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COMPUTATION WITH IMPRECISE PROBABILITIES—A BRIDGE TO REALITY Lotfi A. Zadeh

COMPUTATION WITH IMPRECISE PROBABILITIES—A BRIDGE TO REALITY Lotfi A. Zadeh Computer Science Division Department of EECS UC Berkeley October 24, 2008 RSCTC’08 University of Akron Akron, Ohio

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COMPUTATION WITH IMPRECISE PROBABILITIES—A BRIDGE TO REALITY Lotfi A. Zadeh

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  1. COMPUTATION WITH IMPRECISE PROBABILITIES—A BRIDGE TO REALITY Lotfi A. Zadeh Computer Science Division Department of EECSUC Berkeley October 24, 2008 RSCTC’08 University of Akron Akron, Ohio Research supported in part by ONR N00014-02-1-0294, BT Grant CT1080028046, Omron Grant, Tekes Grant, Chevron Texaco Grant and the BISC Program of UC Berkeley. Email: zadeh@eecs.berkeley.edu LAZ 10/21/2008

  2. PREVIEW LAZ 10/21/2008

  3. KEY POINTS systems analysis information analysis decision analysis • Information analysis is a portal to decision analysis. • In most realistic settings, decision-relevant information is imperfect. LAZ 10/21/2008

  4. CONTINUED Imperfect information: imprecise and/or uncertain and/or incomplete and/or partially true. Second-order uncertainty= uncertainty about uncertainty= uncertainty2. Generally, imperfect information is uncertain2. So is vague information. imprecise probabilities, fuzzy probabilities, fuzzy sets of Type 2 are uncertain2. Decision analysis under uncertainty2 lies in uncharted territory. LAZ 10/21/2008

  5. THE TRIP-PLANNING PROBLEM • I have to fly from A to D, and would like to get there as soon as possible • I have two choices: (a) fly to D with a connection in B; or (b) fly to D with a connection in C • if I choose (a), I will arrive in D at time t1 • if I choose (b), I will arrive in D at time t2 • t1 is earlier than t2 • therefore, I should choose (a) ? B (a) A D (b) C LAZ 10/21/2008

  6. IMPRECISE PROBABILITIESBASIC MODALITIES imprecise probabilities data-based perception-based Walley (1991) et al Described in NL Computing with Words (CW) • What is the probability that there will be a significant increase in the price of oil in the near future? LAZ 10/21/2008

  7. THE CW-BASED APPROACH TO COMPUTATION WITH IMPRECISE PROBABILITIES I/NL: information set q/NL: question ans(q/I) I: p1 . . pn pwk p’s are propositions containing imprecise probabilities, relations, functions and constraints pwk is information drawn from world knowledge LAZ 10/21/2008

  8. EXAMPLE X is Vera’s age p1: Vera has a son in mid-twenties p2: Vera has a daughter in mid-thirties pwk : usually mother’s age at birth of child is between *20 and *30 q: what is Vera’s age? LAZ 10/21/2008

  9. EXAMPLE A box contains about 20 balls of various sizes p1: most are small p2: a few are large q: what is the probability that a ball drawn at random is neither small nor large? This problem and Vera’s age problem have similar deep structures. LAZ 10/21/2008

  10. PROTOFORMAL (ABSTRACT) VERSION protoform= abstracted summary Total number is approximately N Q1 A’s are B’s Q2 A’s are C’s ?Q3 A’s are D’s LAZ 10/21/2008

  11. SAME ABSTRACT STRUCTURE, DIFFERENT SURFACE STRUCTURE A country, A, is known to have approximately 300 missiles. Most are short range. A few are long range. How many have intermediate range? LAZ 10/21/2008

  12. IS PROBABILITY THEORY SUFFICIENT FOR COMPUTATION WITH IMPRECISE PROBABILITIES? The probability community stubbornly holds to the Dennis Lindley view that standard probability theory is all that is needed to deal with uncertainty. I argued to the contrary in many papers, among them my 1978 paper on possibility theory, my 2002 paper on the perception-based theory of probabilistic reasoning, and in my 2005 and 2006 papers on the Generalized Theory of Uncertainty (GTU). LAZ 10/21/2008

