Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 1990 Uncertainty reasoning and representation: A Comparison of several alternative approaches Barbara S. Smith Follow this and additional works at: http://scholarworks.rit.edu/theses Recommended Citation Smith, Barbara S., "Uncertainty reasoning and representation: A Comparison of several alternative approaches" (1990). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Rochester Institute of Technology Department of Computer Science Uncertainty Reasoning and Representation: A Comparison of Several Alternative Approaches by Barbara S. Smith A Thesis, submitted to The Faculty of the Department of Computer Science in partial fulfillment of the requirements for the degree of Master of Science in Computer Science Approved by: Professor John A. Biles Date fj/O/10 , Professor Feredoun Kazemian Date _____________ IO_A~ Professor Peter G. Anderson Date uncertainty Reasoning and Representation: A Comparison of Several Alternative Approaches I, Barbara s. Smith, hereby grant permission to the Wallace Memorial Library of RIT to reproduce my thesis in whole or in part. Any reproduction will not be for commercial use or profit. Date: August 13,1990 ABSTRACT Much the of research in Artificial done Intelligence involves investigating and developing methods of incorporating uncertainty reasoning and representation expert The situations. on strict approaches methods have been handling uncertainty in for attempted based Several systems. theories range probabilistic based on investigates a Probability, of Evidence, Mycin from proposed problem solving numerical reasoning to approaches non-numeric logical reasoning. This study of these approaches including number Fuzzy into and Bayesian Dempster-Shafer Certainty Factors, Set Theory, Possibility Theory and Theory non logic. Each of these theories and their formalisms are explored by means of examples. underlying The discussion concentrates on a comparison of the different approaches, noting the type of uncertainty that they best monotonic represent. CR Categories 1.2.1 1.2.4 and Medicine Science and Artificial Formalisms : Uncertainty, Shafer Theory Intelligence Representation Knowledge Keywords Systems Expert Applications and Methods approximate of reasoning, Evidence, Fuzzy set Dempster- theory Table 1 2 . . 3. of Contents Introduction The Nature 3 of 6 Uncertainty 10 Probability Theory 3 3 3 . . . 1 Background: 2 Bayesian 3 3.4 3 . 5 What are Conditional 3.2.2 Bayes 3.2.3 Statistical 3.2.4 Ignorance - 14 Probability Theory 3.2.1 Mycin 10 probabilities? An ' Probabilities Theorem Early 17 Independence Assumptions Expert Mycin 3.3.2 Combining evidence 3.3.3 Certainty Factors 3.3.4 Ignorance Certainty System 23 in Mycin - 26 Examples 28 31 3.4.1 Representation 3.4.2 Belief 3.4.3 Combining Evidence 3.4.4 Demster's Rule 3.4.5 Ignorance of 22 Factors Dempster-Shafer Mathematical , 19 21 3.3.1 Comparison 15 of Theory Uncertain Evidence of Information Commonality, Plausibility of Representation 3.5.2 Ignorance 3.5.3 Prior - from Multiple Combination - An 33 Example... 38 sources 40 Examples 42 48 Probability-based 3.5.1 33 of Probability 50 approaches 50 Belief 52 - Problem of Subjectivity .... 53 3.5.4 Assumptions: Mutual 3.5.5 4 . Fuzzy 4 . 1 54 56 Inf erencing methods 58 Theory Definition of Fuzzy Sets 59 4 3 Possibility Theory 65 4 Ignorance 67 . . Logic Comparison Non-monotonic 63 of Possibility Theory and Probability 68 74 Logic 5.1 Basic Characteristics 75 5 Modal Logic 76 . 2 5.3 6. and Exclusivity Fuzzy 4.5 . Independence 4.2 4 5 Set Statistical Doyle's Summary and Truth Maintenance Conclusion 77 78 83 REFERENCES ANNOTATED System BIBLIOGRAPHY 85 Chapter One Introduction Every day decisions and domains These information to obtained several questions information required also The also be could information Expert must to * known. not The known but is uncertain, they are not the then answers totally sources the If be can problem the all multiple inconsistent. or in that available from obtained The ways. in that be approximate, conflicting systems have been and The and goal in expert system a domain with of to totally attempted or specifically, based non-numeric is first ability to about The situations. approaches these theories the knowledge proposed people support with conceptualize solving numerical to created be designed expert' reason problem are making environments. several only be partial, could information domains. uncertain combine in of available solved uncertainty. with making are uncertain characterized could answers accurate. only be the dilemma with in problems solving can faced are people approaches supported for domain. * these with design expert' Several is to ability methods range from probabilistic reasoning logical reasoning based predicate by cope handling uncertainty in theories strict on order a in decision on calculus. different Each assumptions of and/or interpretations of existing theories. other uncertainty best handled by is the primary It investigate by proposed in making AI uncertain domains. Bayesian Factors, Dempster-Shafer theory Theory of each represents Secondly, the first area is methods the underlying theory handles ignorance Finally or we some assumptions special conflicting will look information from at each Chapter Two will uncertainty and An situations. how included. Mycin In Evidence will related cases they and How such uncertainty inferences be how along, required. will theory in arise how of task can of problem people each as be discussed. made using the Bayesian of solving process resolution the also information will as and Theory three very distinct probability. Zadeh's be Probability Theory, the Dempster-Shafer and presented interpretations discuss Fuzzy Set are describe different types Factors be of evidence. examined information then Chapter Three Certainty that each uncertain be will how theory. overview for decision making sources or The four on with information for combining used Set logic. non-monotonic concerned be Certainty Fuzzy focus primarily the uncertain will Mycin will information from different with theories that Evidence, decision and solving theory The areas. and of to that have been problem The Possibility Theory discussion this thesis of Probability Theory, are: general for different. quite approaches researchers reviewed Theory, objective different six is each the Therefore, Chapter Possibility of yet Four will Theory. Chapter Five will non-numeric Chapter will be be approach Six the concerned to strong summarized. with reasoning and weak non-monotonic with aspects logic uncertainty- of each In approach as a Chapter Two The Nature The are of Uncertainty domains complex very diverse often usually can be 1987]. The first requirements step alternative to analysis done In selected. steps for a when to tremendous either information has more an steps objectives and necessary solutions are generated. the all these impacts last step or actually and solve on the of costs the most of a the appropriate evaluation problems, needed or is there iteration back to of second alternative further information is effect The interpretation an made amount and [Sage, the decision of the potentially implementation order process all of is situation and in step two is for alternative the In alternatives. is the formulation analysis evaluate each decisions make three primary by identified, are people Though step the desired involves the solutions need step which the underlying decision characterized this In acceptable dynamic. and in many ways, unique problem. within is a earlier new selected previously alternative. Information block of this is without decision doubt the process understanding human reasoning understand the information most model. is to used by important building The first characterize people for step and problem in In resolution. is done, processing particular information is aquired, the and represented, The decisions is solutions are be can within however, the nature In and quickly of the is several not very is often ill-suited a people are forced to make alternative More due to cut or evidence characterized inaccurate, inconsistent, positive clear The make alternative the best cases, being partial, for forced to are information. available inferences. make in that is to how paid little difficulty. with choice to people some available information that used which be must information is how the complex very possible. chosen commonly, the often attention it then is how situation information to understand how order vague otherwise or judgement to be made. decisions based on as Therefore, uncertain information. There information to the a or the the when the a number a could be partial information result of are the high fact that the information has a reasons of fact that the tools required be are time constraint information cannot decision to be The sources electronic imperfect. sensor or cost our is placed be collected on the person, to could tools obtain Often, process, quickly enough, partial information, This adequately. decision a the proper required be due could all obtain obtaining the of on It available. readily been developed another Therefore, to technology based made of not evidential incomplete. or required actual not that all forcing information. whether are they be an inherently information that we obtain from these sources solve the majority with consulting these be problems multiple that the information information from or another an error tests made to resolve can be collected in an we rely a errors effort on from It one could and source the continue conflicts discrepancy, be must additional or to outweigh the is the with has been recognized, the to decision. In order to source. Once be order information from support obtained In actually conflicting recognized. can The to these solving process, face, we sources. combined inconsistent either problem the of is then sources possible in fact be inaccurate. could data more false information. People formal, rational the not this they perceive in for it It expert more simple not information but unknown. person evidence all only also choose required to truths a this its with of the world falsities. and formal state' As representation it becomes less original meaning, or at together. important to to evaluate recognize By understanding may a x current state' development, inconsistent is lost is system the Unfortunately, information is translated to accurate, times how expressed in Their decision making processes manner. by decision making approach information describing the easily scheme a always surrounding them. world perceptual is not strongly biased are of do to take what what this available information information specific eliminate the actions remains to ignorance. remains unknown, obtain the properly describe the decision making process, To imperative to belief ignorance. total and situation where support decision no a none between any hand, situation not all certain the of present alternative representing between limit person alternative to This difference between discussed the further in the following number as it of not to make though be a Based able and to the there a a to is choice On the other exists when completely subset on to make that partial the of this a She will, alternatives ignorance chapters. a solutions. relates bias as required words, belief partial available. will able other will partial characterized solutions. required is available, information is a that a information In available. information information, choice the decision necessary is is Ignorance the relevant of evidence supporting a the difference between understand it is are subset certain however, be feasible. belief formalisms of will be described Chapter Three Probability Theory Probability theory in AI decision making 1987]. [Nutter, At and the theory in reasoning subject. theory is it for dealing however, numbers severe with are or often A subjective. make this argued theory over 10 role proven The not generally same discounted of on time, assumptions stems 1986] from x fuzzy' Background Numerical many different : that the its lack of formalism to probability, available the with evidence that for in of It is a that Theory is order is to also probability expressiveness, especially events. What are probability ways. probabilities? theory has been interpreted in Each of these different 10 is it is Bayesian required inadequacy a requirement the basis contention to be probability opponents is the of years probability mathematical that the At of continues theory computationally feasible. [Zadeh, describing for out that source in in favor fact a role systems uncertainty A active point strong independence the 3.1 to quick an time, the uncertainty. evidence is expert provides shortcoming. available played same with widely debated that has for interpretations * language' theories has or to be frequency A ratios. Each likelihood of statistical from the number of an the of outcomes that of zero represents all weight defined an as assigned limited a (E) event In one of to AI, elements the subjective be noted, studied and available, In AI, a for this with samples to A weight it is true. The space. the as sum An of of to (A) is false, hypothesis sample proportion element set. a the the as the a tests, or in this of all, the data that no strong that large of whereas is event probability all the of weights nature 11 of is in areas a of good probability