Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds

Probabilistic arithmetic involves the calculation of the distribution of arithmetic functions of random variables. This work on probabilistic arithmetic began as an investigation into the possibility o f adapting existing numerical procedures (developal for fixed numbers) to handle random variables (by replacing the basic operations of arithmetic by the appropriate convolutions). The general idea is similar to interval arithmetic and fuzzy arithmetic. In this paper we present a new and general numerical method for calculating the appropriate convolutions of a wide range of probability distributions. An important feature of the method is the manner in which the probability distributions are represented. We use lower and upper discrete approximations to the quantile function (the quasi-inverse of the distribution function). This results in any representation error being always contained within the lower and upper bounds. This method of representation has advantages over other methods previously proposed. The representation fits in well with the idea of dependency bounds. Stochastic dependencies that arise within the course of a sequence of operations on random variables are the severest limit to the simple application of convolution algorithms to the formation of a general probabilistic arithmetic. We examine this dependency error and show how dependency bounds are a possible means of reducing its effect. Dependency bounds are lower and upper bounds on the distribution of a function of random variables that contain the true distribution even when nothing is known of the dependence of the random variables. They are based on the P~dchet inequalities for the joint distribution of a set o f random variables in terms of their marginal distributions. We show how the dependency bounds can be calculated numerically using our numerical representation of probability distributions. Examples of the methods developed are presented, *This work was supported by a grant through the Australian Research Grants Scheme and a Commonwealth Postgraduate Research Award. Address correspondence to: Robert C. WUliamson, Department of Systems Engineering, Australian National University, Canberra, 2601, Australia. International Journal of Approximate Reasoning 1990; 4:89-158 1990 Elsevier Science Publishin 8 Co., Inc. 655 Avenue of the Americas, New York, NY 10010 0888-613X/90/$3.50 89 90 Robert C. Williamson and Tom Downs and relationships with other work on numerically handling uncertainties are briefly described.