# Possibility Theory and its applications: a retrospective and prospective view D. Dubois, H. Prade irit-cnrs, Université Paul Sabatier 31062 toulouse france

 Sana 21.06.2017 Hajmi 445 b. • ## Quantitative possibility theory • ## it is not additive and makes sense on ordinal structures. • ## Permission: It is allowed to do something (deontic) • ## πx(s) = 1 iff x = s is normal, fully plausible, unsurprizing • ## partial ignorance :

• 70 ≤ x ≤ 80 (sets, intervals)

• ## partial ignorance with preferences : May have reasons to believe that 72 > 71  73 > 70  74 > 75 > 76 > 77 • ## But this result is enforced by the setting of exchangeable bets (Dutch book argument). Actual information is often poorer. • ## IGNORANCE : π(s) = 1,  s  S • ## N(A) = 1 – (Ac)=min uA 1 – π(s) The degree of certainty (necessity) that x  A • ## iii) a ≤  ≤ b: then x >  is possibly true or false; N(x >  ) = 0; (x >  ) = 1. • ## Corollary N(A) > 0  (A) = 1 • ## He introduces a notion of conditional possibility • ## The ones and only ordinal counterparts to possibility measures • ## Such degrees of provability coincide with necessity measures. • ## the key axiom of "maxitivity" for possibility measures is highlighted (also for fuzzy events). • ## Qualitative:

• comparative: A complete pre-ordering ≥π on U A well-ordered partition of U: E1 > E2 > … > En
• absolute: πx(s)  L = finite chain, complete lattice...

• ## Theories diverge on the conditioning operation • ## Not the converse!!!! • ## Accepted events are closed under deduction • ## DIFFERENT FROM PROBABILISTIC REASONING BASED ON AVERAGING • ## non-monotony ## PLAUSIBLE INFERENCE WITH A POSSIBILITY DISTRIBUTION

• Given a non-dogmatic possibility distribution π on S (π(s) > 0, s)
• Propositions A, and B

• ## This is a form of inference first proposed by Shoham in nonmonotonic reasoning ## PLAUSIBLE INFERENCE WITH A POSSIBILITY DISTRIBUTION • ## can be expressed by constraints

• (b  f) > ( b ¬f)
• (p  b) > (p  ¬b)
• (p  ¬f) > (p  f)
• the minimally specific π* ranks normal situations first:
• ¬p  b  f, ¬p ¬b
• then abnormal situations: ¬f  b
• Last, totally absurd situations f  p , ¬b p ## Example (back to possibilistic logic)

• material implication

• ## K = {(b f ), (p  b), (p  ¬f)}, with  <  • ## Abductive reasoning for diagnosis under poor causal knowledge (satellite faults, car engine test-benches) • ## This is SAVAGE framework. • ## a qualitative utility function µ: X  U

• µ(x) = 0 totally rejected consequence
• µ(y) > µ(x) y preferred to x
• µ(x) = 1 preferred consequence • ## UPES(f) = minsS max(n(π(s)), µ(f(s)))

• where n is the order-reversing map of V
• Low utility : plausible state with bad consequences

• ## UOPT(f) = maxsS min(π(s), µ(f(s))) • ## in fuzzy expert systems:

• µ = membership function of rule condition
• π = imprecision of input fact
• ## in fuzzy databases

• µ = membership function of query
• π = distribution of stored imprecise data
• ## in pattern recognition

• µ = membership function of attribute template
• π = distribution of an ill-known object attribute • ## There exists a common scale V that contains both L and U, so that confidence and uncertainty levels can be compared.

• (certainty equivalent of a lottery)
• ## If only a subset E of plausible states is known

• π = E
• UPES(f) = minsE µ(f(s)) (utility of the worst consequence in E)
• ## criterion of Wald under ignorance

• UOPT(f)= maxsE µ(f(s)) ## On a linear state space • ## WITH UOPT(f) : replace N(A) by (A) • ## Suppose the preference relation a on acts obeys the following properties:

• (XS, a) is a complete preorder.
• there are two acts such that f a g.
•  A, f, x, y constant, x  a y  xAf  yAf
• if f >a h and g >a h imply f g >a h
• if x is constant, h >a x and h >a g imply h >a xg
• ## then there exists a finite chain L, an L-valued necessity measure on S and an L-valued utility function u, such that a is representable by the pessimistic possibilistic criterion UPES(f). • ## Presupposes that consecutive qualitative value levels are distant from each other (negligibility effects) • ## Membership functions of fuzzy sets

• Natural language descriptions pertaining to numerical universes (fuzzy numbers)
• Results of fuzzy clustering

• ## Upper probability bound

• Random experiments with imprecise outcomes
• Consonant approximations of convex probability sets
• ## Semantics: frequentist, subjectivist (gambles)... • ## P(A| B) ≤ maxx  B P(A| x) ## POSSIBILITY AS UPPER PROBABILITY

• Given a numerical possibility distribution , define P() = {Probabilities P | P(A) ≤ (A) for all A}
• Then, generally it holds that (A) = sup {P(A) | P  P()} N(A) = inf {P(A) | P  P()}
• So  is a faithful representation of a family of probability measures. • ## P is always representable by means of a possibility measure. Its possibility distribution is precisely

• πx = mini max(µEi, 1 – ai) • ## Only in the consonant case can m be recalculated from π • ## It improves the state of information (reduction of focal elements) • ## For inference from generic knowledge based on observations • ## Why ?

