Reichenbachian Common Cause Systems

Reichenbachian Common Cause Systems G. Hofer-Szabó Department of Phiosophy Technica University of Budapest e-mai: gszabo@hps.ete.hu Mikós Rédei Department of History and Phiosophy of Science Eötvös University, Budapest e-mai: redei@udens.ete.hu Forthcoming in Internationa Journa of Theoretica Physics Abstract A partition {C i } i I of a Booean agebra S in a probabiity measure space (S, p) is caed a Reichenbachian common cause system for the correated pair A, B of events in S if any two eements in the partition behave ike a Reichenbachian common cause and its compement, the cardinaity of the index set I is caed the size of the common cause system. It is shown that given any correation in (S, p), and given any finite size n > 2, the probabiity space (S, p) can be embedded into a arger probabiity space in such a manner that the arger space contains a Reichenbachian common cause system of size n for the correation. It aso is shown that every totay ordered subset in the partiay ordered set of a partitions of S contains ony one Reichenbachian common cause system. Some open probems concerning Reichenbachian common cause systems are formuated. 1 Reichenbach s notion of common cause Let (S, p) be a cassica probabiity space with Booean agebra S of random events and probabiity measure p on S. If the oint probabiity p(a B) of A and B is greater than the product of the singe probabiities, i.e. if p(a B) > p(a)p(b) (1) then the events A and B are said to be (positivey) correated and the quantity Corr(A, B) p(a B) p(a)p(b) (2) is caed the correation of A and B. According to Reichenbach [13], Section 19, a probabiistic common cause of a correation such as (1) is an event C (common cause) that satisfies the four conditions specified in the next definition. Definition 1 C is a Reichenbachian common cause of the correation (1) if the foowing (independent) conditions hod: p(a B C) = p(a C)p(B C) (3) p(a B C ) = p(a C )p(b C ) (4) p(a C) > p(a C ) (5) p(b C) > p(b C ) (6) where p(x Y ) = p(x Y )/p(y ) denotes the conditiona probabiity of X on condition Y, C denotes the compement of C and it is assumed that none of the probabiities p(x), (X = A, B, C, C ) is equa to zero. 1

We sha occasionay refer to conditions (3)-(6) as Reichenbach(ian) conditions. It is standard terminoogy to ca (3)-(4) screening-off conditions and to say that C (and aso C ) screens off the correation between A and B. To excude trivia common causes we ca a common cause C proper if it differs from both A and B by more than a measure zero event. In what foows common cause wi aways mean a proper common cause. Reichenbach prooves the foowing proposition. Proposition 1 If the events A, B, C satisfy the Reichenbachian conditions (3)-(6) then there is a positive correation between A and B. The significance of Proposition 1 is that it shows in what sense a common cause expains a correation: from the assumption that A, B and C satisfy the Reichenbachian conditins one can derive (equivaenty: predict) that A and B are (positivey) correated this is an instance of expanation in the sense of Hempe. Reichenbach s proof of Proposition 1 is based on the foowing Lemma, which, for ater purposes, we spe out in a sighty more genera form than used by Reichenbach. Before stating the emma et us reca that the set of events {C i S i I} is a partition of S if i C i = Ω (Ω being the unit in S) and C i C = if i.) Lemma: Let {C i } i I be a partition of S and et A, B S be arbitrary eements. If p(a B C i ) = p(a C i )p(b C i ) for a i I then we have p(a B) p(a)p(b) = 1 p(c i )p(c )[p(a C i ) p(a C )][p(b C i ) p(b C )] (7) 2 i Appying Lemma (proof of which is eft to the reader) with C 1 = C and C 2 = C one obtains p(a B) p(a)p(b) = p(c)p(c )[p(a C) p(a C )][p(b C) p(b C )] (8) which is the formua Reichenbach uses in showing Proposition 1. Eq. (8) impies that Corr(A, B) is indeed positive if (5)-(6) hod. Eq. (8) aso shows, however, that for Corr(A, B) to be positive (5)-(6) are sufficient but not necessary: positivity of Corr(A, B) is impied by the positivity of the right hand side of (8); hence, what is decisive from the point of view of the expanatory power of the comon cause is that [p(a C) p(a C )] and [p(b C) p(b C )] have the same sign. It aso is cear from (8) that a common cause can expain negative correations in ust the same way as it can expain positive ones: if C and C are such that screening off conditions (3)-(4) hod and [p(a C) p(a C )] and [p(b C) p(b C )] have opposite signs, then the right hand side of (8) is negative, hence existence of such a C entais the negative correation. A what foows can be modified triviay in order to cover the case of negative