  13. CONTINUED More concretely, probability theory is not equipped to deal with problems in which the information set includes propositions drawn from natural language, as is the case in preceding examples. LAZ 10/21/2008

  14. THE DENNIS LINDLEY VIEW [Dennis Lindley (1987)] The only satisfactory description of uncertainty is probability. By this I mean that every uncertainty statement must be in the form of a probability; that several uncertainties must be combined using the rules of probability; and that the calculus of probabilities is adequate to handle all situations involving uncertainty…probability is the only sensible description of uncertainty and is adequate for all problems involving uncertainty. LAZ 10/21/2008

  15. CONTINUED All other methods are inadequate…anything that can be done with fuzzy logic, belief functions, upper and lower probabilities, or any other alternative to probability can better be done with probability. LAZ 10/21/2008

  16. SIMPLEST PROBLEM WHICH CONTRADICTS THE DENNIS LINDLEY VIEW X is the value of a real-valued variable. What X is is uncertain. The available information about X is (a) X is larger than approximately a; and (b) X is smaller than approximately b. What is the probability that X is approximately c? The question cannot be answered. A question that can be answered is: What is the possibility that X is smaller than approximately b? LAZ 10/21/2008

  17. CONTINUED X is a real-valued random variable. The available information about X is (a) Usually X is larger than approximately a; and (b) Usually X is smaller than approximately b. What is the probability that X is approximately c? What is the expected valued of X? LAZ 10/21/2008

  18. BACKDROP LAZ 10/21/2008

  19. Science deals not with reality but with models of reality. In large measure, scientific progress is driven by a quest for better models of reality. • The real world is pervaded with various forms of imprecision and uncertainty. To construct better models of reality, it is essential to develop a better understanding of how to deal with different forms of imprecision and uncertainty. LAZ 10/21/2008

  20. PREAMBLE • An imprecise probability distribution is an instance of second-order uncertainty, that is, uncertainty about uncertainty. • Computation with imprecise probabilities is not an academic exercise—it is a bridge to reality. In the real world, imprecise probabilities are the norm rather than exception. LAZ 10/21/2008

  21. CONTINUED • In large measure, real-world probabilities are perceptions of likelihood. Perceptions are intrinsically imprecise. Imprecision of perceptions entails imprecision of probabilities. Generally, numerical probabilities do not exist. • Example: What is the probability that the stock market will experience a sharp decline tomorrow? Can you precisiate the meaning of this probability? LAZ 10/21/2008

  22. IMPRECISION OF MEANING μ middle-age 1 0.8 core of middle-age 0 45 60 40 55 43 definitely not middle-age definitely not middle-age definitely middle-age Imprecision of meaning = elasticity of meaning Elasticity of meaning = fuzziness of meaning Example: middle-age LAZ 10/21/2008

  23. CONTINUED • Peter Walley's seminal work "Statistical Reasoning with Imprecise Probabilities," published in l99l, sparked a rapid growth of interest in imprecise probabilities. • In the mainstream literature on imprecise probabilities, imprecise probabilities are dealt within the conceptual framework of standard probability theory. LAZ 10/21/2008

  24. CONTINUATION • What is widely unrecognized is that standard probability theory, call it PT, has a serious limitation. More specifically, PT is based on bivalent logic—a logic which is intolerant of imprecision and does not admit shades of truth and possibility. As a consequence, the conceptual framework of PT is not the right framework for dealing with imprecision and, more particularly, with imprecision of information which is described in natural language. LAZ 10/21/2008

  25. CONTINUATION • The approach to computation with imprecise probabilities which is described in the following is a radical departure from the mainstream literature. LAZ 10/21/2008

