Generalised Reichenbachian Common Cause Systems

Generalised Reic hen bac hian Common Cause Systems Claudio Mazzola Sch o ol of Historical and Philosophica l Inquiry The Univ er sit y of Queensland F organ Smith Building (1), St. Lucia, QLD 4072, Australia c.mazzola@uq.e du.a u 1 In tro duction Chancy coincidences happ en everyday , but sometimes coinci dences are just to o striking, or to o improbable, not to reveal the pre s ence of some c o ordinating pro cess. T o wit, if all the electr ical applia nces in a building were to shut down at exactly the sa me time, it would not b e unreas onable to s e arch for a brea kdown in their common p ower supply . Similar ly , if the pr ice of p etro l were to simultaneously rise in all o il impo rting countries, it would b e a fair b et that exp orters had concertedly decided to reduce extraction. The principle of the common cause is the inferential rule g ov erning instances of this kind: informally stated, it ass erts that improbable coincidences are to b e put down to the action of a common cause. Reichen bach [15] was the first to pr ovide the pr inciple of the co mmo n cause with a matema tica l charac- terisation. His treatment relied on three ma jor ingredient s : first, he represented improbable coincidences as p o sitive probabilistic cor relations b etw een rando m even ts; second, he demanded that common caus es should incre a se the probability o f their effects; a nd third, he further req uir ed that conditioning o n the presence, or on the a bsence, of a common cause should mak e its effects probabilistically indep enden t from one another. Reichn enb ach’s tr e a tmen t, how ever, was ov erly restrictive, as it r ested on a to o narrow c o nception of improbable coincidences, and on a c orresp ondingly narrow under standing on the expla natory function of common causes. In [11], I acco rdingly prop osed an improv ed interpretation of the principle, along with a suitably revided probabilistic mo del for common cause s , which generalises Reichen ba c h’s origina l formulation in tw o resp ects. On the one hand, it represents i mprobable coincidences not a s pos itive correla tio ns, but r ather as p ositive differences b etw een the c orrelation actually e x hibited by a sp eficied pair of events, and the correla tion that they should exhibit according to historical data, background beliefs, or established theory . On the o ther ha nd, a nd cor resp ondingly , it dema nds that co nditioning on the presence or on the absence of a co mmo n causes should resto re the exp ected correla tion b etw een its effects. Reichen bach’s understanding of the principle is demonstrably a sp ecial case of this interpretation, applying when the exp ected co rrelation b etw een the event s of interest is null. Nevertheless, there is o ne resp ect in which the probabilistic mo del prop ose d in [1 1] is still no t general enoug h. Lik e Reic hen bach’s o riginal account, in fact, it depicts the action of a single common cause, and it is accordingly inadequate to ca pture 1 instances whereby tw o c o o rdinated effects are broug ht a b out b y a system of distinct common c auses. The aim of this pap er is pre c is ely to further expand the mo del in this dire c tion. T o this end, tw o aven ues for the generalisa tion of the mo del will b e explored, ea ch based o n a different pr o babilistic characterisa tion for systems of common causes. The article will b e structured in three main se ctions. Firstly , in §2 the extended in terpretation of the principle elab ora ted in [11] will be br iefly outlined, and g iven forma l treatmen t. Next, in §3 sa id interpre- tation will b e incorp orated in to Hofer-Szab ó and Rédei’s Reichen bachian Common Cause Systems mo del [7]. Finally , in §4 the extended v ers ion of the principle will b e int egrated with m y own revisitation of Reichen bachian Commo n Cause Systems [10]. 2 Generalised conjunctiv e common causes Reichen bach or iginally applied the principle of the common cause to pairs of p ositively correlated, albeit causally unrelated, events. Befor e introducing his probabilistic mo del for co mmon causes, a definition of probabilistic correlation is th us needed: Definition 1. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B , C ∈ Ω such that p ( C ) 6 = 0 , we define: C orr ( A, B | C ) := p ( A ∧ B | C ) − p ( A | C ) p ( B | C ) . (1) Mor e over, C orr ( A, B ) := C orr ( A, B |∪ X i ∈ Ω X i ) . (2) The expr ession C orr ( A, B | C ) deno tes the c orr elation of even ts A and B c onditional on even t C . The expression C or r ( A, B ) , instead, deno tes the absolute c orr elation or unc onditional c orr elation of event s A and B . T wo even ts ar e said to be p ositively ( ne gatively ) c orr elate d (conditional on ano ther even t) if their co rrelation (co nditiona l on said even t) is greater (smaller ) than zer o ; by the same token, they are said to be un c orr elate d or pr ob abili stic al ly indep endent (conditional on another ev ent) if their correla tion (conditional on that even t) is equal to zero. The existence of a pos itiv e correla tion b etw een t wo even ts is often an indication that one o f them is a cause of the other . Howev er, this is not inv aria bly the case: as is well known, cor relation doe s not imply causation. Reic henbach’s in ter pr etation of the common ca use principle could indeed b e s een as an attempt to preser ve a one-one corr esp ondence betw een pro babilistic correla tion and causal dep endence [9]: in his account, acquiring informatio n ab out the o ccurrence of a common cause s hould disso lv e, a s it were, any po sitive co rrelation b etw een c a usally unrelated e vents. Reichen bach g av e formal shap e to this intuition b y demanding that co nditioning on the pre s ence of a common c a use, or on its a bsence, should make its effects probabilistically indep e nden t. The r esult was a proba bilistic mo del for common causes known as c onjunctive fork . With only a slig h t ter minological mo dification and few minor notationa l v aria n ts, we can in tro duce his model as follows: Definition 2. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any thr e e distinct A, B , C ∈ Ω , the event C is a c onjunctive c ommon c au se fo r C orr ( A, B ) 2 if and only if: p ( C ) 6 = 0 (3) p  C  6 = 0 (4) C orr ( A, B | C ) = 0 (5) C orr  A, B   C  = 0 (6) p ( A | C ) − p  A   C  > 0 (7) p ( B | C ) − p  B   C  > 0 . (8) Conjunctiv e c o mmon c a uses, as just defined, ar e intended to explain the o ccur r ence of non-caus al p ositive correla tio ns in tw o wa ys. On the one hand they increase the join t probability of their effects, c o nsequently fav ouring their correla tion, as esta blished b y the followin g prop osition: Prop osition 1. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom event s Ω and pr ob abili ty me asur e p . F or any thr e e distinct A, B , C ∈ Ω , if C is a c onjunctive c ommon c ause for C orr ( A, B ) , then: C orr ( A, B ) > 0 . (9) On the o ther hand, conditions (5)-(6 ) dema nd that the co rrelation betw een the effects o f a co njunctive common cause should disapp ear when co nditioning on the o ccur rence, or on the absence, of said caus e: in ja rgon, we say that the common cause scr e ens- off the tw o effects from one ano ther. This is meant to indicate that the p ositive correla tion be t ween the t wo effects is purely epiphenomenal, b eing a mere b y-pro duct of the underlying action of the common cause. Reichebac h’s conjunctive commo n ca use mo del has exerted co nsiderable influence in b oth probabilistic causal modelling and the philosoph y of scie nce . T o mention but few o f its contributions to the latter field, it anticipated the probabilistic causality pro gram [6, 21, 2, 18, 5], fos tered the development of pro babilistic accounts of scientific expla nation [16, 22], and inspir ed causal interpretations o f Bell’s no -go theor em in quantum ph y sics [24]. Sim ultaneously , the Bay esia n Net works mov ement in proba bilistic ca usal mo delling incorp orated and g eneralised the scr eening-off constraints (5)-(6) in the guise of the so-called Ca us al Marko v Condition, ac c o rding to which any t wo v ariables that are not related as cause a nd effect must b e probabilisitically indep endent conditiona l on the set of their direct cause s [13, 1 4, 19]. Nonetheless, the conjunctiv e common cause model r e lies on a demonstrably r estrictive understanding o f the principle of the common cause, and on a corr esp ondingly narrow co nce ption of the explanator y function p erformed b y common causes. T o fully appreciate this, it will b e instructive to start by taking a deep er lo o k at the v er y thing the principle of