correation, and a statements presented beow remain vaid in the case of negative correations; however, to simpify notation we restrict ourseves to positive correations. To sum up: the intuitive idea behind expaining by a Reichenbachian common cause a correation between A and B in a statistica ensembe is that one shoud be abe to cut the statistica ensembe by a pair of orthogona events (C and C ) into two disoint parts in such a way that (i) the correation disappears in both of the resuting subensembes (this is expressed by the two screening off conditions); and (ii) one of the subensembes shoud increase the probabiity of both A and B (which is the content of the requirement of [p(a C) p(a C )] and [p(b C) p(b C )] having the same sign). 2 The notion of a Reichenbachian common cause system It is easy to see that there exist common cause incompete probabiity spaces (S, p), i.e. probabiity spaces that contain a pair of correated events without containing a (proper) common cause of the correation. Existence of such common cause incompete probabiity spaces can be a threat to what has become caed Reichenbach s Common Cause Principe (RCCP): Given a correation Corr(A, B) > 0, either there is a direct causa infuence between A and B that can be hed responsibe for the correation, or there exists a common cause in the sense of Definition 1 that expains the correation. So if one sees a correation between A and B and has good reasons to think that the correated events A, B in (S, p) cannot infuence each other causay and yet there exists no common cause in S of the correation Corr(A, B), then the suspicion arises that RCCP might not hod. Confronted with a common cause incompete probabiity space (S, p) in which a direct causa infuence between the correated events is excuded, one can have in principe two strategies aiming at saving RCCP: One may try to argue that S is not rich enough to contain a common cause 2

but there might exist a arger (S, p ) that aready contains a common cause of the correation (see Definition 3 beow for what it means to enarge (S, p) into (S, p )). It was shown in a previous paper that this strategy aways works in the sense that it is aways possibe to enarge (S, p) in such a way that the enarged space aready contains an event C that satisfies the Reichenbachian conditions (see [4]). Another natura idea is to suspect that the correation between A and B is not due to a singe factor but may be the cumuative resut of a (possiby arge) number of different partia common causes, none of which can in and by itsef yied a compete common-cause-type expanation of the correation, a of which, taken together, can however account for the entire correation. Expaining a correation by such a system of partia common causes woud mean that one can partition the statistica ensembe into more than two subensembes in such a manner that (i) the correation disappears in each of the subensembes, (ii) any pair of such subensembes behaves ike the two subensembes determined by the pair C, C in the Definition 1 of common cause and (iii) the totaity of partia common causes expains the correation in the sense of entaiing it. A mathematicay expicit formuation of this idea is speed out in the next definition. Definition 2 Let (S, p) be a probabiity space and A, B two events in S. The partition {C i } i I of S is said to be a Reichenbachian common cause system (RCC system for short) for the pair A, B if the foowing two conditions are satisfied p(a B C i ) = p(a C i )p(b C i ) for a i I (9) [p(a C i) p(a C )][p(b C i) p(b C )] > 0 (i ) (10) The above definition is a natura generaization of Reichenbach s origina definition of common cause to the case when more than one singe factor contributes to a correation. The cardinaity of the index set I (i.e. the number of events in the partition) is caed the size of the RCCS. Since C, C with a Reichenbachian common cause C is a RCCS of size 2, we ca a RCCS proper if its size is greater than 2. The next proposition shows that a Reichenbachian common cause system aso has expanatory power exacty in the sense in which a singe common cause does: Proposition 2 Let the partition {C i } i I of S be a Reichenbachian common cause system for the pair A, B. Then the eements A and B are positiey correated. Proof: The statement in the proposition is an immediate coroary of Lemma and the definition of the notion of Reichenbachian common cause system. 