has of amount not be data of available is available statistical amounts offers the would probability approach due to the a resulting from given is defined that First however, which assigned in E. Secondly, very element frequency interpretation to derive cases. the or is defined This weight defines correspond is then defined is space in terms hypotheses space, is to say, given the of of fall represents subset most should that that the application. required of that elements sample one. (A) element possible sample That experiment. all the as to zero likelihood number of occurrence sufficiently large a set in the element ranging have been defined probabilities generally known outcomes, weight specific to express probability judgement used Traditionally, first. own 1986]. [Shafer, of its produced often foundation. that are the majority for of It well statistical method in data are reasoning. domains, large As a has amounts result given event, degree the the a person an this In certain that chance' will belief functions the or is the probability from the the Bayesian degree hypothesis true. * a the probability subjective event is subjective will of measure of The occur. directly attached an to for Mycin, a subjective measure Dempster-Shafer true, not is In this belief of Theory in that the belief is the believes that the person chance' Dempster-Shafer Theory. The approach Theory is the theory Bayesian whether to attached as differs measure evidence proves the the hypothesis itself is the whether evidence the hypothesis. certainty factors Mycin model a to be The supports is the probability belief that observation. The [Shortliffe, indicate that 1976]. a of Confirmation is of a supported by a evidence or to proven, support the as x given factors is confirmation hypothesis a early diagnosis. for certainty of is true in the Dempster-Shafer Theory, 12 is an measure subjective basis as medical probability known hypothesis indicates that there a in use hypothesis a theoretical in the interpretation developed were inexact reasoning for physician's As a occur the of in the Bayesian Theory. In is believes the event as probability approach, evidence, generalization theory, of a approach event. A of belief. of of unavailable. the Bayesian limitations, definition the is generally data statistical these data of adopted measure of does not it merely the hypothesis. chance' is attached to whether All of the three present the 3.3, expert Shafer the are the differences lie in the Theory will of of Mycin, attention The subject to fact that next section much they will considerable features to the calculus and debate. In inexact reasoning implemented in will be discussed 13 ways. standard Probability in presently model system, theory use particular that the hypothesis. approaches in different Bayesian assumptions the Their calculus detail paying section these of probability- apply this supports evidence be presented. further in The section Dempster- 3.4. 3.2 Bayesian The Bayesian Thomas of Bayes (1702 represents the a a sum space would All are be or range of expert approach formalism for 1984], of Bayes with will it occur will that represented a by a A hypothesis zero imply would in hypotheses values a such given that their in the following with provide a uncertainty as a uncertainty good mathematical [Clancy, Shortliffe, inherent limitations early systems were mechanisms of dealing Theory several using heurisitic model for Bayesian due to these Theorem, used in fact, with Unfortunately, mathematical of probability systems does, dealing true Bayesian schemes (H) chance' [0,1J. of value foundation for their designs this is false of one. Many early and * The alternative assigned In this believes person work is believed to be totally one probability absolutely false. a the on subjective measure evidence. is true probability whereas sample a 1986]. [Shafer, which in the value probability to current hypothesis assigned 1761) - the degree given specific probability is based of The probability that hypothesis certainty. true, Theory probability is a approach, occur, Probability Theory but methods. instead 14 and the with ad A discussion the Bayesian Theory sections implemented not will assumptions of be and hoc the presented inherent limitations 3.2.1 this model will of Conditional Probabilities Bayesian probability conditional is based probabilities, which otherwise of probability event a hypothesis (E) has already (H) is event (E) zero read The . and as hypothesis probability is is true. represents fact that we would An set P(It the of always an is known. are often they are set or the formally, fact a as: to a (H) (N) value given between belief that the our following it is lightning. rains when If example. probability, In discussions prior to to any 15 as to in contrast, hypothesis before any referred it prior evidence. of was given other Bayesian the our is lightning, there is raining | It is lightning) evidence most on likelihood that it is raining unconditional probability More hypothesis of Consider the know that belief that it is incomplete is raining | It is lightning) P(It This situations N = set represents in probability is the represented the probability which one, event of occurring based P(H|E) which idea useful occurred. probability is conditional an A conditional uncertain. the on are the information concerning where that be highlighted. we one. is the information or Theory these probabilities, because A prior probability is represented and P(H) as N, is true before any In let probabilities, 1972]. [Lukacs, us trials were made and frequency relative RF(X) N = Nx / the Likewise, a events (X) and event (X) is: What we of are event the Y number respect to space is which event X is a event outcomes of reduced composed events represented = experiment (Y) were observations the of were 3.1 of relative sample space. hypotheses The to This we ) 3.2 frequency to know (Y) reduced with sample set original frequency of Y **xny__ RF(Y|X) = and RFfXOY) RF(X) 16 3.2 we get ( 3.3 in given as from 3.1 substitution . ) Nx by (X) event need event from the relative The is observed correspond n (XHY) event In other words, that all where ( X. ratios observed. /n is true. (X) RF(Y|X) of conditional frequency of interested in is the really given of frequency Nxny zero hypothesis a ( representing that both = of n relative RF(XOY) idea statistical the number represents belief that to the return of between number formal definition a Given of a information is known. other to build order is N where representing the degree one where = ) Given that eguation the 3.3 Let P(Y|X) called occur ed. P(X) > 0 be (.Q.,S,P) trials of the suggests Suppose events. is number a 0 definition. space Now let X P(Y) = probability Y of and 0. From that 3.4 we see P(XHY) = P(X) P(Y|X) P(XOY) = P(Y) P(X|Y) Pm we 3.4 given Y be two events By eliminating P(Xf)Y) Bayes Bayes' ' event X has that suppose obtain PfX|Y) ( 3.5 Theorem can specifies be a way that calculated. From conditional equation 3.5 we see that P(H|E) P(H) = ( PfElHl 3.6 ) P(E) P(H) ) Theorem probabilities Where two ) P(X) 3.2.2 be Y and : and > X and ( conditional P(Y|X) then is large, experiment P(XQY) P(X) = the and following probability > P(X) the of and hypothesis (H) represents the P(E) or represent event (E) conditional 17 prior occur, probabilites respectively. probability of the that P(E|H) event (E) hypothesis given represents the is known. formula by applying denominator. a set let B be events, the total an of of the eguation probability that Theorem The Total Let A be side Equation 3 Bayes' formula. left The . the posterior evidence Bayes' (H) . 6 (H) Rule as from this to the rule probability Probability after is generally known be derived can states: exclusive mutually occurs and exhaustive 3.6 we arbitrary event, P(B)" ^P(Aj) P(B|Aj) = applying this Therefore, Bayes' to equation at arrive Theorem: Assume exhaustive a of mutually events or hypotheses P(Hi) PCElHi) ^PCHi) P(E|Hi) = The Total all hypotheses This hypothesis in the the remainder 3.5 ) be in the means of our by decisions made As are space given space can occur. Bayes' , requirement mutually time any (S) space As only a formula one basis ( for Equation used. characterized evidence. mutually sample at discussion, Most decision making are in and introduces the sample that sample (H) rule Probability exclusive. will exclusive set PfHjjE) that rule new a single based environments piece on a information 18 of are not evidence. collection of is gathered, generally Instead, information and beliefs must be to revised to extended this relect new accomodate Bayes' evidence. multiple pieces of formula evidence be can as follows: P(H|Ei...En) = P(H) (Ei..^!!) P(Ei...En) Unfortunately number of probabilities above concept 3.2.3 equation of events or has situation or a direct a other, is merely represented on of an event (H) are true to occurrence situations or has no of Conditional false. At a bearing In this where another characterize hypothesis. situation on case, the our event (E) totally independent. is thought that events X and Y are it follows that the probability equal primarily belief that our need hypothesis it affect particular the on in describing we occurrence concerned probabilities. however, the If useful based either where in events is extreme, simplify form is to apply the workable hypothesis other making this to One way have been conditional are the Independence Assumptions of belief increases, also ) independence. probabilities evidence X more sections probabilities each a increases, evidence required The preceding other and to statistical Statistical with of computationally impractical. solution the the amount as 3.7 ( to the probability as: 19 of Y of independent Y alone. given This of event is P(Y|X) By substituting this into events be describes relation further to a to true. This probabilities, events the world of be can can be & all new & E2) form, the to order based that the hypothesis 3.8 incorporate the idea of each be 3.8 that independent in in are which = as a ) in of equation P(H) can extended relation hypothesis particular is follows: P(X|H) PfE-jH) probabilities case of is gathered, evidence apparent events: other independent are Y|H) = in the required evaluated relevant ( P(Y|H) Bayes' formula on as the new can be new updated 20 ( required one event are is From received easily. 3.9 ) reduced [Charniak, the hypothesis evidence. evidence P(E2|H) P(E2) PfE^ As P(Y) as: PfHiE! those that two see independence assumptions, these written In this we P(Xn) represented P(X By adopting In world. describe subset that events to the conditional V(XL)... = P(X) = include to extended P^O.^) This 3.4 independent if: are can P(Y) equation P(XOY) This = must 3.9, be to 1984]. re it is the probability of Ignorance 3.2.4 The Bayesian explicitly Probability Theory represent total does ignorance. In not offer to order the total lack is The maximum entropy assumption assumes used. events as a as possible each result case probability of least two would and over is event representing the in the knowledge, independent are evenly of sets then be will be up a amount further a of P(B) = discussed 21 probability that all . As For example, B the A and follows: as = .5 that The validity in 1985] value commitment. events background against. simulate assumption [Cheeseman, events distributed neutral compared all independent be way to then distributes the uncertainty assigned P(A) This the maximum entropy a section any of 3.5.3. new changes these can assumptions 3.3 MYCIN An - Early Mycin medical Expert System diagnosis the Stanford Heuristic Shortliffe Group and in collaboration Mycin was developed to treatment of key is uncertainty that is based Mycin during physician, backward an the towards critical the feature diagnosis, reasoning express This One the of for the account decision clinical and making. the physician condition of is asked their a benefit The questions number hypothesis the to backtrack used attributing to the and system the acceptance was Mycin of is focused are This 1982]. [Michie, of A backward chaing of by asked a patient. is then method with a by physicians. To handle a of particular a in the diagnosis tree to determine the disease treatment. groups 1982]. [Michie, how to of Disease Infectious interactive dialogue an inference and/or appropriate that on which chaining part 1984]. concerning the questions as Edward by infections. was inherent in [Buchanan, Shortliffe, the in 1974 blood in Mycin faced Project physicians and developed was with School aid meningitis challenges through Programming Stanford Medical at system the uncertainty Shortliffe model degree of Buchanan and in Mycin varying degrees belief is 22 involved in of which belief medical developed allows a as implemented physician in facts characterized and and a to hypotheses. probabilistic weight called session of would which an that by entering and part 1984], The definition factors and the units of and of in this some functions Certainty certainty measure: representation of is as read h, hypothesis, Similarly, the used cases factors and belief is the formal measure the are two separate given and interpretation of combine based of = of [Adams, certainty them be will evidence, of independent on independent two The formal x increased belief of = e, is equal disbelief is x e. measures confirmation in the to x. increased disbelief the evidence, 23 however, equivalent disbelief. notation measure to be derived from can is shown, notation to was model Factors MD[h,e] hypothesis, h, 10 sections. belief given representing the and its inherent model MB[h,e] which 1 of to probability converted formal and in the following Mycin between number It has been restrictions. theory Mycin's degree his express 1976]. probability 3.3.1 a interactive an in developing this reasoning substantial presented could Bayesian probability and strict a physician During then be automatically Their goal assumptions a event [Shortliffe, values avoid certainty factor. Mycin, with certainty a The in the requirement is due to theory stating for an that C[h,e] = 1 a - C[not piece does h,e] of not [Shortliffe, that evidence In particular 1-x. Even though measures are negation the of hypothesis the can be avoided to a These measures belief of in terms represented two by using (MB) Bayesian of MB[h,e] = retain MD[h,e] = a independent strong disbelief as (MD) hypothesis prior conditional subjective must be to any probability theory, beliefs the disbelief 3.9 represents the evidence the that ( and all in the P(h|e) represents 3.10 ) in the hypothesis, decrease 1984], hypotheses Therefore, one. h, and a l-P(h) equation in disbelief If the In evidence. alternative to sum [Buchanan, Shortliffe, 24 belief the known proportionate ) P(h|e) subjective given 3.9 ( P(h) - evidence probability assigned represents the be follows: P(h) represents can Pfh) - - P(h) and probability Pfh|e) 1 P(h) degree, in Bayesian probability theory. foundation where a inherent in restrictions factors have stating that the hypothesis to of of negation hypothesis a that states experts uncomfortable of belief, certainty of particular this, many some probability theory a support of this Simply, they believe in degree, x, they they believe in the the support that though expressed supports affect necessarily that hypothesis. 