  26. CONTINUED • Its principal distinguishing features are: (a) imprecise probabilities are dealt with not in isolation, as in the mainstream approaches, but in an environment of imprecision of events, relations and constraints; (b) imprecise probabilities are assumed to be described in a natural language. This assumption is consistent with the fact that a natural language is basically a system for describing perceptions. LAZ 10/21/2008

  27. MODALITIES OF PROBABILITY probability objective subjective measurement-based perception-based NL-based LAZ 10/21/2008

  28. EXAMPLES • I am checking-in for my flight. I ask the ticket agent: What is the probability that my flight will be delayed.  He tells me:  Usually most flights leave on time. Rarely most flights are delayed.  How should I use this information to assess the probability that my flight may be delayed? LAZ 10/21/2008

  29. BASIC APPROACH • To compute with information described in natural language we employ the formalism of  Computing with Words (CW) (Zadeh l999). • The formalism of Computing with Words, in application to computation with information described in a natural language, involves two basic steps: (a) precisiation of meaning of propositions expressed in natural language; and (b) computation with precisiated propositions. Computing with Words is based on fuzzy logic. LAZ 10/21/2008

  30. FUZZY LOGIC (FL) AND COMPUTING WITH WORDS (CW) LAZ 10/21/2008

  31. FUZZY LOGIC—A NEW LOOK There are many misconceptions about fuzzy logic. Fuzzy logic is not fuzzy. In essence, fuzzy logic is a precise logic of imprecision. The point of departure in fuzzy logic—the nucleus of fuzzy logic, FL, is the concept of a fuzzy set. LAZ 10/21/2008

  32. DIGRESSION—THE CONCEPT OF A FUZZY SET class informal (unprecisiated) precisiation precisiation set (a) generalization fuzzy set fuzzy set boundary (b) measure (a): A set may be viewed as a special case of a fuzzy set. A fuzzy set is not a set; LAZ 10/21/2008

  33. CONTINUED (b) Basic attributes of a set/fuzzy set are boundary and measure (cardinality, count, volume) Fuzzy set theory is boundary-oriented Probability theory is measure-oriented Fuzzy logic is both boundary-oriented and measure-oriented LAZ 10/21/2008

  34. CONTINUED A set, A, in U is a class with a crisp boundary. A set is precisiated through association with a characteristic function cA: U {0,1} A fuzzy set is precisiated through graduation, that is, through association with a membership function µA: U [0,1], with µA(u), uεU, representing the grade of membership of u in A. Membership in A is a matter of degree. In fuzzy logic everything is or is allowed to be a matter of degree. LAZ 10/21/2008

  35. EXAMPLE—MIDDLE-AGE μ middle-age 1 0.8 core of middle-age 0 45 60 40 55 43 definitely not middle-age definitely not middle-age definitely middle-age Imprecision of meaning = elasticity of meaning Elasticity of meaning = fuzziness of meaning LAZ 10/21/2008

  36. FACETS OF FUZZY LOGIC Fuzzy Logic (wide sense) (FL) FLl logical (narrow sense) FLs fuzzy-set-theoretic relational FLr fuzzy set epistemic FLe From the point of departure in fuzzy logic, the concept of a fuzzy set, we can move in various directions, leading to various facets of fuzzy logic. LAZ 10/21/2008

  37. FL-GENERALIZATION T FL-generalization = fuzzy T (T+) The concept of FL-generalization of a theory, T, relates to introduction into T of the concept of a fuzzy set, with the concept of a fuzzy set serving as a point of entry into T os possibly other concepts and techniques drawn from fuzzy logic. FL-generalized T is labeled fuzzy T. Examples: fuzzy topology, fuzzy measure theory, fuzzy control, etc. LAZ 10/21/2008

  38. CONTINUED A facet of FL consists of a FL-generalization of a theory or a FL-generalization of a collection of related theories. The principal facets of FL are: logical, FLl; fuzzy set theoretic, FLs; epistemic, FLe; and relational, FLr. LAZ 10/21/2008