3 Existence and uniqueness of Reichenbachian common cause systems It is not obvious that proper Reichenbachian common cause systems exist. It is not difficut however to give an exampe of a probabiity space (S, p) containing a pair of correated events A, B for which there exists no common cause in S but there exists in S a Reichenbachian common cause system of size 3 (see [8]). This exampe aso shows that the intuition mentioned in Section 2 is correct: there are cases when an expanation of a correation with the hep of a singe common cause is impossibe within the bounds of a given event structure; yet the event structure is rich enough to contain a proper RCCS that can expain the correation. It is not difficut however to find probabiity spaces that contain neither a (proper) common cause nor a proper Reichenbachian common cause system. As it was mentioned in Section 2 it is known that such common cause incompete probabiity spaces can aways be extended in such a way that the extension contains a common cause of the given correation. More is true, however: on can show that any probabiity space can be extended in such a way that the arger probabiity space contains a Reichenbachian common cause system consisting of a arge number of events. Before speing out the precise proposition et us reca the definition of an extension (S, p ) of a probabiity space (S, p): Definition 3 (S, p ) is caed an extension of (S, p) if there exists an inective attice homomorphism (embedding) h: S S (preserving aso the orthocompementation) such that p (h(x)) = p(x) for a X S (11) Proposition 3 Let (S, p) be a cassica probabiity space and n be an arbitrary finite natura number greater than 2. There exists then an extension (S, p ) of (S, p) such that (S, p ) contains a Reichenbachian common cause system of size n. 3

(See [8] for the ong and tedious proof of this proposition.) Proposition 3 shows that RCCS s of arbitrary finite size exist. Probem Do Reichenbachian common cause systems of countaby infinite size exist? The answer to the above question is not known; we conecture that RCCS s of infinite size aso exist. A given correation Corr(A, B) in probabiity space (S, p) can possess two or more Reichenbachian common causes. Simiary, a correation can have more than two proper Reichenbachian common cause systems (this being a consequence of Proposition 3); however, the different RCCS s cannot be arbitrariy ocated in the event structure S. To formuate the proposition constraining the ocation of RCCS s in S, consider the set P of a partitions of S. Let P 1 = {Ci 1 } i I and P 2 = {C 2 } J be two partitions in P. The partition P 1 is defined to be finer than P 2 (equivaenty: P 2 is defined to be coarser than P 1 ) (notation P 1 P 2 ) if for every Ci 2 P 2 there exist C 1 P i 1 ( i L J) such that Ci 2 = i LC1. P i 1 is caed stricty finer (coarser) than P 2 if P 1 is finer (coarser) than P 2 and P 1 P 2. The reation is a partia ordering on P, and the terminoogy (stricty) finer and (stricty) coarser ) appies to RCCS s as we since RCCS s are partitions. Proposition 4 If {C i } i=n i=1 is a Reichenbachian common cause system in (S, p) for the pair A, B, then there exists in (S, p) neither stricty finer nor stricty coarser Reichenbachian common cause system for A, B. Proof: Assume that {C 2 } =m =1 is a RCCS stricty coarser than the RCCS {C1 i } i=n i=1 (n > m). There exist then a C 2 such that for some C 1 P i 1 with i L and L having the cardinaity of at east 2, we have C 2 = i. Consider the probabiity measure space (S LC1 i C 2, p( C 2 )) where S C 2 = {X C 2 X S} (12) and where p( C 2 ) is the conditiona probabiity measure of p with respect to the conditioning event C 2. By the definition of {C 2 } =m =1 as a Reichenbachian common cause system, the events A and B are statisticay independent with respect to the probabiity measure p( C 2 ), i.e. p(a B C 2 ) = p(a C 2 )p(b C 2 ) (13) On the other hand, the events C 1 (i i L) form Reichenbachian common cause system in (S C 2, p( C 2 )) with respect to the events (A C 2 ) and (B C 2 ); hence, by Proposition 2 there is a positive correation between (A C 2 ) and (B C 2 ) in the measure p( C 2 ) i.e. we have p((a C 2 ) (B C 2 ) C 2 ) > p(a C 2 C 2 )p(b C 2 C 2 ) (14) which contradicts (13). So the assumption of existence of two, different RCCS s that are in the finer-coarser reation has ed to contradiction, so the proposition is proved. 