the common ca use is in tended to apply to: improbable coincidences . Reichenb ach, as w e saw, understo o d improbable co incidences a s pos itiv e co r relations be t ween ca usally unrela ted even ts. P o sitiv ely correla ted even ts tend to be coinstantiated, so it is clear w h y po sitive co rrelations can be used to give coincidences a pro ba bilistic repre sent a tion. The pro blem is: in what sense, then, ca n coincidences b etw een causally unrelated even ts be deemed impr ob able ? The underlying presupp osition is that in gener al causally unrelated even ts tend to b e uncor related, so in gener al positive corr e la tions b etw een such ev ents are not 3 to be expected. Reic henbach, in other words, applied the principle o f the common caus e to pair s o f even ts that happ en to b e p o sitively correlated, even though we w ould exp ect them to b e not. T o b e even more explicit: he applied the principle to ca ses where the obs erved v alue of the co rrelation betw een t wo ev ents is strictly higher than its exp ected v alue, which is zer o . Once the principle is presented in this wa y , how ever, it b ecomes apparent that there is no r eason not to demand that it s hould equally a pply to al l ca ses in which tw o events are mor e strongly cor related than exp ected, whatever the v alue o f their exp e cted co r relation. The extende d principle of the c ommon c ause is sp e cifically tailo red to meet this demand. Compressed in o ne sen tence, it claims tha t the role of common causes is to explain statistically significant devia tions b etw een the estimated v alue o f a c orrelation and its exp ected v alue, by c o nditionally restoring the latter. T o illustrate, let us consider an economic example. Let us imagine that an econometric analysis revealed a strong positive corr elation b etw een holding a po stgraduate degree and earning a higher-than- av erag e income. This po sitive corr elation, in and o f itself, w ould not b e surprising, as it would b e consistent with bo th c ommon se ns e and micro economic theory: p eople who study more ar e likely to e a rn hig her wages, owing to the comparatively scar ce supply and higher productivity of skilled labo ur. But suppose that, in the ca se at hand, the estimated co rrelation were remark ably strong: strong eno ugh to b e significantly dissimilar from the average correlation rep orted b y other simila r studies. Then, excluding any mistakes in the analysis, it would b e natura l for one to wonder if there were a n y thing ab out the selected s a mple, which could bring ab out s aid discrepancy . The extended principle of common cause urges that the explana tion should b e sought in the presence of some unacknowledged common cause. T o wit, w e may imagine that the econometric analys is in our example were conducted in relatively wealth y subp opulation. People coming fro m wealth y families are more likely to undergo a dditional years of study , since they can mor e easily a fford the opp ortunity costs this in volv es ; moreov er , they are more likely to earn their degr ees fro m renouned but exp ensive academic institutions, whose gr aduates have a higher chance to b e hired in high-earning app ointmen ts. By simul- taneously increasing the pro babilit y of holding a pos tgraduate degree a nd the pro babilit y of earning a higher-than-average income, family wealth would consequently increas e their joint pro babilit y , a nd e x plain their stronger-tha n- usual co rrelation. Remark ably , in this case it w ould be unrea sonable to require that the cor r elation betw een holding a po stgraduate deg ree and o f earning a higher-than-average income should disappear conditional on family wealth: after all, as w e already no ticed, s o me positive corr e lation b et ween wage and qualification is to be exp ected. Rather, co nditioning o n the commo n caus e should re store the expected co rrelation betw ee n the t w o ev ent s , consequently eliminating the appar e nt disa greement be tw een the econometric analysis and the preceding studies. T o provide the e xtended principle of the common cause with so me formal bite, let us first define: Definition 3. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω , t he deviation of C orr ( A, B ) is the quantity δ ( A, B ) := C or r ( A, B ) − C or r e ( A, B ) , (10) wher e C or r e ( A, B ) denotes t he exp ected corr elation b etwe en A and B . 