1976]. piece given of evidence be increases than greater P(h) representing the if a piece P(h|e) of our growth of belief our On the the and other in hand, hypothesis, a of value will increase will belief P(h|e) MD would increased disbelief. our independent these two combines into disbelief and MB of belief. P(h) certainty factor of our decreases evidence increase representing measures the value and be less than would The belief in the hypothesis, single a measure as 3.9 3.10, follows: CF[h,e] Substituting and , into the been the single measure values from the that In Mycin, facts certainty that definition, be MB were prior The probability, P(h|e), Because to combined are it has this, of factor may be expert and a more express reasoning model assumes that the values of MB and are adequate estimates of the calculated MD if the necessary known. certainty contain and an 1984], expert would probabilities and CF[h,e]. intuitive term for implemented in Mycin equations probablity, of that the [Buchanan, Shortliffe, received MD[h,e] that both the shows conditional suggested naturally - the probability ratios, into this formula P(h) MB[h,e] = MD of [0,1]. Therefore, of [-1,1] There are some are CF a factors level of restricted applied of to both uncertainty. to is limited to number 25 are values values special cases rules By in the in the range range that define the properties 1 = MD[h,e] and hypothesis 0 = and negation = CF[h,e] CF[h,e] It is is in a obtained joint out point is that all the 1984] . belief MB^S-l In evidence the that an evidence As in & new a hypothesis case = that or values of of the the all that conclusion MB is & and = 1, by adjusted MD to belief. our are and is important to these functions it is assumed other [Adams, increases that MB[h,S2](l s2] of each adjusted new - It is to say, MB on as incrementally assumption obtained is usually favoring of in Mycin. is + MB since define how That MBtl^SjJ 26 MB[h,e] Mycin, evidence is independent from MDfhjSj^ in the where MD are independently combined underlying In and combination is done evidence absolute contradictory information. of the hypothesis, s2] or MB[h,e] when certainty situation the of in Mycin independence. of absolute In the functions is certain hand, other MB[h,e] where representing -1 or remember evidence following acquired = case undefined. effect independent measures, disfavoring is pieces important to The CF[h,e] Evidence numerous the reflect On the there is conflicting 1, = Our belief evidence 1. = In the is absolutely expert the hypothesis. and on 1, = Combining based the in the hypothesis MD[h,e] 3.3.2 and of evidence 0, = MD[h,e] disbelief these measures. of as - that our follows: MB^S-^) definition MB ( 3.11) will 0. equal &.s2] MD^s-l and stated, MD be disbelief already Mycin, description hypotheses Functions of evidence is our a Our were belief the acquired also defined to disbelief or fact particular defined are values or for both the belief 'and' Boolean defined are in the as and min (MB[h1,e], MB[h2,e]) MDth-L & h2, e] = max (MD[h1#e], MD[h2,e]) in the disjunction or the by represented are of follows: = hypotheses 'or' conjunction e] disbelief the multiple h2, two MB observation. disbelief and in allow & and of or MBth-L belief of MD[h,s1]) - present. two different hypotheses or' MD[h,s2](l + functions given operations. Our follows: MD[h,s1] new disfavoring increased proportionally to the belief are In reflect as adjusted = as to hand, other MD will evidence, Simply the On the boolean following equations. MB[h1 or h2, e] = max (MBCh^e], MB[h2,e]) MD[hx or h2, e] = min (MD[hlfe], MD[h2,e]) In viewing these assumptions functions, some are concerning the relationship the For example, consider are mutually exclusive. the hypotheses either there would hypothesis. case In equal The 27 where this zero of the hypotheses. hypothesis, case, the regardless disjunction underlying of the hi, and conjunction of our h2 of belief hypotheses, on in the hand, other belief in In in in for account facts beliefs our of this, our belief Shortliffe and Buchanan incorporate the the of evidence. ( 0, CF[s,e] ) MD[h,s] = MD'[h,s] * max ( 0, CF[s,e] ) belief MB'[h,s] and totally these the certain [Buchanan, s, are based Below is an on the Factors example 28 - given evidence, rules Shortliffe, the prior that are 1984]. describing sections used actually Certainty of represent in the hypothesis, factor in the following functions MD'[h,s] the decision certainty in the evidence, and disbelief represent experts represents 3.3.3 strength max are examples functions following * that from the actual hypothesis, MB'[h,s] of Mycin, defined the in the reflected the our the h, given In s. acquired CF[s,e] actual evidence, will belief actual our = measure we of to both applied insure that also to order MB[h,s] functions In these are is properly only In observations. disbelief or not hypothesis but particular To our 1984]. [Adams, uncertainty is or evidence larger than number certainty factors of which a separately a evidence in the supporting measure of in Mycin. rules and to equal decision making, beliefs our be hypothesis either our apparent would e. belief The illustrate how these in Mycin. Examples of an actual decision rule that could IF be The 1) 2) 3) in the found strain expert the of Mycin system [Waterman, 1986] . is grampos, and is coccus, and organism is chains organism The morphology of the organism The growth conformation of the THEN is There the In more suggestive is organism generic IF X & Y & THEN In this terms, mathematical decision H with factors belief are are example, the facts certainty factor has Consider evidence supporting certainty Our of is be represented from experts, based rules on the evidence. given absolute In this assigned expert's by Therefore, to this actual the level that the three certain. represented prior X with certainty 0.5 Y with certainty 0.7 Z with certainty 0.3 belief CF'[h,x 29 y & z] = 0.7. a rule. belief in the following in the hypothesis is & as certainty relatively high a is true has been 0.7 0.7. of factors. experts of 0.7 = z] expressed that the now, y & known to be are of form the would supporting belief that the hypothesis underlying rule acquired of expert our & to these assigned in the truth takes certainty this notation rules identity . Z CF[h,x As rule that the (0.7) evidence streptococcos given as of By applying functions the incorporate the the calculate actual experts strength & y & z] = CF'[h,x CF[h,x & y & z] = 0.7 Now the by applying hypotheses we * = 0.7 * = 0.21 & y & z] = CF[h,x & y & z] CF[h,x & y & z] expert the supporting certainty expressed downward. It knowledge CF[x & & y & y & z]) z]) the ( max 0,min(0. 5, 0. 7 conjunction 0. , of 3) ) 0.3 relatively low level a a zero of of of of adjusted expert falsity or value an certainty value has been noting that if assign of consequently the and to the hypothesis worth would can CF[x max(0, describing concerning the truth he evidence is * z] 0, max( evidence assigned we evidence, to at 0.7 CF[h,x & y functions arrive The * & of 3.3.2 section belief in the hypothesis. actual CF[h,x in defined a has no piece of certainty to the evidence. Missing information is simply disregarded from the For rule. case that the is It example as evidence several belief combination consider possible also of actual of MBth^ & s2] information concerning result new , is different obtained in the all these two that to relevant rules: 30 given decision hypothesis the a s2 = MBthfS-J is missing. hypothesis rules. substantiate must be rules. in the is the In this rules, the recalculated For case, example, as a RULE 1 X IF : & Y THEN H with EVIDENCE: RULE 2 IF : certainty X with certainty Y with certainty EVIDENCE: certainty 0.8 Z with First we must calculate for based rule 0.5 0.7 Z THEN H with each 0.6 the actual belief the on certainty 0.5 in the hypothesis in the supporting belief present evidence. RULE 1: 2: RULE The by CF[H, X & Y] = 0.6 CF[H, X & Y] = 0.3 CF[H,Z] = 0.8 CF[H,Z] = 0.4 factors certainty applying equation * max Y) & Z] = CF[H,(X & Y) & Z] = 0.3 CF[H, (X & Y) & Z] = 0.58 3.3.4 Ignorance CF[H,X + & 0,min(0. 5, 0. 7) ) ( 0,0.5) rules must now be 1 CF(H,X combined of ignorance the an expert 31 Y] + CF[H,Z]( - & Y] 0.4(0.7) factors representation that ( 3.11. & event max from both CF[H,(X Mycin certainty * has do or no not total offer an unambiguous lack of knowledge. evidence or knowledge In ) a concerning disbelief would be hypothesis and neither Unfortunately, due to case, the net belief equal evidence disconfirms 0. In CF[h,e]=0 case, is independent evidence confirms this or and disconfirms of the it 1984]. [Shortliffe, arise to equal indicate that the would hypothesis, h, his belief particular a certainty factor non-zero that has been the hypothesis of values zero. 32 by of of MB observed equal zero and MD. both amounts may also In this confirms resulting in and a 3.4 Dempster-Shafer Mathematical The Dempster-Shafer Mathematical first developed further Shafer Mathematical Shafer a on of This not the in the This between lack of theory knowledge 3.4.1 Representation with is of the with based an make First narrow by the this explicit all, can expert set The set. will an the in the hypothesis of it of This evidence. gathered not affect distinction Information evidence frame 33 in certainty. Uncertain of representation begins subset allows and of accumulated hypothesis evidence which that ability to larger a on support has single calculus Theory deals reasoning. evidence a probability this characteristics the on Glenn "A of of was Bayesian probabilities. because solution. Theory degrees accumulation bear calculus approximate theory evidence Dempster- The applies the order The it with set instead bears actual on of published Dempster-Shafer several to approach achieved does but has and standard numerical than rather Dempster-Shafer hypothesis be and Theory in 1976. although The way. theory attractive the Theory, evidence evidence same Evidence of in the 1960 's. theory Evidence" of the uses different weights the extended Bayesian much Arthur Dempster by Theory Theory the as Theory of in the Dempster-Shafer discernment ( e ) [Shafer, 1976] frame The . hypotheses or frame given can be all set is assigned possible subsets possible [Spruce, a the range of our exact belief 8 0. of the over exclusive events 2e by Figure or , 3.4.1 frame to in a and which belief corresponding to the represents of possible discernment all set of - of in the belief [0,1]. in the evidence, evidence The a basic probability in represent the as theory number to used is defined proposition assignment. by assigning function A numerical a Dempster-Shafer basic assignment is follows: as If hypotheses all possibilities mutually represented piece a probability are All of set Pine], indicates defined of subsets Fir, Given domain. a discernment The exhaustive. in events of is the discernment of is frame a is m:2e->[0,l] discernment, of the function basic probability the called then function whenever 1) m(^) = 0 2) m(0) = 1 implies that This subset will represents A. This total a assigned sum to cannot only the belief total belief value that 34 is of be m such to committed by all assignment the the to A committed to one. that exactly to subdivided is committed equal probability committed be to ought belief The 1. the belief belief represents The is 0 of numbers one's and set empty belief no to the Each the m(A) , proposition subsets of A; alone. a hypothesis is it [ Spruce [ Pine, [ Pine The Pine, ] Spruce, [ Pine, Fir Fir ] ( ] ] subsets of the set of