  39. BASIC STRUCTURE OF FL fuzzy logic graduation/precisiation granulation/imprecisiation applied fuzzy logic theoretical fuzzy logic epistemic facet relational facet set-theoretic facet logical facet LAZ 10/21/2008

  40. PRINCIPAL FACETS OF FL The logical facet of FL, FLl, is fuzzy logic in its narrow sense. FLl may be viewed as a generalization of multivalued logic. The agenda of FLl is similar in spirit to the agenda of classical logic. The fuzzy-set-theoretic facet, FLs, is focused on FL-generalization of set theory. Historically, the theory of fuzzy sets (Zadeh 1965) preceded fuzzy logic (Zadeh 1975c). The theory of fuzzy sets may be viewed as an entry to generalizations of various branches of mathematics, among them fuzzy topology, fuzzy measure theory, fuzzy graph theory and fuzzy algebra. LAZ 10/21/2008

  41. CONTINUED The epistemic facet of FL, FLe, is concerned in the main with knowledge representation, semantics of natural languages, possibility theory, fuzzy probability theory, granular computing and the computational theory of perceptions. The relational facet, FLr, is focused on fuzzy relations and, more generally, on fuzzy dependencies. The concept of a linguistic variable—and the associated calculi of fuzzy if-then rules—play pivotal roles in almost all applications of fuzzy logic. LAZ 10/21/2008

  42. NOTE—SPECIALIZATION VS. GENERALIZATION Consider a concatenation of two words, MX, with the prefix, M, playing the role of a modifier of the suffix, X, e.g., small box. Usually M specializes X, as in convex set Unusually, M generalizes X. The prefix fuzzy falls into this category. Thus, fuzzy set generalizes the concept of a set. The same applies to fuzzy topology, fuzzy measure theory, fuzzy control, etc. Many misconceptions about fuzzy logic are rooted in misinterpretation of fuzzy as a specializer rather than a generalizer. LAZ 10/21/2008

  43. CORNERSTONES OF FUZZY LOGIC graduation granulation FUZZY LOGIC precisiation generalized constraint The cornerstones of fuzzy logic are: graduation, granulation, precisiation and the concept of a generalized constraint. LAZ 10/21/2008

  44. THE CONCEPT OF GRADUATION • Graduation of a fuzzy concept or a fuzzy set, A, serves as a means of precisiation of A. Examples • Graduation of middle-age • Graduation of the concept of earthquake via the Richter Scale • Graduation of recession? • Graduation of civil war? • Graduation of mountain? LAZ 10/21/2008

  45. THE CONCEPT OF GRANULATION • The concept of granulation is unique to fuzzy logic and plays a pivotal role in its applications. The concept of granulation is inspired by the way in which humans deal with imprecision, uncertainty and complexity. • Granulation serves as a means of imprecisiation. LAZ 10/21/2008

  46. GRADUATION / GRANULATION A graduation/precisiation granulation/imprecisiation A* *A • graduation = precisiation • granulation = imprecisiation LAZ 10/21/2008

  47. GRANULATION granulation forced deliberate Forced: singular values of variables are not known. Deliberate: singular values of variables are known. There is a tolerance for imprecision. Precision carries a cost. Granular values are employed to reduce cost. LAZ 10/21/2008

  48. CONTINUED • In fuzzy logic everything is or is allowed to be granulated.  Granulation involves partitioning of an object, A, into granules. More generally, granulation involves an association with A of a system of granules. Informally, a granule is a clump of elements drawn together by indistinguishability, equivalence, similarity, proximity or functionality. Example: body parts • A granule, G, is precisiated through association with G of a generalized constraint. LAZ 10/21/2008

  49. GRANULATION / PARTITION object granule • Graduated granulation = fuzzy granulation LAZ 10/21/2008

  50. GRANULATION / SYSTEM OF GRANULES object granule granule • Graduated granulation = fuzzy granulation LAZ 10/21/2008

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