4 Concuding remarks We can express the content of Proposition 4 in the foowing way: any totay ordered subset of partitions in (S, p) contains ony one Reichenbachian common cause system for a given fixed pair of events A, B S. So whie there may exist many RCCS s for a given correated pair, the different RCCS s provide different sorts of expanations of the correation between A and B. In particuar, different Reichenbachian common cause systems cannot be put together to form a finer RCCS that woud provide a more detaied expanation of the correation. This aso impies that the partition of (S, p) generated in the natura manner by different Reichenbachian common causes C i S does not yied a RCCS; hence the non-uniqueness of Reichenbachian common causes cannot be expained by saying that the different common causes are ust coarse-grained manifestations of a deeper, finer underying Reichenbachian common cause system. Given two correations Corr(A i, B i ) > 0 (i = 1, 2) in (S, p), the event C S is caed a common common cause of the two correations if it is a common cause of both Corr(A 1, B 1) > 0 and Corr(A 2, B 2 ) > 0. It is known that common causes are not in genera common common causes, i.e. that there exist two correations in a probabiity space that cannot have a common common cause (see [7] and [1] for resuts concerning necessary and sufficient conditions impying the existsence of common common causes). The notion of a Reichenbachian common common cause system aso is a meaningfu concept and it woud be interesting to find necessary and sufficient conditions for a set of correations to have a Reichenbachian common common cause system. 4

Motivated by considerations somewhat different from the one in this paper, the probem of a common cause system is raised aso in [9], where a common cause system (caed mutipe common cause ) is defined to be a finite partition possessing the screening off property (eq. (9)) ony (pus some mathematicay not expicit requirement concerning the spatiotempora ocation of the events C i (i = 1, 2,... n) and A, B, see Definition 3.3 in [9]). Thus the notion of RCCS defined in the present paper is different from the one proposed in [9]. We wish to point out in this regard that without some requirement in addition to (9) such as (10), the notion of common cause system becomes trivia: for instance, the set of atoms in any finite Booean agebra form a partition for which the screening off condition (9) hods; hence a probabiity measure spaces with a finite Booean agebra possess a common common cause system, which seems counterintuitive. A Reichenbachian common cause system as defined in the present paper is a much stronger notion. Reichenbach s notion of common cause can naturay be adapted to quantum probabiity spaces (L, p), where L is a non-distributive, orthomoduar attice and where p is an additive (generaized) bounded measure on L (see [12], [10], [11], [4], [6]; for some other attempts see [2] and [3]). The notion of Reichenbachian common cause system aso can easiy be generaized to the non-commutative case aong the ideas foowed in this paper. Probems and questions concerning non-commutative Reichenbachian common cause systems paraeing the ones treated here aso can be formuated, no resuts are known, however, on the non-commutative case. Acknowedgement: Work supported by OTKA, contract numbers: T 035234, TS 04089, T 024841, T 043642 and T 025880. References [1] D. Danks and C. Gymour: Linearity properties of Bayes nets with binary variabes, in J. Breese and D. Koer (eds.), Uncertainty in Artificia Inteigence: Proceedings of the 17th Conference (UAI-2001), Morgan Kaufmann, San Francisco, 2001, 98-109 [2] G. Hofer-Szabó: The forma existence and uniqueness of the Reichenbachian common cause on Hibert attices Internationa Journa of Theoretica Physics 36 (1997) 1973-1980 [3] G. Hofer-Szabó: Reichenbach s common cause definition on Hibert attices Internationa Journa of Theoretica Physics 37 (1998) 435-443 [4] G. Hofer-Szabó, M. Rédei and L.E. Szabó: On Reichenbach s common cause principe and Reichenbach s notion of common cause The British Journa for the Phiosophy of Science 50 (1999) 377-3999 [5] G. Hofer-Szabó, M. Rédei and L.E. Szabó: Common cause competabiity of cassica and quantum probabiity spaces Internationa Journa of Theoretica Physics 39 (2000) 913-919 [6] G. Hofer-Szabó, M. Rédei and L.E. Szabó: Reichenbach s Common Cause Principe: Recent resuts and open probems Reports on Phiosophy Nr. 20, (2000) 85-107 [7] G. Hofer-Szabó, M. Rédei and Szabó, L.E. Common-causes are not common common-causes manuscript, submitted, preprint: http://phisci-archive.pitt.edu [8] G. Hofer-Szabó and M. Rédei: Reichenbachian common cause systems of arbitrary finite size exist manuscript, submitted [9] T. Pacek: Is Nature Deetrministic?, Jageonian University Press, Cracow, 2000 [10] M. Rédei: Reichenbach s Common Cause Principe and quantum fied theory Foundations of Physics 27 (1997) 1309-1321 [11] M. Rédei: Quantum Logic in Agebraic Approach, Dordrecht: Kuwer Academic Pubishers, 1998 [12] M. Rédei and S.J. Summers: Loca primitive causaity and the common cause principe in quantum fied theory Foundations of Physics 32 (2002) 335-355 [13] H. Reichenbach: The Direction of Time, Los Angees: University of Caifornia Press, 1956 5