• fusion of heterogeneous data
• decision-making : betting according to a possibility distribution leads to probability.
• Extraction of a representative value
• Simplified non-parametric imprecise probabilistic models • ## PROB POSS: Confidence intervals Replacing a probability distribution by an interval A with a confidence level c.

• It defines a possibility distribution
• π(x) = 1 if x  A,
• = 1 – c if x  A • ## Informational criteria:

• from  to P: Preservation of symmetries
• (Shapley value rather than maximal entropy)
• from P to : optimize information content
• (Maximization or minimisation of specificity • ## The result is i = j=i,…n pi

• (case of no ties) • ## In other words the -cut of a STFN contains the (1)-confidence interval of any such p. • ## The most specific possibility distribution dominating p is π such that L > 0, π(aL) = π(aL+ L) = 1 – P(IL). • ## The symmetric triangular fuzzy number (STFN) defines a possibility distribution that optimally dominates any symmetric density with given mode and bounded support. ## From possibility to probability

• Idea (Kaufmann, Yager, Chanas):
• Pick a number  in [0, 1] at random
• Pick an element at random in the -cut of π.
• a generalized Laplacean indifference principle : change alpha-cuts into uniform probability distributions.
• Rationale : minimise arbitrariness by preserving the symmetry properties of the representation.
• ## The resulting probability distribution is:

• The centre of gravity of the polyhedron P(
• The pignistic transformation of belief functions (Smets)
• The Shapley value of the unanimity game N in game theory. • ## A critique: The agent is forced to be additive by the rules of exchangeable bets.

• For instance, the agent provides a uniform probability distribution on a finite set whether (s)he knows nothing about the concerned phenomenon, or if (s)he knows the concerned phenomenon is purely random.
• ## Idea : It is assumed that a subjective probability supplied by an agent is only a trace of the agent's belief. • ## Assumption 1: Beliefs can be modelled by belief functions

• (masses m(A) summing to 1 assigned to subsets A).

• ## Method : reconstruct the underlying belief function from the probability provided by the agent by choosing among the isopignistic ones. ## SUBJECTIVE POSSIBILITY DISTRIBUTIONS

• There are clearly several belief functions with a prescribed Shapley value.

• ## RESULT : The least informative belief function whose Shapley value is p is unique and consonant. • ## It gives results that are less specific than the confidence interval approach to objective probability. • ## Defuzzification based on Choquet integral (linear with fuzzy number addition) • ## Non-parametric conservative modelling of imprecision in measurements (Mauris) • ## Quantitative possibilistic expectations : decision-theoretic characterisation ? • ## A bridge between logic-based AI and probabilistic reasoning • ## -If A  B and C |=π A then C |=π B

• (right weakening rule RW)
• -If A |=π B and A |=π C then A |=π B C
• ## (Right AND) • ## If A |=π B and A  B |=π C then A |=π C

• (cut, weak transitivity )
• (But if A normally implies B which normally implies C, then A may not imply C)

• ## If B is normally expected when A holds,then B is expected to hold when both A and C hold, unless it is that A normally implies not C • ## Theorem: If |= satisfies restricted reflexivity, right weakening, rational monotony, Right AND and Left OR, then A > B is the strict part of a possibility relation on events.

• So a consequence relation satisfying the above properties is representable by possibilistic inference, and induces a complete plausibility preordering on the states. ## A POSSIBILISTIC APPROACH TO MODELING RULES • ## •ne may compute  : the least specific possibility distribution in (∆) • ## Cautious inference

• ∆  A  B iff
• For all (∆), (AB) > (Ac B).
• ## Possibilistic inference

• ∆  A  B iff *(AB) > *(Ac B) for the least specific possibility measure in (∆).
• Leads to a stratification of ∆ according to N*(Ac B) • ## Captures the idea of uncertain knowledge in an ordinal setting • ## Syntactic inference K |(p ) is well-defined • ## Inference with a partially inconsistent knowledge base becomes non-trivial and nonmonotonic K |nt p iff K | (p ) and  > inc(K) • ## = 1 – if p is false in state s • ## The p. d. πK is the least informed state of partial knowledge compatible with K • ## For the non-trivial inference under inconsistency: {(p 1)}  K |nt q iff (q  p) > (¬q  p) • ## Possibilistic logic

• Formulas are Boolean
• Truth is 2-valued
• Weighted formulas have fuzzy sets of models
• Validity is many-valued
• degrees of validity are not compositional except for conjunctions
• Represents uncertainty ## Example: IF BIRD THEN FLY; IF PENGUIN THEN BIRD; IF PENGUIN THEN NOT-FLY

• • K = {b  f, p  b, p  ¬f}
•  = material implication
• K  {b} | f; K  {p} |  contradiction

• ## Hence K  {(p, 1), (b, 1)} |nt ¬f Do'stlaringiz bilan baham:

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