Proceedings of Machine Learning Research 103:324–326, 2019 ISIPTA 2019 Dilation and Asymmetric Relevance Arthur Paul Pedersen APP@ARTHURPAULPEDERSEN.ORG Gregory Wheeler G.WHEELER@FS.DE Centre for Human and Machine Intelligence, Frankfurt School of Finance & Management, Germany Abstract 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. Keywords: dilation, sets of probabilities 1. Introduction and Preliminaries A characterization result specifying necessary and sufficient conditions for dilating sets of probabilities expressed in terms of witnesses to positive and negative association in lower and upper conditional probability neighborhoods was given in [5] and generalized in [6]. This result admits an extremal counterexample, presented in Section 2. A minor modification to the conditions is shown to re-establish the characterization 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 algebra over Ω, P denotes a set of probability functions on A , and P denotes the lower probability function over A determined by P by the requirement that P(A) = inf{p(A) : p ∈ P} for each A ∈A . The value P(A) is called the lower probability of A. The upper probability function P over A is accordingly defined, as usual, by stipulating that P(A) = 1−P(Ac) for each A ∈A ; the value P(A) is called the upper probability of A. Given B ∈A for which P(B)> 0, conditional lower 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 events B from A a positive measurable partition (of Ω) if B is a partition of Ω such that P(B)> 0 for each B ∈B. Let B be a positive measurable partition of Ω. Say that B dilates A if each B ∈B: P(A | B) < P(A) ≤ P(A) < P(A | B).1 In other words, B dilates A just in case the closed interval [P(A), P(A) ] is contained within the open interval 1. While this terminology agrees with that of [3, p. 252], it differs from that of [8, p. 1141] and [4, p. 412], who call dilation in this sense strict dilation. (P(A | B), P(A | B)) for each B ∈B. Examples of dilation are discussed in [7, 5, 6] and [9, §6.4.3] 1.1. Measures of Dependence Given a probability function p on algebra A and events A,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 stochastic independence between events. The value Sp(A,B) expresses in ratio form the covariance between events A and B, cov(A,B) = p(A∩ B)− p(A)p(B). Events A and B are stochastically independent if Sp(A,B) = 1; positively correlated if Sp(A,B) > 1, and negatively correlated if Sp(A,B) < 1. Given a set of probabilities P on A 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 are stochastically independent is called the surface of independence for A and B with respect to P. In what follows, subscripts are dropped when there is no danger of confusion. 1.2. Characterizing Dilation Given 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 upper neighborhoods of A conditional on B, respectively, with radius ε . A probability function p is a member of the lower neighborhood of A conditional on B with radius ε if p(A |B) © 2019 A.P. Pedersen & G. Wheeler. DILATION AND ASYMMETRIC RELEVANCE is within ε of P(A | B), and similarly for an upper neighborhood. Corollary 5.2 of [5] reports that B dilates A just in case there 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 counterexample to be presented in the next section. 2. Counterexample The following example, due to Michael Nielsen and Rush Stewart, was conveyed to us in correspondence. Suppose Ω := {ω1,ω2,ω3,ω4} supports two probability functions, p0 and p1, such that: ω1 ω2 ω3 ω4 p0 1/8 1/2 1/3 1/24 p1 5/24 1/24 1/24 17/24 Let P be the convex hull of {p0, p1}. Consider events A := {ω1,ω2} and B := {ω1,ω3} and partition 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 1 0 0.2 0.4 0.6 0.8 1 Figure 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 numbers 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 Result Return to the example from Section 2 and observe that the partition C := {A,Ac} nevertheless dilates event B. That is, P(B) = 1/4 and P(B) = 11/24, while P(B | A) = 1/5 < 1/4 and 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. While the results reported in [5] and [6] do indeed identify conditions which suffice to establish dilation between variables associated with A and C, they not provide for conditions determining its direction. Yet since relevance might be asymmetric in this setting [1, 2], the indices S− and S+ of association are symmetric, so specifying the target event for dilation is important to rule out cases, like this one, where asymmetric relevance is in play. Given a probability function p, a set of probability functions 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 and B=(Bi)i∈I be a positive measurable partition for a given set of probability functions P 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. Select ε := min(|P(A)−P(A | Bi)|: i ∈ I). For i ∈ I, suppose |p(A | Bi)−P(A | Bi)|< ε . Then by hypothesis it follows that p(A | Bi) < P(A). So p(A∩Bi)/P(A)p(Bi) < 1, thus Sp(A,B) < 1. Therefore, P(A | Bi,ε) ⊆ S−P (A,Bi). Similarly, P(A | Bi,ε)⊆ S + P (A,Bi). For (ii)⇒ (i), suppose that condition (ii) holds for some positive ε and assume for reductio ad absurdum that B fails 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. Acknowledgments We would like to thank Michael Nielsen and Rush Stewart for sharing their example with us. Gregory Wheeler's research is supported in part by the joint Agence Nationale de la Recherche (ANR) & Deutsche Forschungsgemeinschaft (DFG) project "Collective Attitudes Formation" ColAForm, award RO 4548/8-1, 325 DILATION AND ASYMMETRIC RELEVANCE References [1] Inés Couso, Serafín Moral, and Peter Walley. Examples of independence for imprecise probabilities. In Gert de Cooman, editor, Proceedings of the First Symposium on Imprecise Probabilities and Their Applications (ISIPTA), Ghent, Belgium, 1999. [2] Fabio Cozman. Sets of probability distributions, independence, and convexity. Synthese, 186(2):577–600, 2012. [3] Timothy Herron, Teddy Seidenfeld, and Larry Wasserman. The extent of dilation of sets of probabilities and the asymptotics of robust bayesian inference. 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