On joint modelling of random uncertainty and fuzzy imprecision

1. On joint modelling of random uncertainty and fuzzy imprecision Olgierd Hryniewicz Systems Research Institute Warsaw

2. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Coping with uncertainties of different types is the difficult problem of systems analysis! Does exist one general methodology for the formal description of different types of uncertainty? Is probability the only method for the description of uncertainty?

3. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Description of uncertainty in plain language • probable • possible • plausible • hopeful • to be expected • ……. These words do not mean the same! Does this imply that their meaning has to be formally described using different models?

4. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Merging of imprecise information Consider the sum of two independent costs reported imprecisely in the form of intervals of their possible values. + = ?

5. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Merging of imprecise information Consider the sum of two independent costs reported imprecisely in the form of intervals of their possible values. + ? || No !

6. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Merging of imprecise information Consider the sum of two independent costs reported imprecisely in the form of intervals of their possible values. + ? || Yes !

7. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Decision making in presence of imprecise or partial information People usually do not make decisions that guarantee the maximal expected utility! [H.Simon, A.Tversky and D.Kahneman, many examples of „paradoxes”] Are they irrational?

8. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Generalizations of probability [P.Walley] • Possibility measures and necessity measures • Belief functions and plausibility functions • Choquet capacities of order 2 • Coherent upper an lower probabilities • Coherent upper and lower previsions • Sets of probability measures • Sets of desirable gambles • Partial preference orderings • …… and others!

9. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Probability • There exists a sample space W consisting of disjoint elements • Borel-field set B is defined on W; elements of B are observable events • Probability is a finite-additive measure, and is defined only on elements of B • For each element of B its probability can be precisely evaluated [Betting in favor of an event A against its complement ACwill not lead to sure loss only if betting odds are P(A) to 1-P(A)]

10. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Probability (counterarguments) • Observable states may be not disjoint [quantum physics; Heisenberg Uncertainty Principle: Bordley] • In presence of partial information coherent probabilities may not be precise [Betting in favor of A is not complementary to betting in favor of AC: Walley] • Probability theory fails in case of modelling ignorance • In decision making processes it is necessary to consider the whole power set of W.

11. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Possibility There exist diiferent interpretations of this term: objective feasibility (related to the idea of preference), plausibility (consistency with available knowledge), …… . The basic notion of the possibility theory is a possibilitydistribution function defined on the possibility space W consisting of not necessarily disjoint elements. Possibilitydistribution is formally equivalent to the membership function of a fuzzy set!

12. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Dempster – Shafer theory of evidence This theory generalizes classical probability. Probability measures, called probability mass asingments are defined on the whole power set of W. To each subset of W we can assign belief measure and plausibility measure Possibility theory is a special case of the Dempster-Shafer theory (when subsets of W form a nested set)

13. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Imprecise probabilities and their generalizations Consider the following information about possible outcomes [win(W), draw(D) or loss(L)] of a football game (Walley) • Probably not W • W is more probable than D • D is more probable than L This information cannot be used for the coherent and precise evaluation (without making additional assumptions!) of such probabilities like P(W)! Only interval probabilities can be coherently used for this purpose!

14. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Imprecise probabilities (P.Walley) Imprecise probabilities proposed by P.Walley directly generalize the subjective probabilities defined using de Finetti’s betting concept. The lower (upper) probability of an event A can be interpreted by specifying acceptable betting rates for betting on (against) A. If betting odds on A are x to 1-x one will bet on A if and against A if The choice is not determined if is between and

15. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Imprecise probabilities – further generalizations Further generalizations of lower and upper probabilities are needed if we want to avoid some problems with • decision making [lower and upper previsions (expectations)] • updating probabilities after observing events with prior probabilities equal to zero • distinguishing between preferences and weak preferences

16. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Merging of randomness with fuzzy imprecision When we have to cope simultaneously with randomness (having interpretation in terms of frequencies) and possibilistic imprecision (described in terms of fuzzy sets) the only satisfactory (as for now) theory is based on the concept of fuzzy random variable. There exist two definitions of a random variable: • Proposed by Kwakernaak and extended by Kruse and Meyer (fuzzy perception of an ordinary random variable) • Proposed by Puri and Ralescu (based on the concept of random sets).

17. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Fuzzy statistics For the K-K-M concept of the random variable there exists a straightforward generalization of classical statistics – fuzzy statitistics. All concepts of classical statistics (estimators, confidence intervals, statistical tests, Bayesian decisions) have their fuzzy counterparts. However, in statistical tests and similar decision problems it is necessary to introduce additionalrequirements, e.g. based on possibilistic measures. For the P-R concept of the random variable statistical procedures are much more complicated (no underlying probability distribution!) and are based on the concept of distance in certain metrics.

18. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Application of fuzzy random variables (assessment of grenhouse gases emission) The emissions of greenhouse gases are not directly measurable. We can directly measure only so called activitiesaij(i denotes a current year of assessment, and j denotes the type of activity). The measures of activity are transformed to related emissions using certain emission factorscij. For the simplest we have the following evaluation of the amount of emission The values of aijand cij may be both uncertain. Especially in the case of emission factors a certain part of uncertainty is not of a random character (e.g. uncertainty related to partial knowledge about the nature of physical and chemical reactions). In general, the total amount of GHG emission may be described by fuzzy random variables!

19. IFIP/IIASA/GAMM Workshop on Coping with Uncertainty (CwU) – Olgierd Hryniewicz Thank you for your attention !