Figure 35 coniferous 3.4.1 e ) [ Spruce, [ Fir trees (2W Fir ] ) ] represented by the belief hypothesis given the hypotheses function. is the (A) that imply sum of A and The total the all the belief in beliefs exact in in A belief exact a itself. BEL(A) = ^_^ m(B) BCA To the total obtain to m(A) the The belief in the hypothesis A, quantities for m(B) all proper basic probability assignment from the belief one B subsets then be can must of add A. recovered function: r = . m(A) </ . (-1)IA-BI ,-, Bel(B) BCA for It one follows that for a all AC0 belief given basic probability assignment, the Therefore, belief there assignment probability function and is only that one Corresponding to belief functions, other The functions that Doubt evidence function refutes represented as a are The of A function. by The the function. there the degree to hypothesis A. three are which Doubt evidence. the function is follows: DOU(A) Our doubt given in characterizing useful expresses a conveyed the basic probability and for belief information is being same is only function there is equal to = our Plausibility function, 36 BEL (A) belief on the in the other negation hand, of A. expresses the extent of hypothesis A. which This fails to doubt one function is PL(A) PL (A) PL(A) the , also extent terms of function called the to which = 1 = 1 upper expresses be stated BEL(A) DOU(A) - - probability function, /_^ /_ - m(B) <C^ < BCe PI function (A) S\ = that m(B) m(B) is <p useful is defined function In the plausibility when dealing is the commonality function. functions commonality plausible. BCA BOA Another expresses follows: as = follows: as the basic probability assignments, can the refute finds the hypothesis one P1(A) belief or Q(A) = The follows: as <C-~ with m(B) BC9 ACB The commonality Q(A) that have A Q(tf) = as their , is equal to the sum of all subsets It is interesting to can be represented in terms subset. note that 1- The belief commonality Q(A) function now . Bel(A) = * T (-1) BCA 37 lBl Q(B) of assignment now be recovered. can commonality functions are PL(A) also 2_ = <__ The plausibility related (-1) follows: as |B|+1 (-1)IBI+1 and Q(B) BCA it Therefore, correspondence For each which function body belief our and of the plausibility function our knowledge a the BEL (A) and case, this 3.4.2 Belief, In an section, on one consider to Based on probability sum to shown one. that are to the again Bayesian Plausibility - we Spruce, Pine evidence, to in Figure 3.4.2. 3.4.2 for the belief, 38 is of previous also could Based take of represented by a 6, assignments Figure in the number assign members all The probability we special An Example The Fir. or When probability. (X) variable belief assign will assignment the and precise, in Figure 3.4.1. scenario is gathered, and In this one. belief bound. upper certain general and The an within interval, to clarify the definitions available calculated values to this the is (A) equal reduces values: which the (A) of 1984] lie. must represents hypothesis Commonality attempt three subsets PL theory evidence of of information. same lower bound the to one these functions define hypothesis a in are [ Garvey , Lowrance , Wesley, about represents the represent evidence, interval" "evidential belief, plausibility, basic probability assignment and commonality one follows that the for on 2e. basic that such each shows plausibility set the and they are Set m Bel PI Pine 0.2 0.2 0.8 0.8 Spruce 0.1 0.1 0.4 0.4 Fir 0.1 0.1 0.5 0.5 Pine, Spruce 0.2 0.5 0.9 0.3 Pine, Fir 0.3 0.6 0.9 0.4 0.0 0.2 0.8 0.1 0.1 1.0 1.0 0.1 0.0 0.0 0.0 1.0 Spruce, Pine, Fir Spruce, Fir t Calculations for a the set given of for the belief, body of evidence coniferous plausibility, pertaining to the trees. Figure 39 and 3.4.2 commonality subsets of functions, based commonality As assignments. their is shown, is due to the This probability assignments [Spruce] plausibility summation do intersect intersect equal to the exception it. all the by sets the of that set [Pine] of Therefore, the Plausibility value is 0.8. as the of a assignments value for Q order multiple to sources, provides are combined. body the of joint effect, and different bodies of the need second evidence 40 to [Spruce, Fir, Pine] . is combination when to be need two or is to be method a evidence. of evidence combined scenario from obtained developed where scenarios scenario the values only Sources information facility for evidence the and from Multiple Dempster-Shafer has first sum that have sets case, to the equal is 0.1. accomodate generally two The all [Spruce, Fir] Evidence Combining for In this subset. is [Spruce, Fir] set resulting given for with for There subsets represented sets those that is all , are In set In the case with summed 3.4.3 given set. the probability The a that [Spruce, Fir] be of with Commonality of for the basic to the the probability assignments of [Spruce, Fir] than greater non-zero values (m) pertaining for evidence [Fir]. and The direct no though the belief function is [Spruce, Fir] zero. these probability on must more to when facts obtain two combined. be in their entirely Given a a &, specified concentrate on hypothesis combining from two of totally belief function that frame same is combined not commutative = <8 ml m2 is the Y and new combining are 1981]. it and ml can Let evidence length in a new combined be must and frame of of m2 over evidence which combination Let ml result of a the of their orthogonal k = (A- one. all subsets basic probability In order to rule, evidence. computes order same ml(X) XHY X a is rule be basic discernment sum. combining The ml 6. new and m2 , as: m(A) Where negates pieces combined the since the as probability assignment, is defined The associative. defining , two on impact the function to be on or formal mechanism to a Combination of important and of now evidence. represents assignments probability Let m of discernment. of supports different bodies functions based belief The have already we We will that provides Dempster's Rule evidence. both model two belief evidence, of evidence, evidence different bodies Given the body given Dempster-Shafer combine is a how to combine elementary facts. shown The and be = m2(Y)) A whose intersection is A. assignment as a understand this result m(A) of m2. intuitively represented in a the basic probability be depicted Now * 41 a unit model assignments portions as Consider geometric of square a combination line with for [Barnett, body each segment of two sides, one side Figure 3.4.3 vertical in mass for committed a is 0 between square. The one with intersection m2 strip the (A) in the we of results of combination therefore, subset; . m2 and rectangle to between us look square sum must due ml Rule at and in the impossibility certain impossibility of of some Ml hypothesis to the empty a of as shows of c. a of The m2 where K is the sum The weight log(K) Dempster's large on of . rule are two amount the a and other hypothesis result of interesting There hypothesis M2, e Example An particularly 3.4.4). a. - ml as , are values set. is defined possibility hypothesis by K, <f states, setting k is defined of examples (Figure 42 by = aob therefore, are new all value other Combination consider m2. values and =0, sources possibility in the of value Dempster set. accomplished each two Zadeh to evidence, certainty is intersection, no assigned (^) assigned First combination. example The values Dempster's Let m2 This l/(l-k). conflict 3.4.4 The 0. = is to the empty multiplying non-zero very made 1. and and =0 equal of ml horizontal a there where the ml so normalized of a m2 representing segments associated particular that m($) however, all and m2(Aj) rectangles committment (6) (A) from side values. For a ml * ml(Aj) other There may be more than AOB. all to combined strip the and the two line shows orthogonally a ml representing c shows sources of also in the hand, and is the hypothesis ml (m2) m2 1 probability mass (ml(Aj) * m2(Aj)) m2 (Aj ) Figure 43 3.4.3 Normalized Orthogonal orthogonal sum Set m2 + ml sum m2 (ml + m2)/(l-k) a 0.9 0.0 0.0 0.0 b 0.1 0.01 1.0 c 0.0 0.1 0.9 0.0 0.0 a (0.9) ml (0.0) b (m2) (0.1) c(0.9) a (ml) k = * ml(X) < m2(Y) = (1-k)) = 0.99 xny=log(K) = log(l = 2 / Figure 44 3.4.4 weight of conflict b the as possible only it to be considered answer highly improbable assignment probability 0.1. of for this source has distributed the entire a,b,and of discrepency between the beliefs If questionable. the reliable, the not interesting to note assignments do not reflect probability of the sources. is combination sources the of same ml evidence This shows example narrow in on frame of be of seen sources a as particular 3.4.6 45 have b a, rule and for are committed c sources hypothesis. each is c, have The reinforced. reinforces the two the conflicts. is collected, we This can hypothesis. illustrates discernment evidence fully is there in that the two for to be in the seen sources discards and more It Again combination of is slightly Dempster's Both . it probability conflict assignments rule implies that Figure the the between agreements m2 large considered that the dominant hypothesis that shows and can identical probability result not belief to hypotheses of This respectively. source resultant example has any source unexpected. the is the two sources, in Figure 3.4.5. shown amount so that the interesting Another are an Each source Due to the each sources is result of of the among the weight neither m2 by shown provided implies that the ignorance. or that the reliability seems unit In other words, uncertainty and counter-intuitive result. which c, reliable. entirely degree rather ml is which Shafer has answer hypothesis though both even a combination [Spruce, Pine, of Fir]. beliefs This over Normalized Orthogonal orthogonal sum Set m2 + ml sum m2 (ml + m2)/(l-k) a 0.3 0.3 0.09 b 0.3 0.3 0.09 0.26 0.26 c 0.4 0.4 0.16 0.48 a (0.3) ml (0.3) (m2) (0.3) b c(0.4) a 0.09 a 0.09 f 0.12 <f (ml) (0.3) b 0.09 <ji 0.09 b 0.12 <f (0.4) c 0.12 <f 0.12 ^ 0.16 c k = <_ * ml(X) m2(Y) = (1-k)) = 0.66 *ny=^ log(K) = log(l = 47 / Figure 46 3.4.5 weight of conflict Normalized Orthogonal orthogonal sum Set Pine (P) (S) (F) ml m2 0.2 0.0 0.3 Spruce 0.1 Fir 0.1 Pine, Spruce (PS) Pine, Fir (PF) Spruce, Fir (SF) Pine, Spruce, Fir 0.2 0.1 0.3 0.3 0.0 0.0 0.2 0.1 0.1 + ml sum (ml + m2)/(l-k) m2 0.17 0.24 0.22 0.14 0.315 0.20 0.11 0.16 0.04 0.03 0.03 0.02 0.01 0.015 (PSF) (m2) PS (.2) P 0 .06 t (.1) S 0 .03 s (.1) F 0 .03 ( . 2 ) PS 0 .06 ( . 3 ) PF 0 .09 (.0)SF 0 0 (.1)PSF 0 .03 k = r S f ^ log(K) SF PSF .02 9I .06 P 0 .04 *S .02 P .01 <f> .03 S 0 .02 S .01 S .01 F .03 (fl 0 .02 F .01 F .06 PS 0 .04 S .02 PS .09 P 0 .06 F .03 PF 0 0 .02 .03 p F 0 0 s PF .01 ml(X) = log(l = 0.154 F .03 * m2(Y) = / (1-k)) = Figure 47 PS 3.4.6 0 .02 0 SF .01 0.3 weight of conflict PSF completes 3.4.5 discussion our Ignorance The Dempster-Shafer theory distinction between explicit ignorance and high a ignorant concerning the hypothesis level its case set of discernment all lack no consists [Male, = possible BEL of knowledge or a person belief In terms - an When he has PL (A) = for allows of disbelief. of determining of 9 The total a Theory ignorance is defined Consider the of evidence negation. Ignorance frame of hypothesis, a in or Dempster-Shafer The the coniferous tree example. of either in the of as is follows: (A) the sex of male and female: a person. Female] is subsets 2" [0,^, Male, Female] = Bel (Male) male likewise Bel is female. In and person was available the person would Bel be evidence called be all zero ignorance. would was and vacuous of the case, male It of be very Bel belief Generally (9) 48 person belief the supported the belief that female, follow In the low. Bel the (Male) be would that one. Bel and Bel This m(Female) (Male) case represents vacuous evidence and m(Male) is that the little function, stated, that the that very would available, the belief represents specifically either also at (Female) very low. set the that was (Female) would the degree represents that and no (Female) situation, total belief function is obtained by setting: m(0) = 1, m(A) Bel(O) = 1, Bel 49 = (A) 0, = for 0, A f 0 all A 4 all for 0 3.5 Comparison In three an attempt theory. x A functions theory. is decision people to truly of a make psychological therefore maintaining The these Dempster-Shafer intuitive way to restrictive. 