Probabilistic Arithmetic by Rob ert Charles Williamson BE QIT MEngSc Qld A thesis submitted for the degree of Do ctor of Philosophy Department of Electrical Engineering University of Queensland August Statement of Originality To the b est of the candidates knowledge and b elief the material presented in this thesis is original except as acknowledged in the text and the material has not b een submitted either in whole or in part for a degree at this or any other university Rob ert Williamson ii Abstract This thesis develops the idea of probabilistic arithmetic The aim is to replace arithmetic op erations on numb ers with arithmetic op erations on random variables Sp ecicallyweareinterested in numerical metho ds of calculating convolutions of probability distributions The longterm goal is to b e able to handle random prob lems such as the determination of the distribution of the ro ots of random algebraic equations using algorithms whichhavebeendevelop ed for the deterministic case To this end in this thesis we survey a numb er of previously prop osed metho ds for calculating convolutions and representing probability distributions and examine their defects We develop some new results for some of these metho ds the Laguerre transform and the histogram metho d but ultimately nd them unsuitable We nd that the details on how the ordinary convolution equations are calculated are secondary to the diculties arising due to dep endencies When random variables app ear rep eatedly in an expression it is not p ossible to determine the distribution of the overall expression by pairwise application of the convolution relations We prop ose a metho d for partially overcoming this problem in the form of dependency bounds These are b ounds on the distribution of a function of random variables when only the marginal distributions of the variables are known They are based on the Frechet b ounds for joint distribution functions Wedevelop ecientnumerical metho ds for calculating these dep endency b ounds and showhow they can b e extended in a number of ways Furthermore weshowhow they are related to the extension principle of fuzzy set theory which allows the calculation of functions of fuzzy variables Wethus provide a probabilistic interpre tation of fuzzy variables We also study the limiting b ehaviour of the dep endency b ounds This shows the usefulness of interval arithmetic in some situations The limiting result also provides a general law of large numb ers for fuzzy variables In terrelationships with a numb er of other ideas are also discussed Anumb er of p otentially fruitful areas for future research are identied and the p ossible applications of probabilistic arithmetic which include managementofnu meric uncertainty in articial intelligence systems and the study of random systems are discussed Whilst the solution of random algebraic equations is still a long way o the notion of dep endency b ounds develop ed in this thesis would app ear to b e of indep endentinterest The b ounds are useful for determining robustness of indep en dence assumptions one can determine the range of p ossible results when nothing is known ab out the joint dep endence structure of a set of random variables iii Acknowledgements Iwould liketoacknowledge the help and supp ort of the following p eople who con tributed to this thesis in various ways Most imp ortantly I would like to publicly thank my wife Angharad who has assisted me greatly in numerous ways over the years Without her help and supp ort this thesis would not have b een p ossible I am deeply grateful for all the assistance she has provided Iwould also liketothank My sup ervisor Professor Tom Downs for continued supp ort assistance and encouragement and for b eing a solver of administrative problems rather than a generator of them Former head of department Professor Mat Darveniza who help ed me as a b e ginning graduate student and later assisted in my obtaining a CPRA scholarship in order to pursue my PhD studies The head of department Professor Tom Parnell for the use of departmental facilities in the preparation of this thesis Dr Phil Diamond of the Department of Mathematics University of Queensland for useful advice and discussions and for the loan of some pap ers An anonymous referee of the International Journal of Approximate Reasoning for a question which led to section of chapter Professor Brian Anderson of the Australian National University whose simple question as to whether I had lo oked at any limiting results was part of the motivation for chapter Dr Helen McGillivray of the Department of Mathematics University of Queens land for a discussion on condence intervals and other matters