4 Notice that the notions of deviation and exp e ctation , as they ar e understo o d here, a re not necessar ily restricted to the corr espo nding sta tistical concepts: in particular, the exp ected correla tion b etw een tw o v alues may b e determined by non-statistical means, e.g. on the ba sis of logical or mathematical rules, or simply on the basis of entrenc hed prior b eliefs. On this basis, we can no w define: Definition 4. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any thr e e distinct A, B , C ∈ Ω , t he event C is a gener alised common cause for δ ( A, B ) if and only if: p ( C ) 6 = 0 (11) p  C  6 = 0 (12) C orr ( A, B | C ) = C orr e ( A, B ) (13) C orr  A, B   C  = C orr e ( A, B ) (14) p ( A | C ) − p  A   C  > 0 (15) p ( B | C ) − p  B   C  > 0 . (16) Just like conjunctive common ca uses do for p ositive corr elations, generalised co mmon causes explain po sitive devia tions in tw o ways. On the one ha nd, (13)-(14) demand that c o nditioning on the presence of a co mmon cause, or on its absence, should restor e the exp ected correlatio n b etw een its effects. On the other hand, g eneralised commo n causes increase the unco nditional corr elation b etw een their effects, consequently generating the o bserved discr epancy b etw een the estimated v alue of said corr elation and its exp ected v alue: Prop osition 2. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom event s Ω and pr ob abili ty me asur e p . F or any thr e e distinct A, B , C ∈ Ω , if C is a gener alise d c ommon c ause for δ ( A, B ) then δ ( A, B ) > 0 . (17) Quite evidently , conjunctive common causes ca n b e thought of gener alised common causes who se effects are exp ected to b e uncorrelated. Nonetheless, the genera li sed common cause model is demonstrably imm une from some of the mo st common ob jections to the standar d interpretation of the common cause principle. F or one thing, it has b een ob j ected that the screening-off con ditions (5)-(6) ar e to o restrictive, either bec a use they ar e only sa tisfied by deterministic common causes [2 3, 4], o r b ecause they exclusiv ely apply when the effects of a common cause are indep endently pro duced [1 7, 3]. Thi s ob jection is ea s ily met b y the generalised common c a use mo del, which dro ps (5)-(6) i n fav o ur o f the mor e general c onstraints (13)-(14). F or another thin g, it has b een con tended that no n - causal p ositive correlations that result from logical, mathematical, semantic, or nomic relations do not g enerally admit of a conjunctive common ca use [1, 25]. The existence of s imila r corre la tions is clearly detrimental to the common undertanding o f the principle of the common cause, but it is p erfectly co nsistent with its extended version. The reason is that, according to the extended principle, simila r correla tions simply do no t c al l for a co mmon cause explanation: by h yp othesis, they are determined by logical, mathematica l, semantic, or physical laws, so they must b e ex pected, and as such they fall o utside its prop er domain of a pplication. They are, 5 accordingly , no co un ter example to it. The following sections will b e dedicated to further enrich the gener alised common cause mo del, so as to cov er systems of mul tiple common causes. 3 Generalised HR-Reic hen bac hi an Common Cause Systems The first attempt to extend the co njunctive co mmon cause mode l to compris e systems o f multiple common causes w as made by Hofer - Szab ó and Rédei in [7], who prop osed, to this end, the following definition: Definition 5. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω , a Reichen ba chi an Co mmon Cause System (HR-R CC S) of size n ≥ 2 for C orr ( A, B ) is a p artition { C i } n i =1 of Ω such that: p ( C i ) 6 = 0 ( i = 1 , ..., n ) (18) C orr ( A, B | C i ) = 0 ( i = 1 , ..., n ) (19)  p ( A | C i ) − p ( A | C j )  p ( B | C i ) − p ( B | C j )  > 0 (1 , ..., n = i 6 = j = 1 , ..., n ) . (20) Hofer-Szab ó and Rédei refer to Reichenb achian Common Cause Systems using the acronym R CCS. The acronym HR-R CCS is here employ ed to distinguish their model from the one utilized in the nex t section. The notion of a HR-RCCS is meant to generalise the notion o f a conjunctive common cause in tw o resp ects. On the one hand, Hofer-Sza bó and Rédei demonstrate