3.5.1 Representation hypotheses to one. Bayesian the are The varies theory, a define An concept that of functional approach to describes uncertainty offering a the requirements problem restriction of uncertainty, actual' require among the three by x manner. each approach always situations level probability is not the of approaches. more appears imposed naturally to be the Belief of theory, probability assigned expert specific actual express least In all decision that a these theories of rational a the requirements in decisions to their However, do rational three as that structures however, truly a All . adhere solving. provide limitations and data and People, in more 1971] [Carnap, by decision making, is defined decision requirements decision making expert to describe rational rational' the by underlying how Dempster-Shafer Approaches formalism primarily based in probability mathematical specific and these approaches to uncertainty of abides Bayesian, Mycin of forced 50 all values so alternative that to divide his belief they among sum individual hypothesis in the each relatively easy task of of the propositions. belief our in very difficult belief. This often In However, limit forced to expert In the of discernment possible we is so a a In this restricted to our doubtful of probability approach. actual belief. by independently avoided the case, to all expert is alternative in his belief. confidence point assigning that the so upper one. where theory our offers Dempster-Shafer of that the total more expert a good longer forced to to commit therefore providing advantage 51 a a of alternative space belief will approach, the assigned the or to sum to ways frame one. express supporting only the belief that he better incomplete, theory, belief is alternative In this is knowledge hypothesis the individual hypotheses. This is to given his of all it find number a in a confident might using the Bayesian longer restricting him to no not either probabilities his of subsets the provides when assign is sum the In all beliefs. are to assign required overcommittment situation to in, is individual hypothesis to their beliefs, we assign expert disbelief. regardless the an beliefs our however, restriction and Dempster-Shafer This an this of approach. his still leads to probabilities the that of that case proposition, event measuring belief hypotheses sure uncomfortable to is he Mycin, longer are m the to the hypothesis value we is This space. in fact ignorant concerning or hypothesis, a or In the his belief no when sample expert is representation Dempster-Shafer not of is no confident his true theory is most in the evident 3.5.2 a no is both In this described in ignorant concerning a belief to the frame of entire set possible assign his belief to any of a allowing and is known Though this established by in found it does not offer express his lack assign a point the of in is his the forced to argues that this events all a therefore express his Even are amount a clear and probability to person amount more of than is independence and is further theoretically, least A by independent. approaches the knowledge. least assumes though that neutral statistical three all expert 52 not representing the argue represents entropy all way to assumptions 3.5.4. section maximum natural is justified, argument is person represents which He a assign Cheeseman assuming that assumption discussed entropy Opponents when overcommitting his belief. Bayesian probability, without way to a individual hypothesis, In committment. an hypotheses. support Dempster-Shafer can (9) to surpasses offering 3.4.5, discernment knowledge background by there where available In the hypothesis he of maximum situation Dempster-Shafer section comfortable more applying the truly the ignorance. represent as approach, lack a knowledge or case, as Bayesian probability and Mycin explicitly to total ignorance. characterized information relevant decision. is of Ignorance Ignorance is case of committment, unambiguous is individual still way to required hypotheses, therefore forcing In person has be naturally provides a more in maximum neither 3.5.3 approach Prior All theory beliefs Prior than because to any each tremendously. a a value of probability approach ignorance than and ignorance, simulate Problem - of experts. also requires as of explicit the as evidence Mycin, 53 on rely The integrity of They differ in that prior reason to be being probabilities highly are more often subjective known the may be the and to be or that prior quite probabilities though the considered are expert prior Subjectivity probabilities valid. stated In to he ambiguous very approaches individual therefore their non-zero a - considered are of a assigning representation conditional being as prior a probabilities probabilities world his Given that hypothesis, a express still methods these Bayesian probability disregarded to was it is offers of subjective subjective hypothesis. Even though this entropy. Probability three beliefs. to contrast reasonable Dempster-Shafer the exceeds Although both Bayesian probability representation. offer a comfortable intuitive more Bayesian probability Mycin to zero information concerning no representing to equal to his belief zero belief that of ignorance is represented by assigning Mycin, certainty factor would statement belief. actual a a * is state' different of the and may vary theoretical basis of MB included MD and experts practice, which conditional probabilities are both combine are probability theory. in ways estimates as Many concept that new still of not problem 3.5.4 Statistical that discussed here. well sufficient systems and prior to reason can refine be designed our [Nutter, strict must more be initial 1987]. statistical idea Independence can In a to the cases is simply approach subjective support or decision be most found actually known to be on which the Exclusivity mutual exclusivity in all situations, these outcome assumptions is actually known The invalid. of part are three theories involving is due in 54 & Mutual and information than assumptions this of applied of solving. especially those documented support the independence Statistical these of quality and These value. in the majority Therefore, and are one gathered that the data to probability many cases, or Expert is However, statistical assumptions In probability is the only legitimate making predicate into improve can maintain available. personal they evidence interpretation. required not (P(H)) probability in themselves not abandon such prior (P(H|E)) probability probabilities are to estimate the certainty asked concerning the validity questions experts to estimate their prior beliefs. asked specifically factors, prior medical and diagnosis, repercussions these approaches to the in limited of is not number of implementations. actual A number verify the was of integrity designed to of compare known probability data certainty factor known and One factor P(H|E) Mycin to on results. the certainty ( P(H) and conducted the program computed for MB values have been studies such from computed known are ) study and using the combining functions MD. The outcome of this that the largest descrepancies between the two CF values were impact their overall to be rather short reasoning A small number accomodate independent of piece note due to the predominant use evidence or be a rules. There techniques help the by increasing also A Caduceus. good together It the a the solve actual have been proposed exclusivity. system makes suggests rule. single to offered independence grouped that though this may unmanageable have been of of that into mutual all a non- single is important to problem implementation much more difficulty in obtaining number of heuristic the restrictions example can be seen [Charniak,McDermott, 55 relatively help and avoid to appears 1984]. [Adams, solutions such evidence However, in Mycin restrictions theoretically it 1984]. assumptions solutions One exclusivity. longer and these of chains. of the evidence [Buchanan, Shortliffe, chains reasoning interrelated to and study showed due the of in the 1984]. mutual expert 3.5.5 Methods Inferencing A function key decisions information. falls not In the making. offering any are through the combination The represented BEL and the bounds defining statement There is presently based on these bounds. could be used these values The should ignorance in the on The . reasoning. reasoning cases, prior will in are a make number inductive-style of People and Inductive reasoning usually perception 56 of the loss of 1981]. our of on in of a implemented style of to deductive In Mycin. how most people incorporate their -state' in making [Thompson, were represent tend to results which masks used deductive success reasoning based intuition be can drawbacks not PI or The probabilities in Mycin a of have limited the decisions. [Barnett, expected with Bel However, in that it used methods for making. mechanism the each for decision making value probabilites. minimize and interval. evidential unknown deductive reasoning does actually the differs conjunction There which is used approach reasoning basis this be be calculated for mechanism Possibly the updating by decisions that 1985] accepted can for decision solely Bayesian calculated no and [Barnett, process the of ignorance in belief functions directly PL functions for decision means Dempster-Shafer approach, are 1981]]. make available effective uncertainty carried is the ability to systems is here that the Dempster-Shafer Theory It by short expert inferences from the draw and of a more personal their solution situation. that best represents Deductive or is most reasoning, shortcomings with Typicality-based individual its kind has also [Nutter, the data has been uncertainty, a of shown default reasoning such generalizations typical as "Birds which Fly", or 57 evidence. to have severe typicality. is involved in centers property that is typical 1987]. or of on whether other an things of Chapter Four Fuzzy Set The Zadeh Theory theory in 1965, complex [Kandel, fuzzy of Possibility Theory a development approaches, sets due to his systems development fuzzy of Since has theory Possibility Theory Fuzzy set language as tall, is a applications young short, generalization presented basis in fuzzy of sections Possibility Theory model for ignorance be included. be A sets 4.1 be will and comparison Probability Theory apply to and will 58 of as of hypotheses or exist. for natural variables Fuzzy theory set in that such the all fuzzy sets also logic will be hold 1982]. fuzzy 4.2, respectively. in examined represented suited theory, set method not linguistic section 4.3 A discussion uncertainty. representing can that [Kandel, sets well do prevalent. are abstract of theorems and for non-fuzzy The etc. concepts definitions vague where the significantly. formal a vague theory is particularly definitions true or of probability-based offers representing uncertainty due to distinct boundaries that time, grown Unlike sets. where analysis introduced by Zadeh in 1978 later was fuzzy of strong interest in the 1982]. set introduced by Lotfi first was in possiblity theory both Possibility be discussed in the as of a how also will and final section. Definition 4.1 Since it is Fuzzy of fuzzy a Sets is set a generalization first necessary to understand the In a crisp set, assigned a value of either within or is strictly set. contained set. In describing universal set are A non-members. a element is defined by of example temperature set value 1 seen of (X) crisp set, of a the universal indicating all a concept outside crisp set, a of 0 or the of elements discrimination set in specified of the members a is it is and delineates boundary Formally, crisp whether into two groups, non-members. crisp the set each function: 1 if and only if XA 0 if and only if XC A all integer = crisp is the set in the values universal a a 1 sharp distinctive from the An element separated members uA each of set of [50F, range X = (0,30,60,90,120), or 0 indicting it each membership in figure 4.1. (T) Elements 0 0 30 0 60 1 90 1 120 0 Figure 59 100F] 4 . l . Given element in the is set the assigned (T) , as It can be determined with certainty that either contained the set. and non-members. A In is set a In fuzzy a assigned the membership fuzzy a a a each delineates of members is boundary sharp element specified grade of each This is formally set. outside boundary. set, in value this set, is element is strictly or boundary * fuzzy' gradual defining specified set sharp distinctive by replaced the within each element in of range the universal representing respect by represented to the the following membership function: Let X X : uA in As a to 0 a 1. of the from a values in the position within represents the uncertainty the elements elements of or of a 0 to an the an the case vagueness 60 specified set it is of allows Clearly, a exists set is 0. between of specified universal universal A fuzzy the fuzzy being a concerning differs set of elements either crisp set assigned assignment an set outside strictly representing set. numbers the value 1 real of element also of 1988]. element boundary special in the the of vague outside or if in that it range in the if within assigned set crisp interval the [Klir,Folger, Likewise, it is set 1 crisp set, contained of [0,1] -> is clearly value set, [0,1] denotes Where from Universal = when wholly set no the membership set. In this would be assigned case, of all values of 0 or 1, and To no elements help illustrate the consider universal membership Elements X set, a assigned HOT and = fuzzy (set of of sets: hot -fuzzy' fuzzy a temperatures) in the range particular (Temperatures, fuzzy to 0 of 1 of cold element representing its 4.2). HOT COLD F) the Given . each (Figure set boundary. set, (set COLD (0,30,40,50,60,70,80,100), value in the in the exist the concepts two following temperatures) is would 10 0 10 30 40 0.8 0.1 50 0.5 0.3 60 0.3 0.4 70 0.1 0.7 80 0 1 100 0 1 4.2 FIGURE In the fuzzy boundary set vagueness uncertainty or fuzzy This inherent or assigned representing the element The all the belongs support elements It to of a indicated is that degree in the of concept element each a are of fuzzy that have set by is non-zero The the the that believes equal amount size subjective fuzzy in the vagueness hot. note person particular 61 level some is due to the important to is is there concerning their membership uncertainty vagueness boundary- grades that in the are (40,50,60,70) elements in the linguistic uncertainty fuzzy the indicating area, set. HOT, of of the