and the loan of a b o ok Dr Guy West of the Department of Economics University of Queensland for discussions and the loan of some pap ers The following p eople assisted me by sending comments on some questions and reprints of their pap ers Professor Alice Agogino University of California at Berke ley Dr Piero Bonissone General Electric Research Labs New York Dr Jo el Bren ner Palo Alto Professors Didier Dub ois and Henri Prade UniversityofToulouse I I I Professor Jurgen Garlo Institut fur Angewandte Mathematik Universitat iv Freiburg Professor Robin Giles Queens UniversityOntario Professor VL Girko Kiev University Dr Ellen Hisdal University of Oslo Professor M Sambandham Atlanta University Dr Elie Sanchez Marseille Professor Berthold Schweizer UniversityofMassachusetts Amherst Professor Eugene Seneta Universityof Sydney Professor Ross Shacter Stanford University Dr Oskar Sheynin Moscow Professor Ushio Sumita UniversityofRochester and Professor Klaus Weise PhysikalischTechnische Bundesanstalt Braunschweig West Germany Iwould also like to thank the indulgence of the Electrical Engineering librarian Mrs Barbara Kormendy and the Mathematics Librarian Mrs Freida Kanowski It is a pleasure to acknowledge a debt to Donald Knuth and Leslie Lamp ort A for the developmentofT X and L T X under which this thesis was prepared and E E to James Alexander for his TIb bibliographic prepro cessor These to ols were most useful Financial supp ort from a Commonwealth Postgraduate ResearchAward and a grant from the Australian Research Grants Scheme is gratefully acknowledged FinallyI would liketoacknowledge an intellectual debt to Karl Popp er who through his writings has taught me the imp ortance of clarity of exp osition and honesty of thought Publications Most of the material in this thesis has b een or will b e published elsewhere The following is a list of pap ers either published submitted or nearing submission which rep ort material on the topic of this thesis Some of these pap ers rep ort further material which is not included in this thesis Rob ert C Williamson and Tom Downs Probabilistic Arithmetic and the Dis tribution of Functions of Random Variables Pro ceedings of the st IASTED Symp osium on Signal Pro cessing and its Applications Brisbane August Parts of chapter including further details and examples on the histogram metho d Rob ert C Williamson and Tom Downs The Inverse and Determinantofa Uniformly Distributed Random Matrix Statistics and Probability Letters Chapter Rob ert C Williamson and Tom Downs Probabilistic Arithmetic Numeri cal Metho ds for Calculating Convolutions and Dep endency Bounds accepted for publication in the International Journal of Approximate Reasoning Chapter Rob ert C Williamson An Extreme Limit Theorem for Dep endency Bounds of Normalised Sums of Random Variables accepted for publication in Infor mation Sciences Chapter Rob ert C Williamson The LawofLargeNumb ers for Fuzzy Variables under a General Triangular Norm Extension Principle under revision for resubmission to Fuzzy Sets and Systems Chapter Rob ert C Williamson Interval Arithmetic and Probabilistic Arithmetic to app ear in the pro ceedings of SCAN IMACSGAMMGI International Symp osium on Computer Arithmetic and SelfValidating Numerical Metho ds Basel Octob er A summary of chapters and and a discussion of the relationship b etween probabilistic arithmetic and interval arithmetic Rob ert C Williamson and Tom Downs Probabilistic Arithmetic Relation ships with Other Ideas to b e submitted to the International Journal of Ap proximate Reasoning Chapter vi Rob ert C Williamson and Tom Downs Numerical Metho ds for Calculating Convolutions of Probability Distributions to b e submitted to Mathematics and Computers in Simulation Chapter Rob ert C Williamson The QuotientofTwo Normal Random Variables An Historical Study in preparation Traces the history of a simple problem related to probabilistic arithmetic Rob ert C Williamson The Discrete Tconjugate Transform in preparation To b e submitted to Information Sciences Extends the Tconjugate transform see section and chapter Contents Probabilistic Arithmetic The Very Idea Motivation Outline of Results Dierent Metho ds for Calculating Convolutions and the La guerre Transform Metho d The Dep endency Bounds and Numerical Metho ds of Calcu lating Them Precursors Multiple Discoveries and Relationships with Fuzzy Sets The Inverse and Determinant of a Random Matrix A Limiting Result for Dep endency Bounds Thesis Structure Notational Conventions Numerical Metho ds for Calculating Convolutions of Probability Distributions Intro duction Aim and Analytical Metho ds History Motivation and Outline Exact Analytical Results