that o nly positively cor related pairs admit o f a HR-R CCS. On the other ha nd, co nditions (18), (19 ) and (20) ar e intended to g eneralise, resp ectively , conditions (3)-(12), (5)–(6), and (7)–(8) from Definition 2. Sp ecifically , (19) demands that ea ch elemen t of a HR-R CCS screen-o ff its commo n effects from one a no ther. This means that HR-RCCSs increa se the correla tio n b etw een otherwise uncorrela ted pairs, em ulating a s a consequence the explanatory function of conjunctiv e common causes. Mo difying the a b ove definition in accordance with the extended in terpretation of common cause principle only requires replacing the sc r eening-off condition (19) with a s uita bly generalised v ariant of (13)-(14). Let us according ly define: Definition 6. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω , a Gener alised Reichen bachian Commo n Cause System (GHR-RCCS) of size n ≥ 2 for δ ( A, B ) is a p artition { C i } n i =1 of Ω such that: p ( C i ) 6 = 0 ( i = 1 , ..., n ) (21) C orr ( A, B | C i ) = C or r e ( A, B ) ( i = 1 , ..., n ) (22)  p ( A | C i ) − p ( A | C j )  p ( B | C i ) − p ( B | C j )  > 0 (1 , ..., n = i 6 = j = 1 , ..., n ) . (23) This definition generalises at once Definition 4 a nd Definition 5: it extends the former by admitting systems of any numb er of common causes; it extends the la tter b y requiring that every common cause in a system should restore the exp ected corr elation betw een its tw o effects whatever its value . 6 Not sur prisingly , every GHR-R CCS increa ses the corre la tion betw een its effects, consequently emulating the explanatory function of genera lised common causes. Prop osition 3. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom event s Ω and pr ob abili ty me asur e p . F or any A, B ∈ Ω and any { C i } i ∈ I ⊆ Ω , if { C i } i ∈ I is a GHR -RCC S of size n ≥ 2 for δ ( A, B ) , then (17) obtains. T o show this, let us first prov e the follo wing lemma: Lemma 1. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . L et A, B ∈ Ω and let { C i } i ∈ I b e a p artition of Ω satisfying c onditions (21)-(22). Then: δ ( A, B ) = 1 2 X i,j ∈ I p ( C i ) p ( C j ) [ p ( A | C i ) − p ( A | C j )][ p ( B | C i ) − p ( B | C j )] . (24) Pr o of. Let (Ω , p ) be a classic a l pr obability spa c e with σ -alg ebra of rando m ev ent s Ω and pr obability measure p . Let A, B ∈ Ω and let { C i } i ∈ I be a partition of Ω satisfying (21). F rom the theorem of total probability it follows that: C orr ( A, B ) = 1 2 n X i,j =1 p ( C i ) p ( C j )  p ( A | C i ) − p ( A | C j )  p ( B | C i ) − p ( B | C j )  + 1 2 " n X i =1 p ( C i ) C or r ( A, B | C i ) + n X j =1 p ( C j ) C orr ( A, B | C j ) # . (25) On the other hand, by h yp othesis { C i } i ∈ I is a partition of the giv en probability space, which implies that: n X i =1 p ( C i ) = 1 . (26) F urther assuming (22) will therefor e produce the following equality: C orr ( A, B ) − C or r e ( A, B ) = 1 2 n X i,j =1 p ( C i ) p ( C j ) [ p ( A | C i ) − p ( A | C j )][ p ( B | C i ) − p ( B | C j )] , (27) which in the lig h t of (10) is but a different fo r m ulation of (24). Demonstrating Prop os itio n 5 on this basis would be straightforward, so we are omitting the details of the pro of. One interesting thing to notice ab out this demonstra tion, howev er, is that setting C or r e ( A, B ) = 0 in (2 7) w ould reduce it to the equatio n employ ed by Hofer-Szab ó a nd Rédei to demonstra te that HR- R CCSs pro duce po sitiv e corr elations. This fact, in itself, is further confirmation o f the adequacy o f GHR-R CCSs as a generalisa tion of HR-RCCSs. 7 3.1 Existence of GHR-RCCSs Hofer-Szab ó and Rédei [8] ar gue that a HR-RCCS of arbitrary finite size exists fo r every p ositively correla ted pair of even ts, in some suitable extension of the original probability space. The discussio n to follow will b e dedicated to es ta blish a similar result for GHR-RCCSs. Remark ably , it will turn out that not all pos itiv e deviations admit of a GHR-RCCS. Hofer-Szab ó a nd Rédei’s pro o f pro ceeds b y noticing that, in g eneral, set { C i } n i =1 is a HR-RCCS of s ize n ≥ 2 for C orr ( A, B ) in probability space (Ω , p ) if and only if the v alues o f p ( A | C 1 ) , ...., p ( A | C n ) , p ( B | C 1 ) , ...., p ( B | C n ) , and p ( C 1 ) , ...., p ( C n ) satisfy some sp ecified constraints. They call