membership in nature, that an set. to values. the crisp From set Figure of the 4.2, support the of SUPP(HOT) A fuzzy element the membership set (40,50,60,70,80,100) fuzzy sets, height grade HOT of fuzzy a to any assigned and COLD, are at least one possible is set of its normalized to the equal elements. with height a l. of Fuzzy can be modifiers or from fuzzy subsets. as illustration, HOT, a Elements very, really, are more modifiers if the the subset, support of *very' is 1982]. the applied to to created. subset, VERYHOT, HOT (Temperatures, F) resulting less in the As is VERYHOT,- fuzzy therefore, referred [Kandel, etc., new subsets and, often usually, create the concise are modifier fuzzy to variables linguistic hedges new in Figure 4.3, as Generally speaking, Fuzzy literature such linguistic modifiers uncertain. set, to applied sets an the fuzzy As shown is half VERYHOT 0 0 30 0 0 40 0.1 0 50 0.3 0 60 0.4 0 70 0.7 4 80 1 8 100 1 1 FIGURE of is HOT, is assigned the highest The grade. highest membership Both = set, is considered to be normalized if set of fuzzy the support for the subset, 62 0 4.3 HOT, indicating that the new is fuzzy set found in the only two less 4.2 a more fuzzy boundary uncertain or It a value goal approximate true either is 1988]. with In in the value important to note, fuzzy a as and is = 1 fuzzy of - Intersection uAnB(X) = a fuzzy two no X subsets min[uA(X),uB(X)] 63 false. propositions is is It uncertainty to two-valued subset fuzzy and to do false. or uA(X) of clear representing the that of other hypothesis have been defined subsets In . method each true either reduced a fuzzy or [0,1] case (0) completely provide is hypothesis fuzzy logic, the functions B be Complement uA(X) 2) of is complement and or follows: Let A 1) for be can true fuzzy logic, in the that logic basis intersection theory include subset to be very imprecise range false or considered degree that the hypothesis As fuzzy new logic, every (1) logic to fuzzy of reasoning [Klir,Folger, a of being completely either is the exists, that the two-valued or hypothesis each assigned decreased to also vague. In classical precise, has elements of number Logic Fuzzy words, The set. indicating elements assigned concise of logic. union, in fuzzy set 3) Union two of UAUB<X) The = fuzzy max[uA(X),uB(X) ] intersection complement, illustrated in Figure 4.4, subsets, TEMP HOT HOT subsets and as functions union they pertain are the to fuzzy VERYHOT. and VERYHOT HOT U VERYHOT HOT VERYHOT HOT D 0 0 0 0 0 30 0 0 0 0 1 40 0.1 0 0.1 0 0.9 50 0.3 0 0.3 0 0.7 60 0.4 0 0.4 0 0.6 70 0.7 0.4 0.7 0.4 0.3 80 1 0.8 1 0.8 0 FIGURE In fuzzy logic, consist (very, of fuzzy the fuzzy of value the propostion and Consider for depends the values was on a yesterday the of cold day is very x a cold day is fairly of these claims will 64 result The the grade of fuzzy truth two truth being claims: true' a was consider day. in the following Yesterday was Yesterday (very true, example, cold can modifiers the membership strength example proposition fuzzy , For 1988]. recorded on also cold) truth ^Yesterday temperature COLD, claimed. Each this fuzzy [Klir,Folger, proposition actual subet, also and fuzzy or (hot, predicates usually) fairly false) 4.4 imprecise an 1 true' in different truth values based on claim expressed. the Fuzzy fuzzy of truth subsets real can fuzzy be as numbers of absolutely true in the range, by these and is false. unit Given [0,1], represented 0. can be applied subsets as shown now U True Fuzzy to modifiers these fuzzy for as value false would create False Very 1 true of or very of 0. of grade to sets True Fairly new fairly fuzzy False 0 0 0 1 1 0.2 0.2 0.3 0.8 0.8 0.4 0.4 0.5 0.6 0.5 0.6 0.6 0.7 0.4 0.2 0.8 0.8 1 0.2 0 1 1 1 0 0 4 ,.5 Possibility Theory Possibility Theory 1978 that in Figure 4.5. Figure 4.3 such U, linguistic value the membership grade set, varying example the by own assuming that truth to the for truth universal numerically by the represented as a by their values, assigned increases towards 1 the membership decrease toward and Consider interval, appropriate represented numerical be can absolutely false is Within this be can described below. membership true variables, and values represented degrees representing the set as behind a development was of Possibility Theory 65 initially fuzzy was set to proposed theory. provide a The good by Zadeh main in purpose formalism to deal information theory inherent fuzziness found in the with is is that centered to used make the around much decisions. concept of a the of Possibility possibility distribution. A possibility as X. fuzzy a constraint Zadeh defines fuzzy distribution, restriction F be a fuzzy X be a variable Then with distribution = the In an example, day'. In that values formal order U be and taking values the Universal set. in U, and let F act "X is F" translates a as into F. this proposition is is equal to R(X) = = a possibility R(X) uF consider this can terms, the example be to possibility relate the above proposition the assigned R (Temperature In of possiblity distribution function of X is defined numerically equal to the membership function of F. TTX cold in terms which TTX As subset the proposition Associated to be to assigned restriction. R(X) This that may be values is defined TTX, follows: as Let fuzzy the as the possibility distribution Let a a on denoted fuzzy to this distribution, 66 idea of a consider COLD, yesterday's propostion (Yesterday) ) set, 'Yesterday = restricts temperature. written as restriction to Cold fuzzy the be can was temperature, 40F, a degree whose The degree of of combining this day' yesterday we concept state The in the values for the 0.5/50 fuzzy + set, 0.3/60 term is read cold is equal as to of The is TTX = 1/0 0/80 + 0/90 0.1/70 + + Although possibility distribution may probability distribution, 4 section . 4 a a possibility that was 4 OF with value that temperature is numerically , equal assigned lies 1/30 + a + 0.8/40 + 0/100 is 40 given where given set, to be very a number each that X to will Possibility Theory, total ignorance possibility distribution consisting = Generally speaking, (1,1,1 the 67 less of is all represented l's. 1} specific the a fundamental investigated in be which is a similar of + 4.5. TTx(u) a the Ignorance In by for appear there differences between the two the was possibility distribution the possibility that X 0.8. of by Yesterday therefore may be and [0,1]. COLD, degree is true. function range of of distribution, TTx(u) the membership x a Now cold. the of becomes the possibility cold of as 0.8. fact that yesterday 0.8, compatibility, The possibility to the given that the proposition given concept that the degree is 0.8 40F the with is COLD, interpreted the proposition that with can was day. cold 4 OF of is 0.8, membership, compatibility cold in the fuzzy set, membership evidence or information, the representing the high degree ignorance is represented by the x largest' consisting of all l's distribution, the other and consequently larger the possibility distribution the hand, of situation no uncertainty. where uncertainty, Hence, total possibility [Klir,Folger, 1988]. have perfect we evidence be represented would On by the following possibility distribution. TTx(u) It relevant evidence, Unfortunately, address all propositions Comparison Both to of to problem is quite represent and solving and best possible. are still as we of forced to were in Probability found decision given with or methods situations However, making. Probabilities the in our evidence This being and are well belief the faced with On a information many the suited of in the uncertainty lies solutions. represent 68 in offer is best handled by these two available. hypotheses possibilities are Possibility Theory different. associated alternative and the uncertainty inherent is currently ambiguity we singletons Possibility hypothesis, particular that of uncertainty that theories as uncertainty that is represent type propositions absence - Probability involving all in the that state in possibility theory probability theory 4.5 (1,0,0,0...0) naturally intuitive to seems any = well other the uncertainty that defined hand, results from the use of vague indistinct or concepts linguistic or terms. As must clear type and which most of available a the alternative in diagnostic prevalent This choose. and environments, is best handled by For suppose instance, to determine this as example, the have a set of person's person two problem be all people viewed as 16 of person's age the we may assign our belief uncertainty, approach. faced are with solutions, sets is in such information The old. our are solving we is limited to the person drives is currently attending highschool. information with defined crisp well years observations; we is true. solutions probabilistic that person hypotheses, Our alternative age. can concerning this following the type problem a a hypothesis the uncertainty lies to alternative alternative particular environment, defined, well the decision process, of their belief that on In this part between the choose based of iterative an a car Given this beliefs probabilistic the and as follows: Age In this 17 18 19 20 T05 75 725 725 TI 705 our the highest information available the 16 example, assigned the 15 to us alternative xbest The value. or lack to support of solutions 69 is guess' 16 and are decision. very clear we On and is inherent in is uncertainty information that our consequently have the other certain. hand, If we were given perfect the person's definitions do on the actual measure is given concept of subjective The young. Zadeh for eggs 1977]. breakfast" can also eat be u Han's of possibility and look as X taking , For a young. and possibilites example given "Hans ate may be associated = ease with Pr(u), with X by interpreting Pr(u) eating u eggs as for breakfast. associated TT(u) 1111 0.1 70 0 the The follows: 0.8 X which 12345678 0.1 X with u Pr(u) by (1,2,3,4..). distribution, probability distributions is in the in U of person vague values A probability eggs. uncertainty to the statement to be the degree associated probability may probabilities distribution, P(u) by interpreting P(u) can relates concept Consider the with of the problem example, illustrated by the following A possibility Hans age the of difference between [Zadeh, than the whether the uncertainty lies interpretation be best can This type In this person's Here concentrates rather determining of certainty distinct language applications. age. to determine how this information 1987]. the task their where with the uncertainty Possibility theory the of natural consider is young, exist. [Klir,Folger, in prevalent example, not predict represent found in situations meaning it of could Possibilities age. due to vagueness we evidence, 0.8 0.6 0.4 0.2 0 0 0 0 with X From this possibility does does possibility- exist a of link between he which principle'. observation that is of impossible it is understanding possibility can be probability and the possibility measure based on represents certain represents Probability probabilities whereas value of measure of is similar possibility a are a and the on decreases if further A concepts and of the A probability will occur, measure is feasible. the by sum the of in the distribution space, are represented fact, special of Specifically, may be 71 so an by the maximum A possibility of measure it has been subset of the proposed that plausibility 1988]. introduction approach does words, hypothesis distribution. In Theory. event the measure. represented debate concerning the validity probabilistic other to the plausibility [Klir,Folger, applicability. In hypothesis a measures measures With the an A possibility events the possibility Dempster-Shafer measures the possibility that are measures of between probability that chance evidence. the degree of by comparing gained the is based improbable. also nor -possibility/probability event. the differences of low degree a This principle the of probability, probabilities the possibility as the probability event the called consistency does imply of Zadeh did propose that there heuristic possibilities high degree a high degree a probability However, weak that observe can imply not low degree a we example, possibility theory began the theory and of new Cheeseman used claims that to describe the a its a uncertainty due to eliminating the Cheeseman -fuzzy' that argues the probability that instead with our In event space support of his the underlying consider fuzzy does in the later in it the case as following = "event X represented One possible assumptions severe proposition values for the of X, 1986]. identifies further rules max/min For shortcomings. intersection for the rule problem [Bhatnagar,Kanal, Cheeseman as occurs" X event occurs". all it does example of two and instead be suffers also B are defends from as the minimum certain the underlying with assumptions. contends uncertain In that the equal particular the defense vagueness of Although verification 72 fact, assumptions to make forced to Possibility Theory, about fuzzy found is one surrounding to the uncertainty context. empirical information order a theory value lacks make Zadeh particular its truth set and Probability In assumptions. independence rule the two of However, zero. as stated this mutually exclusive, in probability by stating that in decisions not min[uA(X),uB(X)] their possibility should found that A case Cheeseman the that possible include not dependency fuzzy logic state Theory is is "it be can 1986]. sets, the will theory [Cheeseman, possibility arguments, uAnB(X) In new interpretation is that in the first this found a a the proposition of whereas for need therefore and concepts, concept in a much in probability theory, it of offers that is a good method inherent in to reason natural 73 with a language. type of uncertainty Chapter 5 Non-Monotonic Logic The preceding to approaches based set by his on of unique mathematical as a be should approaches for modeling methods Non-monotonic as formalisms based One specific many have such further offers numerical approaches. systems logics logic offer and [Shoham, unique consideration. include logic default Some McDermott 1987]. the non-monotonic been people into hard more realistic each purposes logic 74 developed of approach, represents by represented the the better known circumscription, and Each in the following discussions of McCarthy's approaches For the the has non-numeric it best uncertainty that of previous monotonic This non-numeric good a from the uncertainty non-monotonic as shared express must that own uncertainty. logic the type were their fact that investigated different Reiter's on numbers. argued numeric requirement expert quite is various approaches shortcoming due to the Consequently, numbers. these is that the of with it difficult to translate their beliefs find often severe of assumptions. uncertainty in terms viewed and and these methods, of Each uncertainty- requirements each have dealt sections of Doyle's these worthy this sections non logical of thesis, however, will my McDermott be and limited to Doyle. 