Probabilistic . . . Need to Take . . . Which Representation . . . Analysis of the Problem From p-Boxes to What Is Needed p-Ellipsoids: Towards an Main Result and Its . . . Auxiliary Result Optimal Representation of Ellipsoids Are Better . . . If we Reconstruct a p- . . . Imprecise Probabilities Home Page Title Page Konstantin K. Semenov 1 and Vladik Kreinovich 2 ◭◭ ◮◮ 1 Saint-Petersburg State Polytechnical University ◭ ◮ 29, Polytechnicheskaya str. Saint-Petersburg, 195251, Russia, semenov.k.k@gmail.com Page 1 of 19 2 Department of Computer Science Go Back University of Texas at El Paso Full Screen El Paso, TX 79968, USA, vladik@utep.edu Close Quit
Probabilistic . . . Need to Take . . . 1. Probabilistic Information Is Important Which Representation . . . • It is very important to take into account information Analysis of the Problem about the probabilities of different possible values. What Is Needed Main Result and Its . . . • This is especially true in many engineering applica- Auxiliary Result tions, when we have a long history of similar situations. Ellipsoids Are Better . . . • There are several mathematically equivalent ways to If we Reconstruct a p- . . . represent information about a random variable X : Home Page def • cdf F ( x ) = Prob( x ≤ X ); Title Page Prob( x ≤ X ≤ x + ∆ x ) def • pdf ρ ( x ) = lim ; ◭◭ ◮◮ ∆ x ∆ x → 0 � ◭ ◮ def x k · ρ ( x ) dx ; instead of = E [ X k ] = • moments M k M 2 , we can describe the variance V = M 2 − M 2 1 ; Page 2 of 19 • characteristic function Go Back � E [exp(i · ω · X )] = exp(i · ω · x ) · ρ ( x ) dx ; � Full Screen • expected values E [ u ( X )] = u ( x ) · ρ ( x ) dx of the Close utility functions u ( x ) that describe user preferences. Quit
Probabilistic . . . Need to Take . . . 2. Need to Take Imprecision into Account Which Representation . . . • In practice, we rarely have full knowledge of the prob- Analysis of the Problem ability distribution. What Is Needed Main Result and Its . . . • In terms of cdf, this means that we only know the Auxiliary Result bounds uncertainty means that [ F ( x ) , F ( x )] ( p-box ). Ellipsoids Are Better . . . • Instead of the exact value ρ ( x ) of the pdf, for each x , If we Reconstruct a p- . . . we know an interval [ ρ ( x ) , ρ ( x )] of possible values. Home Page • Instead of the exact values of the moments M k , we Title Page know intervals [ M k , M k ] of possible values, etc. ◭◭ ◮◮ • When we have the exact knowledge of the probabilities, ◭ ◮ all representations are mathematically equivalent. Page 3 of 19 • However, in the presence of uncertainty, these repre- Go Back sentations are no longer equivalent. Full Screen Close Quit
Probabilistic . . . Need to Take . . . 3. Taking Imprecision into Account (cont-d) Which Representation . . . • Let us show that in the presence of uncertainty, differ- Analysis of the Problem ent representations are no longer equivalent. What Is Needed Main Result and Its . . . • Example: if we know the bounds ρ and ρ on ρ ( x ) on Auxiliary Result [ x − , x + ], we can deduce bounds on F ( x ): Ellipsoids Are Better . . . F ( x ) = ( x − x − ) · ρ and F ( x ) = ( x − x − ) · ρ. If we Reconstruct a p- . . . Home Page • However, these bounds contain a distribution for which: Title Page – first the cdf F ( x ) is equal to F ( x ) and ◭◭ ◮◮ – then at some point x 0 ∈ [ x − , x + ], it jumps to F ( x ). ◭ ◮ • For this distribution, the probability density ρ ( x ) is Page 4 of 19 infinite at x = x 0 , hence ρ ( x 0 ) = ∞ �∈ [ ρ, ρ ]. Go Back • So which of these non-equivalent representations of im- Full Screen precise probability should we use? Close Quit
Probabilistic . . . Need to Take . . . 4. Which Representation Is the Best? Which Representation . . . • One of the main objectives of data processing is to Analysis of the Problem make decisions. What Is Needed Main Result and Its . . . • Standard approach: select the action a with the largest Auxiliary Result expected utility E [ u a ( x )]. Ellipsoids Are Better . . . • In many cases, the utility function u a ( x ) is smooth: If we Reconstruct a p- . . . u a ( x ) ≈ c 0 + c 1 · ( x − x 0 ) + c 2 · ( x − x 0 ) 2 . Home Page Title Page • So, to compute E [ u a ( x )], it’s sufficient to know M k . ◭◭ ◮◮ • Sometimes, utility function is discontinuous: e.g., there ◭ ◮ is a fine is pollution is beyond a threshold x 0 . • When u = u − for x < x 0 and u = u + = 1 for x ≥ x 0 , Page 5 of 19 then E [ u a ( x )] = u − + ( u + − u − ) · F ( x 0 ) . Go Back • So, depending on the application, different representa- Full Screen tion are optimal: moments M k or cdf F ( x ). Close Quit
