A characterization result of dilation in terms of positive and negative association admits an extremal counterexample, which we present together with a minor repair of the result. Dilation may be asymmetric whereas covariation itself is symmetric. Dilation is still characterized in terms of positive and negative covariation, however, once the event to be dilated has been specified

Pedersen, Arthur Paul

Wheeler, Gregory

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Proceedings of Machine Learning Research 103:324–326, 2019 ISIPTA 2019Dilation and Asymmetric RelevanceArthur Paul Pedersen APP@ARTHURPAULPEDERSEN.ORGGregory Wheeler G.WHEELER@FS.DECentre for Human and Machine Intelligence, Frankfurt School of Finance & Management, GermanyAbstractA characterization result of dilation in terms of positiveand negative association admits an extremal counterex-ample, which we present together with a minor repairof the result. Dilation may be asymmetric whereascovariation itself is symmetric. Dilation is still charac-terized in terms of positive and negative covariation,however, once the event to be dilated has been speci-fied.Keywords: dilation, sets of probabilities1. Introduction and PreliminariesA characterization result specifying necessary and sufficientconditions for dilating sets of probabilities expressed interms of witnesses to positive and negative association inlower and upper conditional probability neighborhoods wasgiven in [5] and generalized in [6]. This result admits anextremal counterexample, presented in Section 2. A minormodification to the conditions is shown to re-establish thecharacterization result in Section 3.A lower probability space is a quadruple (Ω,A ,P,P)such that Ω denotes a set of states, A denotes an algebraoverΩ, P denotes a set of probability functions onA , and Pdenotes the lower probability function over A determinedby P by the requirement that P(A) = inf{p(A) : p ∈ P} foreach A ∈A . The value P(A) is called the lower probabilityof A. The upper probability function P over A is accord-ingly defined, as usual, by stipulating that P(A) = 1−P(Ac)for each A ∈A ; the value P(A) is called the upper proba-bility of A. Given B ∈A for which P(B)> 0, conditionallower and upper probabilities are defined as P(A | B) =inf{p(A | B) : p∈ P} and P(A | B) = sup{p(A | B) : p∈ P},respectively. In the following, call a subcollection of eventsB from A a positive measurable partition (of Ω) if B is apartition of Ω such that P(B)> 0 for each B ∈B.Let B be a positive measurable partition of Ω. Say thatB dilates A if each B ∈B:P(A | B) < P(A) ≤ P(A) < P(A | B).1In other words, B dilates A just in case the closed in-terval [P(A), P(A) ] is contained within the open interval1. While this terminology agrees with that of [3, p. 252], it differs fromthat of [8, p. 1141] and [4, p. 412], who call dilation in this sensestrict dilation.(P(A | B), P(A | B)) for each B ∈B. Examples of dilationare discussed in [7, 5, 6] and [9, §6.4.3]1.1. Measures of DependenceGiven a probability function p on algebra A and eventsA,B ∈A , define:Sp(A,B) :=p(A∩B)p(A)p(B)if p(A)p(B)> 0;1 otherwise.Thus the quantity Sp is an index of deviation from stochas-tic independence between events. The value Sp(A,B) ex-presses in ratio form the covariance between events Aand B, cov(A,B) = p(A∩ B)− p(A)p(B). Events A andB are stochastically independent if Sp(A,B) = 1; positivelycorrelated if Sp(A,B) > 1, and negatively correlated ifSp(A,B) < 1.Given a set of probabilities P onA and events A,B ∈A ,define:S+P (A,B) := {p ∈ P : Sp(A,B) > 1};S−P (A,B) := {p ∈ P : Sp(A,B) < 1};IP(A,B) := {p ∈ P : Sp(A,B) = 1}.The set of probability functions IP for which A and B arestochastically independent is called the surface of inde-pendence for A and B with respect to P. In what follows,subscripts are dropped when there is no danger of confu-sion.1.2. Characterizing DilationGiven lower probability space (Ω,A ,P,P), events A, B ∈A with P(B) > 0, and ε > 0, define:P(A | B,ε) := {p ∈ P : |p(A | B) − P(A | B)|< ε};P(A | B,ε) := {p ∈ P : |p(A | B) − P(A | B)|< ε}.Call the sets P(A | B,ε) and P(A | B,ε) lower and upperneighborhoods of A conditional on B, respectively, withradius ε . A probability function p is a member of the lowerneighborhood of A conditional on B with radius ε if p(A |B)© 2019 A.P. Pedersen & G. Wheeler.DILATION AND ASYMMETRIC RELEVANCEis within ε of P(A | B), and similarly for an upper neighbor-hood.Corollary 5.2 of [5] reports that B dilates A just in casethere is (εB)B∈B ∈ RB+ such that P(A | B,εB) ⊆ S−(A,B)and P(A | B,εB) ⊆ S+(A,B), which Theorem 1 of [6]generalizes. The right-to-left implication admits a coun-terexample to be presented in the next section.2. CounterexampleThe following