5.1 Basic Characteristics The revise information is current that ability to are or changing essential uncertain. the underlying uncertainty contradictory information Classical traditional or does therefore, not conclusions as traditional logic problem new solving beliefs based our of the environments In truly order be must to revision of in and nature and, previous obtained. be effectively Consequently, to dynamic applied having incomplete situations reflect considered. monotonic information is cannot most in modeling logic is allow the hypothesis, incomplete a also on or uncertain information. Non-monotonic inconsistencies that monotonic and logic Consequently, value. either contradictory front to resolve It assigned. there each is presently method. In previous chapters, In the case of variable in the or no be noted way to a that could logic, Non calculus to a single is evidence assumptions and previous on predicate that the single must value due to this approaches non-montonic based restricted represent hypotheses 75 order is allows information. new case missing, numerical revised first any conflicts, should the on hand, other or from arise is based that requires the on invalidated to be conclusions logic, made must up be restriction ignorance discussed be with this in the be partially believed. each hypothesis is either believed or that applied. are disbelieved based suppositions The made not process. in decision strongly depends underlying 5.2 Modal The Infer modal this In 1985] be the as confidence nature inability as "p is consistent drawn based As conclusions operators the on logic to infer representing the with may be ~ BIRD 2) PENGUIN allow a 3 BIRD 4 ) theory M( CAN-FLY) we can BIRD prove CAN-FLY 76 and idea of extended As an -> A NOT (NOT p) "consistency" contingent fact that there withdrawn. -> is theory" is no information is received, new 1) In this these the represent the CAN-FLY (CAN-FLY) and [Maselli, conclusion evidence example following. ) of have we follows: operator modal to classical modal The contrary- logic, the read . of introduced by McDermott operators from the (Mp) Mp is be logic non-monotonic by adding modality, M, can level on in the Logic "consistency". Here masked assumptions. incorporates Doyle the Therefore, assumptions through the decision carried making our is uncertainty is and initial the on to the previous consider to the the premises information is 5) and be must previous Doyle's on the The obtained. tree each structure each to the revises of user monotonic the necessary to the logic with for truth each changes the its child In this The in the decision reflect contain may system receives or TMS also systems 77 new theorem, in the tree in order inform will made. the chains, amount In may degrade the a non by TMS, reached assumptions. inferencing the and justifications the hypothesis were the structure. node a information new maintains various consistency. that as hypothesis a assumptions explicitly maintenance tree provides to effectively demonstrate able longer a If the with changes system. not models represents turn, underlying uncertainty systems, in (TMS) that program represent system all does however, node In has been TMS a inconsistent the restore operator System justifications. and input that is TMS inconsistent now situation. revising reasoner node assumptions. hypotheses a and is TMS of of the of Truth Maintenance knowledge base and new Truth Maintenance System framework for updating children The modal context is (4) conclusion re-evalutated. Doyle's is following received our meaning based 5.3 if the Now 3. and PENGUIN find that we 1,2 of of large system expert response significantly. Chapter Summary Conclusion and Probability theory has played modeling uncertainty in This years. many Dempster-Shafer theories, of but theories problem be can Probability Theory subsequent in seen also such Theory as of probabilities belief, 6 as not in role for environments only in the Bayesian the basis for Mycin certainty defined to the contrast solving Evidence. are important an In as number factors three all a a traditional frequency the and these of subjective of measure ratio approach. that In Bayesian probability, an event [0,1], range large volume computational Its systems. lack model data limited the that when in the first incorporates weight limited the due a in representation uncertainty. to of to its partial of events. breakthrough certainty due expert part 'fuzzy' In general, resulting is and In success in large also in the value event. implementations called 78 a the and tremendous workable by 'chance' or applied ignorance a represented probabilistic required success for to experienced complexity especially Mycin of of represented directly attached expressiveness of beliefs, one is occur, probability has Bayesian the will the probability a Mycin, factor by offering reasoning a is used to represent degree one's belief. of certainty factors lies in Consequently, and probability model consequently it such has as chains have mutual exclusivity The on theory is bear subsets that the on of a explicit effective restrictions. theory, the Much same of It's part this of and assumptions, Mycin's success in model, inferencing short independence the of the is One based to and accumulated Theory of ignorance. method 79 and and is based In of other this expert all on a also Drawbacks does possible truer allows of include the lack difficulties it evidence the provides beliefs evidence by instead This set. evidence. the same however, advantage notable expert's of on the approach whether but uses theory, This way. hypothesis the Evidence Bayesian attached evidence inferencing of support representation Dempster-Shafer and simplified Theory the hypothesis of more as of single representation developing substantial of effects the hypothesis. supports not 'chance' the some different much degrees numerical words, a evidence assumptions. probability in it applies the minimized of a probability- Dempster-Shafer calculus that from its to in goals assumptions independence. to Bayesian comparison the of from probability suffers attributed certainty factor a to avoid strict Bayesian however, statistical been One was be derived can theory. belief that the available its inherent has been shown, to assigned of model reasoning theoretical basis for confirmation the hypothesis. supports It value the degree represents this the The an the of an in defining frames discernment of in Probability-based theories partial in beliefs domains. complex evidence are hypotheses and for expressing effective uncertainty lies in the approaches be best applied to domains that would defined and realize that reasonable and assumptions set offer formal vague concepts natural the methods of or fuzzy and feasibility of the actual of the assumptions such empirical from max/min testing Non-monotonic a its rules. since found logic 80 information. used turn, to relate whereas a hypothesis set own of hypotheses offers probability the measure occur. will and restrictions dependency it presently lacks with reasoning represents Many have been this In fuzzy logic measure the underlying as Possibility Theory that for suited imprecise or approximate A possibility chance of In concepts. hypothesis, well very unclear are methodology to do suffers in the restrictions Possibility theory meaning membership Possibility found are applications. a give representing uncertainty due to consequently propositons. represents of 'fuzzy' functional to expected the underlying of well is important to It be are Possibility theories, developed by Zadeh, and degrees vague all can the uncertainty due to definitions to model These followed. are language represents theory, unless the hypotheses. of available. mathematical results Fuzzy is the data where no context the where restrictions skeptical much of of the in probability theory. offers a non-numeric approach to handling translations Doyle's as beliefs of "consistency" in the conclusions logic Non-monotonic in reasoning absence is are with to used to the and revised modal represent to us allowing evidence jump the to contrary. in modeling default effective environments to be conclusions thereby of McDermott by incorporating operators in the model, the difficult avoids numbers. allows received These modal operators. into hard logic non-monotonic information is new therefore, uncertainty and, incomplete or missing information. In conclusion, provide a panacea Probability-based our hypothesis. most The expressive ignorance. In order systems, we uncertainty domains types need set is to human into one must that make the of is these allow when the our information. modes in expert of In the majority of all evidential In order misrepresented, experts based reasoning. by incorporating partial not a to be completely generalizations. information where generalizations system. decisions and to be developed that used all combine including 81 be reasoning reasoning vagueness our to of appears for default method can in describing in applicable situations context Theory vagueness that theory uncertainty. in best are should a need of especially allowing model global evidential theory uncertainties information, ensure in the approach, would people of approaches logic typicality one situations is due to the Non-monotonic on all not Dempster-Shafer Fuzzy uncertainty for lies uncertainty is there to represent to systems all their different types appropriate of The method. combining the inferences to be rules allowing good these types systems The best developed. theory for the best limitations It alternative is with that the Many the difficulty in acquiring the full Consequently, usually the have global are All in key 82 to not currently is to the choose aware of the the made. all not the the the of the of that handle systems information limitations information world and into Unfortunately, present the all most methods made. base that is extemely diverse the in successfully that of all expert's knowledge incomplete. reasoning will methods. the lies representation inferencing of are reasoning and state at remember difficulty in developing uncertainty the domain, being particular trade-offs and ideal though important to is difficulty with from these different numbers of uncertainty or we due to describing domain. begin realm complex, answer. are of with is human and no one theory REFERENCES [Adams, 1984] Adams, J. Barclay, "Probabilistic Reasoning Certainty Factors" in Rule Based Expert Systems The and . Mycin Experiments the of Stanford Heuristic Protect, edited by Buchanan and Shortliffe, Addison-Wesley Publishing Co., 00 263-271, 1984. [Barnett, 1981] for a pp. 868-875, Barnett, Jeffrey A. "Computational Methods , Mathematical Theory of Evidence", Seventh International Joint Conference on Artificial Intelligence, 1981. [Buchanan, Shortliffe, Shortliffe, Edward H. 1984] Buchanan, Bruce G. and "A Model of Inexact Reasoning in Medicine" in Rule Based Expert Systems The Mycin Experiments of the Stanford Heuristic Programming Project. edited by Buchanan and Shortliffe, Addison-Wesley , . Co. Publishng , 1984. 233-263, pp [Carnap, 1971] Rudolf Carnap, in Inductive Logic Studies California Press, [Charniak, 1984] pp. Eugene, and [Cheeseman, pp. 1985] Intelligence. 457,482, 1984. Cheeseman, Probability," International Intelligence, pp. [Cheeseman, 1986] Reasoning" Fuzzy Kanal by edited (North Holland) Cheeseman, Peter, Lemmer, 85-101, Elsevier Drew, of Artificial on "Probabilistic Science Publishers William J., and Shortliffe, 1984. 1984] Garvey, Thomas D. , Leonard P., "Reasoning about Lowrance, [Garvey,Lowrance, Wesley, D. Wesley, Evidential Approach", ppl-40, July 1984. [Kandel, and 1982] [Klir,Folger, Fuzzv Set. 1988. [Lukacs, 1972] Statistics. 