility transformation of the sum of two symmetric unimodal independent/comonotone random variables

“…They point out that the notion of non interactive variable introduced in the Zadeh's extension principle fuzzy theory, and widely used in the fuzzy community, is indeed equivalent to the comonotonicty of random variables (this connection is mentioned in the Ph. D. thesis of Williamson [19]). Accordingly, the Zadeh's extension principle is conservative for unimodal symmetric random variables with bounded support, but not necessarily if the support is indefinite.…”