example, due to Michael Nielsen and RushStewart, was conveyed to us in correspondence.Suppose Ω := {ω1,ω2,ω3,ω4} supports two probabilityfunctions, p0 and p1, such that:ω1 ω2 ω3 ω4p0 1/8 1/2 1/3 1/24p1 5/24 1/24 1/24 17/24Let P be the convex hull of {p0, p1}.Consider events A := {ω1,ω2} and B := {ω1,ω3} andpartition B := {B,Bc}. Observe that P = (pα)α∈[0,1],where pα := (1 − α)p0 + α p1 for each α in [0,1].0 0.2 0.4 0.6 0.8 100.20.40.60.81Figure 1: Graphing pα ’s against values of α from 0 to 1.Hence:• pα(A) = 5−3α8• pα(A | B) = 3+2α11−5α , and• pα(A | Bc) = 12−11α13+5α .Figure 1 plots the values thus parametrized by α in [0,1].It is readily established that there are positive real num-bers εB and εBc satisfying the requirements P(A | B,εB) ⊆S−(A,B) and P(A | Bc,εBc) ⊆ S+(A,Bc), while P(A) =1/4 < 3/11 = P(A | B), so B does not dilate A.3. Repaired ResultReturn to the example from Section 2 and observe that thepartition C := {A,Ac} nevertheless dilates event B. Thatis, P(B) = 1/4 and P(B) = 11/24, while P(B | A) = 1/5 < 1/4and P(B | Ac) = 1/8 < 1/4, as well as 11/24 < 5/6 = P(B | A)and 11/24 < 8/9 = P(B | Ac).The foregoing example illustrates a key insight. Whilethe results reported in [5] and [6] do indeed identify condi-tions which suffice to establish dilation between variablesassociated with A and C, they not provide for conditionsdetermining its direction. Yet since relevance might beasymmetric in this setting [1, 2], the indices S− and S+ ofassociation are symmetric, so specifying the target event fordilation is important to rule out cases, like this one, whereasymmetric relevance is in play.Given a probability function p, a set of probability func-tions P, and events A and B, define:Sp(A,B) :=p(A∩B)P(A)p(B)and Sp(A,B) :=p(A∩B)P(A)p(B),Likewise define:S+P (A,B) := {p ∈ P : Sp(A,B) > 1};S−P (A,B) := {p ∈ P : Sp(A,B) < 1}.The following result is easily established:Theorem 1 Let A be an event andB=(Bi)i∈I be a positivemeasurable partition for a given set of probability functionsP over an algebra. The following statements are equivalent(i) B dilates A;(ii) There exists ε > 0 such that for every i ∈ I:P(A | Bi,ε)⊆ S−P (A,Bi) and P(H | Bi,ε)⊆ S+P (A,Bi)Proof For (i) ⇒ (ii), suppose that B dilates A. Se-lect ε := min(|P(A)−P(A | Bi)|: i ∈ I). For i ∈ I, suppose|p(A | Bi)−P(A | Bi)|< ε . Then by hypothesis it followsthat p(A | Bi) < P(A). So p(A∩Bi)/P(A)p(Bi) < 1, thusSp(A,B) < 1. Therefore, P(A | Bi,ε) ⊆ S−P (A,Bi). Simi-larly, P(A | Bi,ε)⊆ S+P (A,Bi).For (ii)⇒ (i), suppose that condition (ii) holds for somepositive ε and assume for reductio ad absurdum that Bfails to dilate A. Without loss of generality, suppose P(A)≤P(A | Bi) for some i ∈ I. Then there is a p ∈ P(A | Bi,ε)⊆S−P (A,Bi) such that Sp(A,Bi)< 1 and P(A)≤ p(A | Bi)<P(A), yielding a contradiction.AcknowledgmentsWe would like to thank Michael Nielsen and Rush Stewartfor sharing their example with us. Gregory Wheeler’s re-search is supported in part by the joint Agence Nationale dela Recherche (ANR) & Deutsche Forschungsgemeinschaft(DFG) project “Collective Attitudes Formation” ColAForm,award RO 4548/8-1,325DILATION AND ASYMMETRIC RELEVANCEReferences[1] Inés Couso, Serafín Moral, and Peter Walley. Examplesof independence for imprecise probabilities. In Gertde Cooman, editor, Proceedings of the First Sympo-sium on Imprecise Probabilities and Their Applications(ISIPTA), Ghent, Belgium, 1999.[2] Fabio Cozman. Sets of probability distributions, inde-pendence, and convexity. Synthese, 186(2):577–600,2012.[3] Timothy Herron, Teddy Seidenfeld, and Larry Wasser-man. The extent of dilation of sets of probabilitiesand the asymptotics of robust bayesian inference. InPSA 1994 Proceedings of the Biennial Meeting of thePhilosophy of Science Association, volume 1, pages250–259, 1994.[4] Timothy Herron, Teddy Seidenfeld, and Larry Wasser-man. Divisive conditioning: further results on dilation.Philosophy of Science, 64:411–444, 1997.[5] Arthur Paul Pedersen and Gregory Wheeler. Demysti-fying dilation. Erkenntnis, 79(6):1305–1342, 2014.[6] Arthur Paul Pedersen and Gregory Wheeler. Dilation,disintegrations, and delayed decisions. In Proceedingsof the 9th Symposium on Imprecise Probabilities andTheir Applications (ISIPTA), pages 227–236, Pescara,Italy, 2015.[7] Teddy Seidenfeld. When normal and extensive formdecisions differ. In D. Prawitz, B. Skyrms, and D. West-erstahl, editors, Logic, Methodology and Philosophy ofScience. Elsevier Science B. V., 1994.[8] Teddy Seidenfeld and Larry Wasserman. Dilation forsets of probabilities. The Annals of Statistics, 21(9):1139–154, 1993.[9] Peter Walley. Statistical Reasoning with ImpreciseProbabilities. Chapman and Hall, London, 1991.326

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