1988] International, Abraham, and Wiley Klir, Uncertainty 107-131, SRI Kandel, John Recognition. versus Intelligence in Medical Artificial Intelligence, Addison-Wesley Publishing Co., John of 1986. Clancy, 1984] Readings , "In Defense in Uncertainty in Artificial and , Addison-Wesley Conference 1985. [Clancy, Shorliffe, Edward H. Peter, Joint 1002-1009, pp. C. University McDermott, Introduction to Artificial Publishing Co., Richard Probability. 1971. 7-19, Charniak, Jeffrey, and and and Sons George Technical Fuzzy Technicrues Inc., J., pp. and Information, 23-91, Folger, Note and An 324, in Pattern 1982. Tina Prentice-Hall, Lukacs, Eugene, Probability 1972. Academis Press, 83 Control: A., pp. Mathematical [Maselli, Achieve 1985] Maselli, A. Raul, Common Sense [Michie, Reasoning", "Basic Characteristics to March 1985. 1982] Michie, Donald, Introductory Expert Systems, 25, 1982. Gordon and in Readings Breach Science Publishers, [Nutter, 1987] Nutter, Jane T. "Uncertainty Probability", International Joint Conference Intelligence, pp. 373-378, August 1987. , pp. *^ 3- and on Artificial [Shafer, 1976] Shafer, Glenn, A Mathematical Theory Evidence, Princeton University Press, 1976. of [Shafer, 1986] Shafer, Glenn, "Probability Judgement in Artificial Intelligence," in Uncertainty in Artificial Intelligence f edited by Kanal and Lemmer, Elsevier Science Publishers (North Holland), pp. 127-135, 1986. [Shoham, 1987] Shoham, Yoav, "Nonmonotonic Logics: Meaning Utility", International Joint Conference on Artificial Intelligence, August 1987- and [Shortliffe, Medical Co. 1976] Shortliffe, Consultations j_ Mycin. Inc., pp 159-194, Edward Hance, Computer-Based American Wlsevier 1985] Thompson, Terence Theories," Evidential-Reasoning "Parallel [Thompson, R. of International Conference on Publishing 1976. Artificial , Intelligence, pp. Formulation Joint 1985. 321-327, [Waterman, 1986] Waterman, Donald A., A Guide to Expert Addison-Wesley Publishing Co., pp 22-40, 1986. Systems [Zadeh, Theory . 1977] Zadeh, L. Possibility" of "Fuzzy A., Sets in Fuzzy Sets and as a Basis for a Applications: by L A^ Zadeh. edited by Yager, Ovchinnikov, Tong and Nguyen, pp. 193-218, 1987. Papers Selected [Zadeh, 1986] Sufficient View," Kanal for Zadeh, Lofti dealing with in Uncertainty in and Holland), Lemmer, pp. Artificial Elsevier 103-116,. A., "Is Probability Theory Uncertainty in AI: A Negative Science 1986. 84 Intelligence Publishers edited (North by ANNOTATED BIBLIOGRAPHY Adams, J. Barclay, Factors" Certainty |^ fxperijentg "Probabilistic Reasoning and in Rule Based Expert SvstLs The - the of Si^EAi^Ei^^^a Buchanan and Addison- lr01*Ct' ^f? bv Shortliffe, Wesley Publishing Co., pp 263-271, 1984. Adams demonstrates that model is based on a substantial part of the Mycin time equivalent to probability theory. Consequently he contends the Mycin model assumes statistical mdependens=ce and is subjected to the inherent and at limitations. Barnett, Jeffrey A., "Computational Methods for a Mathematical Theory of Evidence," Seventh International Joint Conference on Artificial Intelligence, pp 868-876 1981. Barnett presents the mathematical basis for DempsterHe compares the Dempster-Shafer Theory with Bayesian statistics. He presents all the current drawbacks for Dempster-Shafer TheoryHe then presents Shafer Theory. computational theory from exponential to Bhatnagar, Raj Uncertain that reduces the calculation time linear. K. , Information: and Kanal, Laveen N. , "Handling A Review of Numeric and Non- " in Uncertainty in Artificial Intelligence , edited by Kanal and Lemmer, Elsevier Science Publishers (North Holland), pp. 3-25, 1986. numeric Methods They briefly , compare and contrast several methods reasoning with uncertainty including: Bayesian statistics, Dempster-Shafer theory, Possibility Theory of Endorsements comparison, they focus Non-monotonic and on three information is represented, and how inferences Buchanan, of Inexact Bruce are G. 85 Theory. In their How uncertain how information is combined made. and Reasoning in areas: logic. of Shortliffe, Edward H. , "A Model in Rule Based Expert Medicine" Systems The Mycin Experiments - Proqramminq Project, of the Stanford Heuristic edited by Buchanan and Shortliffe, Addison-Wesley Publishing Cp, pp 233-262, 1984. Buchanan and Shortliffe present the model they developed for approximate reasoning in medicine using certainty factors in Mycin. A brief discussion of Bayesian is included as a basis of the decision model. probability They full describe the model, including definitions, notations terminology and of all discussion of how this in Mycin is included. general the functions. A is then implemented nodel Inductive Rudolf and Jeffrey, Richard C, Studies in Logic and Probability, University of California Press, 7-19, Carnap, pp 1971. A quick review of inductive logic and rational decision Statistical probability and personal (subjective) probability are discussed as two different theories of probability. Carnap states the personal concept of probability must be used in decision theory. Personal probability is divided into two version: actual degree of belief and rational degree of belief. making. Charniak, Conference Charniak argues disease multiple heuristic Artificial A Eugene, were Bayes and Thus for basis statistics work medical doesn't well with work various McDermott, Drew, Introduction to Addison-Wesley Publishing Co., and combined of the use how Bayesian to overcome statistics. 86 of a discussion statistics the in plausible its limitations statistics is defined Theorem Includes discussed. Caduceus independence make it 1984. discussion reasoning. are 1983. to necessary. it does Sense 70-73, pp. realistic Bayesian case, intelligence. 457-482, good a not Common of solutions. Charniak, pp Theorem are offers Eventhough, Basis the National statistical Bayes feasible statistics diagnosis. for on of Intelligence, that the imposed computationally Bayesian Bayesian Proceedings Artificial on assumptions "The Eugene, Diagnosis," Medical and of and limitations Mycin and heuristic of rules Bayesian Cheeseman, Peter, "In Defense of Probability ' International Joint Conference on Artificial Intelligence, pp 1002-1009, 1985. " Cheeseman argues that probability is the only scheme needed for reasoning about uncertainty. He defines a probability as a measure of belief rather that a On this basis he refutes most claims frequency ratio. presently held against probability - Cheeseman, Peter, "Probabilistic versus Fuzzy Reasoning, "in Uncertainty in Artificial Intelligence edited by Kanal and Lemmer, Elsevier Science Publishers (North Holland) pp. 85-102, 1986. , Cheeseman demonstrates how probability theory can be used to solve "fuzzy" problems. He maintains the view that probabilities are a measure of belief in a proposition. He presents a detailed example using this view of He concludes that probability to solve a fuzzy problem. in many cases the fuzzy set approach is identical to probability therefore the and fuzzy is approach not necessary. William Clancy, J., and Edward Shortliffe, in Medical Artificial Intelligence. Readings H., Addison- 1984. Wesley Publishing Co., survey of artificial intelligence work in medicine from to 1981. A collection of works describing much of Reviews of Internist and the work done in this period. Puff are included. A 1971 Garvey, Thomas D. , Lowrance, John D. , P., "Reasoning About Control: A Leonard Approach." 40, International, Technical Wesley, Note 324, pp. 1984. July Present SRI and Evidential a uncertain for reasoning with evidential or The computational model is information. model dependence inference graph engine extrapolate dependent the of model used mass is one support. developed distributions propositions. 87 evidential by to the a The Lowrance remaining to 1- Thomas Garvey, Reasoning: Society, A D. and , Developing 6-9, pp. Lowrance, Concept," October 1982. They discuss their for evidiential method knowledge based systems. Dempster-Shafer approach. methods John D. , "Evidential IEEE Cybernetics and reasoning in Their work is based on the They contrast this to Bayesian . David Heckerman, E., Horvitz, and Rule-Based Systems AAAI proceedings, pp. expressivenes of Uncertainty," J., "On the for reasoning with Eric 121-126, July 1987- argue that "certain" beliefs can be represented by based systems whereas "uncertain" beliefs can not. This is due to the fact that uncertain beliefs are not They rule and the rule based approach cannot express dependencies. They present a modificaiton to rule based systems that will accomodate dependencies and therfore, uncertain beliefs. modular Edward, Hoenkamp, Experiments Conference Joint August This "An Analysis Non-monotonic on on of Artificial pp. 115-117, 1987. is based on psychological that initial inferences made by change International Intelligence, paper show Psychological Reasoning," eventhough the basis original experiments a person for the that may not inferences is discredited. D. Kahneman, Under , Slovic, Uncertainty: University Press, This article P- and , Heuristics Biases, describes three heuristics with uncertainty. making judgements heuristics often these that people using that It people is exhibit in their judgements. biases Lukacs, Judgement Cambridge 1985. when and A., Tversky, and Eugene, Probability Statistics. Academic Probability and Press, Statistics 88 and Mathematical 1972. fundamentals. use shown errors *-,??* MaCmilllan k? Probability and MoroneY< Richard, introduction ^lishing Company Inc., New Statistics fundamentals. H-' and a^Y?JS;-R^n2 WalPle/ Statistics for Engineers |Dg Publishing Company, Probability and Inc., 1972. "Uncertainty pp. and 373-378, August Nutter argues that their is Were involved, approaches incorporated. Andrew Sage, P., all ignorance and to uncertainty of be must Information Systems," in Uncertainty Bouchon and by though of Yager, 1987. pp3-29, research systems was concerned that can deal knowledge. It with effectively encompassed all designing knowledge with phases information for Shafer, Princeton of design: presents theory of Shafer theory. Conference of Glenn, on process of Evidence. description fo the mathematical is the basis for the DempsterDetailed derivations of all formulas that full a evidence the basis with solving. Glenn, A Mathematical Theory University Press, 1976. Shafer Shafer, problem base imperfect acquisition, representation, and inferencing Emphasis on how humans imperfect knowledge. are for place uncertainty, vagueness edited . a problems "On the Management in Knowledge-Based Systems This with in Knowledge-Based Imperfections 1987. definitely typicality, other Probability," Artificial on probability theory in reasoning it cannot provide the answer to are Probability Macmillian Statistics fundamentals. Nutter, Jane T., uncertainty. Ronald E. , Scientists. and International Joint Conference Intelligence, to _ which the theory. "Hierarchical Artificial 89 Evidence," Intelligence IEEE Second Applications, pp. December 16-21, 1985. Shafer reviews the mathematics of the Dempster-Shafer theory. He proposes an algorithm for using this theory in the case of hierarchical evidence. Shafer, Glenn, "Probability Judgement in Artificial Intelligence,"in Uncertainty in Artificial Intelligence. by Kanal Holland) edited (North and , pp. Lemmer, Elsevier 127-135, 1986. Science Publishers Shafer presents two theories of probability judgement: Bayesian Theory and the Theory of Belief functions. Three examples are presented detailing the differences between the Bayesian approach the Dempster-Shafer approach. Shortliffe in the : from system R. on Shafer, Kyburg differences Turner, fuzzy Bayes, Convex categories: the of He to of Joint 321-327, 1985. evidential Bayes, Dempster- reviews Background mechanism each approach elements. and Decision assumptions, similarities approaches. Raymond, Logics for Artificial Intelligence, Limited, pp. 101-114, 1984. Horwood describes set logic. applicaitons and approaches reports, Updating He identifies key Observation mechanism. Formulation International Intelligence, pp. Possibility. and four according to Turner five Classical reasoning: "Parallel , Theories," Artificial discusses Thompson 1984. Shortliffe, Buchanan, Evidential-Reasoning Conference nodel jointly Terence Thompson, for inexact reasoning in developed with Buchanan for use Mycin. Much of the article is the presents that he expert exerpts Ellis of Hance, Computer-Based Medical Mycin, American Elsevier Publishing Co., pp 159-194, 1976. medicine and function Edward Shortliffe, Consultations Inc., the belief and expert of the basics A general this systems. 90 for fuzzy discussion theory in set of artificial theory and the intelligence Lofti Zadeh, dealing with Uncertainty and pp. A. "Is Probability Theory Sufficient for in Uncertainty in AI: A Negative in Artificial Intelligence edited by Kanal , View," Lemmer, Elsevier Science Publishers, 103-116, 1986. (North Holland) , the idea that classic probability can treat properly any kind of uncertainty, especially for Zadeh refutes fuzzy events number of probabilistic papers and A by fuzzy logic in A., fuzzy would Fuzzy Zadeh. Wiley and of works by of Fuzzy set study expert Sets Also environment. systems. 91 methods be edited John collections detaled a L.A. Hguyen, probabilities. where methods Lofti, Zadeh, fuzzy and examples He details instead a of required. and Applications: Selected by Yager, Ovchinnikov, Tong Sons Inc., Zadeh. theory looks 1987. Included at is a good decision making the role of fuzzy and in