a n y set { a i , b i , c i } n i =1 of 3 n n umbers satisfying said constra in ts admissibl e for C orr ( A, B ) and demonstrate th at, for any tw o po sitiv ely correlated e vents A and B and an y n ≥ 2 , a set of n admissible n um ber s for C orr ( A, B ) ca n be found. On this basis, they fina lly sho w how an extension of the given pr obability space can alwa ys b e constructed, in w hich so me pa rtition { C i } n i =1 exists such that n ≥ 2 and the v alues o f p ( A | C 1 ) , ...., p ( A | C n ) , p ( B | C 1 ) , ...., p ( B | C n ) , and p ( C 1 ) , ...., p ( C n ) are admissible for C orr ( A, B ) , thereby eta blishing the existence of a HR-RCCS o f size n for C orr ( A, B ) in that space. The following pro of will follow the broa d lo gical structure of Hofer-Sza bó and Rédei’s arg umen tatio n. Our first step will cons is t in ident ifying the necessary a nd s ufficient conditions that must b e sa tisfied by the v alues of p ( A | C 1 ) , ...., p ( A | C n ) , p ( B | C 1 ) , ...., p ( B | C n ) , p ( A ∧ B | C 1 ) , ...., p ( A ∧ B | C n ) , and p ( C 1 ) , ...., p ( C n ) to ma ke { C i } n i =1 a GHR-RCCS of size n ≥ 2 for δ ( A, B ) . This, how ever, will b e done in t w o s tages, as some of the c o nditions that we are g oing to s ingle out will b e sha red by the mo del to be developed in §4. Let us begin by iso lating these. Definition 7. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω satisfying (17) and any n ≥ 2 , the set { a i , b i , c i , d i } n i =1 of r e al n umb ers is c al le d quasi-admissible* for δ ( A, B ) if and only if the fol lo wing c onditions hold: n X i =1 a i c i = p ( A ) (28) n X i =1 b i c i = p ( B ) (29) n X i =1 c i = 1 (30 ) d i − a i b i = C orr e ( A, B ) ( i = 1 , ..., n ) (31) 0 < a i , b i , d i < 1 ( i = 1 , ..., n ) (32) 0 < c i < 1 ( i = 1 , ..., n ) . (33) The attentiv e reader will hav e noticed that, for each n ≥ 2 , quasi-admissible se ts include 4 n num b ers, whereas a dmissible sets, as defined b y Hofer-Szab ó a nd Rédei, include only 3 n num b ers. Moreov er , while (28)-(30) and (32)-(33) a re e ither identical to or straig htforward generalizations of s ome of Hofer-Szab ó 8 and Rédei’s or iginal conditions for admissible num b ers, constraint (30) is not. Similar changes a re needed to av oid a logical mistake in their orig inal proof, along the lines illustrated in more detail in [12]. T o complete this par t of the proof, w e need to supplement quasi-admissible sets with one mo re co ndition: Definition 8. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω and any n ≥ 2 , a set { a i , b i , c i , d i } n i =1 of r e al n umb ers is c al le d HR-a dmiss ible for δ ( A, B ) if and only if it is quasi-admissible for δ ( A, B ) and it further satisfies [ a i − a j ][ b i − b j ] > 0 (1 , ..., n = i 6 = j = 1 , ..., n ) . (34) The a deq uacy of the ab ov e definition is testified by the following lemma, whose pro of is straig htforward and whic h can consequently b e o mitted: Lemma 2. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω and any { C i } n i =1 ⊆ Ω wher e n ≥ 2 , the set { C i } n i =1 is a GHR- RC CS of size n for δ ( A, B ) if and only if ther e exists a set { a i , b i , c i , d i } n i =1 of HR-admissible num b ers for δ ( A, B ) such that p ( C i ) = c i ( i = 1 , ..., n ) (35) p ( A | C i ) = a i ( i = 1 , ..., n ) (36) p ( B | C i ) = b i ( i = 1 , ..., n ) (3 7) p ( A ∧ B | C i ) = d i ( i = 1 , ..., n ) . (38) The next s tep in our pr o of will b e to establish the necessary and sufficien t conditions for the ex istence of HR-admissible nu m ber s for δ ( A, B ) . T o this purp ose, how ever, we shall need the following lemma: Lemma 3. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . Mor e over, let A, B ∈ Ω and let { C i } n i =1 ⊆ Ω with n ≥ 2 . Then, any set { a i , b i , c i , d i } n i =1 of r e al numb ers satisfying identities (35)-(38) is quasi-admi ssible for δ ( A, B ) if and only if it further satisfies (32)-(33) as wel l as a n = a − n − 1 X k =1 c k a k 1 − n − 1 X k =1 c k (39) b n = b − n − 1 X k =1 c k b k 1 − n − 1 X k =1 c k (40) c n = 1 − n − 1 X k =1 c k (41) 9 d n = ε + " a − n − 1 X k =1 a k c k # " b − n − 1 X k =1 b k c k # " 1 − n − 1 X k =1 c k # 2 (42) d k = ε + a k b k ( k = 1 , ..., n − 1) , (43) wher e a = p ( A ) (44) b = p ( B ) (45) ε = C orr e ( A, B ) . (46) Pr o of. