Compatibility, coherence and the running intersection property - PowerPoint PPT Presentation

Introduction Unconditional case Conditional case Conclusions Compatibility, coherence and the running intersection property Enrique Miranda Marco Zaffalon University of Oviedo IDSIA WPMSIIP ’2018 E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Goals of the work • To determine necessary and sufficient conditions for the compatibility of a number of marginal models with some joint. • To extend the result based on the running intersection property to the conditional case. • To get a more efficient manner to compute the natural extension. E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Disjoint assessments A simple scenario is when we have disjoint sets of variables: if we are given marginal probability measures P 1 , P 2 , P 3 on X 1 , X 2 , X 3 , then we can find a joint P on X 1 × X 2 × X 3 simply by applying independence: we make P := P 1 × P 2 × P 3 . In the finite case, we just make the product of the mass functions. E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions What if they are not disjoint? If we consider assessments P 12 on X 1 × X 2 and P 23 on X 2 × X 3 , then a necessary condition for the existence of a compatible joint is that P 12 , P 23 induce the same marginal on X 2 : P 12 ( A ) = P 23 ( A ) ∀ A ⊆ X 2 . In fact, we could always define in the finite case P ( x 1 , x 2 , x 3 ) = P 12 ( x 1 , x 2 ) · P 23 ( x 3 | x 2 ) , where P 23 ( x 3 | x 2 ) is derived from P 23 using Bayes’ rule. E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions What if they are not disjoint? If we consider assessments P 12 on X 1 × X 2 and P 23 on X 2 × X 3 , then a necessary condition for the existence of a compatible joint is that P 12 , P 23 induce the same marginal on X 2 : P 12 ( A ) = P 23 ( A ) ∀ A ⊆ X 2 . In fact, we could always define in the finite case P ( x 1 , x 2 , x 3 ) = P 12 ( x 1 , x 2 ) · P 23 ( x 3 | x 2 ) , where P 23 ( x 3 | x 2 ) is derived from P 23 using Bayes’ rule. ֒ → So is this enough? E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Pairwise compatibility � global compatibility Actually it is not: consider X 1 = X 2 = X 3 = { 0 , 1 } and the marginals P 12 , P 13 , P 23 given by: P 12 (0 , 0) = P 12 (1 , 1) = 0 . 5 , P 12 (0 , 1) = P 12 (1 , 0) = 0 P 13 (0 , 0) = P 12 (1 , 1) = 0 . 5 , P 12 (0 , 1) = P 12 (1 , 0) = 0 P 23 (0 , 0) = P 12 (1 , 1) = 0 , P 12 (0 , 1) = P 12 (1 , 0) = 0 . 5 They are pairwise compatible (all of them have uniform marginals), but not globally compatible: P 12 implies X 1 = X 2 , P 13 implies X 1 = X 3 and P 23 implies X 2 � = X 3 , and these three things are incompatible! E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Running intersection property (Beeri et al.) The key is in the running intersection property (RIP): we say that indices S 1 , . . . , S r satisfy RIP when S i ∩ ( ∪ j < i S j ) ⊆ S j ∗ for some j ∗ < i . Then if we have marginals P S 1 , . . . , P S r on X S 1 , . . . , X S r , � P S 1 , . . . , P S r pairwise compatible P S 1 , . . . , P S r globally compatible ⇔ S 1 , . . . , S r satisfy RIP . E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Why the hell?? The key here is that the RIP condition allows us to establish an order in the marginals we are given, and then we can apply the law of total probability by adding some assumptions of independence between sets of variables. E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions First generalisation Now we are going to try to generalise the result in a number of ways: ◮ When the possibility spaces are infinite. ◮ When the marginals are imprecise. ◮ When we have conditional information. E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Sets of desirable gambles A gamble on X is a bounded real-valued function f : X → R . We denote the set of all gambles on X by L ( X ), and let L + := { f � 0 } , the set of positive gambles. Given X 1 , . . . , X n and S ⊆ { 1 , . . . , n } , we let X S := × j ∈ S X j . A gamble f on X n is S -measurable if f ( x ) = f ( y ) for every x , y ∈ X n such that π S ( x ) = π S ( y ), and we denote by K S the set