Probabilistic . . . Need to Take . . . 5. Analysis of the Problem Which Representation . . . • Reminder: we can use several moments M 1 , M 2 , . . . , Analysis of the Problem or several values F ( x 1 ), F ( x 2 ), . . . , of cdf F ( x ). What Is Needed Main Result and Its . . . • In each case, we use several values v 1 , . . . , v n to describe Auxiliary Result a distribution. Ellipsoids Are Better . . . • In general, all formulas are linear in ρ ( x ), so relation If we Reconstruct a p- . . . between different representations is linear: Home Page n � v i → v ′ Title Page i = a i + a ij · v j . j =1 ◭◭ ◮◮ • Imprecision is usually represented by bounds v i and v i ; ◭ ◮ so, possible values of v = ( v 1 , . . . , v n ) form a box Page 6 of 19 [ v 1 , v 1 ] × . . . × [ v n , v n ] . Go Back Full Screen • Alas, in general, a linear transformation transforms a box into a parallelepiped – and not into a box. Close Quit
Probabilistic . . . Need to Take . . . 6. What Is Needed Which Representation . . . • Reminder: what was a box in one representation be- Analysis of the Problem comes a different objects in another one. What Is Needed Main Result and Its . . . • So, different box representations of imprecise probabil- Auxiliary Result ity are not equivalent. Ellipsoids Are Better . . . • We therefore need a family F of sets which remains of If we Reconstruct a p- . . . the same type after a linear transformation T : Home Page def if V ∈ F then T ( V ) = { T ( v ) : v ∈ V } ∈ F. Title Page ◭◭ ◮◮ • In many situations (e.g., in automatic control), when ◭ ◮ we need to make decision very fast. Page 7 of 19 • In general, the more parameters we need to process, the longer our computations. Go Back • It is therefore desirable to select a family F with the Full Screen smallest possible number of parameters. Close Quit
Probabilistic . . . Need to Take . . . 7. Main Result and Its Corollary Which Representation . . . Main Result: Analysis of the Problem What Is Needed • Let F be a linear-invariant r -parametric family of con- Main Result and Its . . . R n . nected bounded closed domains from I Auxiliary Result • Then r ≥ n ( n + 3) ; and if r = n ( n + 3) , then: Ellipsoids Are Better . . . 2 2 If we Reconstruct a p- . . . – either F is the the family of all ellipsoids E , Home Page – or, for some λ ∈ (0 , 1) , F is the family of all sets Title Page E − λ · E. ◭◭ ◮◮ ◭ ◮ Discussion: Page 8 of 19 • If we restrict ourselves to convex sets (or only to simply connected sets), we get ellipsoids only. Go Back • So, to describe imprecision, we should use p-ellipsoids : Full Screen ellipsoid-shaped regions in the space of all cdf f-s F ( x ). Close Quit
Probabilistic . . . Need to Take . . . 8. Towards Auxiliary Result: What Does “Opti- Which Representation . . . mal” Mean? Analysis of the Problem Let A be a class of families of sets, and let G be a group What Is Needed of transformations defined on A . Main Result and Its . . . • By an optimality criterion , we mean a pre-ordering Auxiliary Result (i.e., a transitive reflexive relation) � on the class A . Ellipsoids Are Better . . . If we Reconstruct a p- . . . • An optimality criterion is G -invariant if for all g ∈ G , Home Page and for all B, B ′ ∈ A , B � B ′ implies g ( B ) � g ( B ′ ) . Title Page • An optimality criterion is final if there exists exactly ◭◭ ◮◮ one B opt ∈ A for which B � B opt for all B � = B opt . ◭ ◮ Explanation: • If there are no optimal B opt , the criterion is useless. Page 9 of 19 • If there are several optimal B opt � = B ′ opt , we can use Go Back this non-uniqueness to optimize something else. Full Screen • So, if B opt � = B ′ opt , the original criterion is not final. Close Quit
Probabilistic . . . Need to Take . . . 9. Auxiliary Result Which Representation . . . Result: Analysis of the Problem What Is Needed • Let A be the family of all r -parametric families of con- R n . Main Result and Its . . . nected bounded closed domains from I Auxiliary Result • Let � be a linear-invariant final opt. criterion on A . Ellipsoids Are Better . . . • Then r ≥ n ( n + 3) ; and if r = n ( n + 3) , then: If we Reconstruct a p- . . . 2 2 Home Page – either the optimal family F opt is the the family of Title Page all ellipsoids E , ◭◭ ◮◮ – or, for some λ ∈ (0 , 1) , F opt is the family of all sets ◭ ◮ E − λ · E. Page 10 of 19 Discussion: Go Back • If we restrict ourselves to convex sets (or only to simply connected sets), we get ellipsoids only. Full Screen • So, to describe imprecision, we should use p-ellipsoids. Close Quit
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