of X S -measurable gambles. There exists a one-to-one correspondence between L ( X S ) and K S . E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Coherent sets of gambles ∈ D and D = posi( D ∪ L + ), where D ⊆ L ( X ) is coherent when 0 / posi denotes the set of positive linear combinations. In particular, we say that a set D ⊆ K S is coherent relative to K S when the set D ′ ⊆ L ( X S ) that we can make a one-to-one correspondence with, is coherent. D avoids partial loss when it is included in some coherent set of gambles. The smallest such set is called its natural extension, and it is E = posi( L + ∪ D ). E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Compatibility for sets of desirable gambles Consider subsets S 1 , . . . , S r of { 1 , . . . , n } , and let D j ⊆ L ( X n ) be coherent with respect to K S j := K j . Given i � = j in { 1 , . . . , r } , we say that D i , D j are pairwise compatible if and only if D i ∩ K j = D j ∩ K i . ◮ If S 1 , . . . , S r satisfy RIP and D 1 , . . . , D r are pairwise compatible, then there exists a coherent set of desirable gambles D ⊆ L ( X n ) that is globally compatible with D 1 , . . . , D r , in the sense that D ∩ K j = D j ∀ j = 1 , . . . , r . E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Coherent lower previsions A lower prevision on is L ( X ) a functional P : L ( X ) → R . P is called coherent when for any f , g ∈ L and any λ > 0: (C1) P ( f ) ≥ inf x ∈X f ( x ); (C2) P ( λ f ) = λ P ( f ); (C3) P ( f + g ) ≥ P ( f ) + P ( g ). When K = L ( X ) and (C3) holds with equality for every f , g ∈ L ( X ), P is called a linear prevision and is denoted by P . A coherent set of desirable gambles D induces a coherent lower prevision P on L ( X ) by means of the formula P ( f ) = sup { µ : f − µ ∈ D} . E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Corollary: compatibility for coherent lower previsions Consider subsets S 1 , . . . , S r of { 1 , . . . , r } satisfying RIP and for every j let P j be a coherent lower prevision on X S j . ◮ There exists a coherent lower prevision P on X n such that P ( f ) = P j ( f ) ∀ f ∈ K j , ∀ j ⇐ ⇒ P i ( f ) = P j ( f ) ∀ f ∈ K i ∩ K j , and for every i � = j ∈ { 1 , . . . , r } . ...so in particular we obtain the classical result. E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Conditional information More generally, we may have unconditional and conditional information. However, the meaning of compatibility is not as clear as in our previous results, in the sense that such a joint may necessarily induce additional assessments that are not in the original ones. Taking this into account, given D 1 , . . . , D r , we shall investigate to which extent these sets avoid partial loss, meaning that they have a joint coherent superset; but we are not requiring anymore that D ∩ K j = D j for every j = 1 , . . . , r . E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions First simplification: remove isolated variables ◮ For every i = 1 , . . . , r , let D ∗ i be the restriction of D i to K S i ∩ ( ∪ j � = i S j ) . i =1 D ∗ ∪ r ⇒ ∪ r i =1 D i avoids partial loss ⇐ i avoids partial loss. We may try to simplify further to pairwise compatibility: i =1 D i avoid partial loss ⇒ ∪ i � = j D j ∪ r i avoid partial loss , where D j i is the restriction of D i to K S i ∩ S j . E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Just look at pairwise intersections? ....but it will not work: ∪ i � = j D j i avoid partial loss � ∪ r i =1 D i avoid partial loss . E. Miranda � 2018 c Compatibility, coherence and RIP

Introduction Unconditional case Conditional case Conclusions Second simplification: coherence graphs It was proven by Miranda and Zaffalon (2009) that the verification of coherence can be simplfied by means of coherence graphs: ✲ B X 8 X 1 s B X 7 s ✠ ❄ ✛ ✲ X 2 X 3 X 4 X 6 X 5 s s s s s s ❘ ✠ ❘ ✠ ✠ ✲ ❄ X 7 X 8 X 9 X 10 s ✻ s ❄ s ❄ s ❄ ❄ ✲ ✛ ✛ X 11 X 12 X 13 X 14 s s ✠ ❘ X 15 X 16 B X 14 Figure: Example of a coherence graph. The assessments in different superblocks are automatically coherent, so can focus on each superblock separately. E. Miranda � 2018 c Compatibility, coherence and RIP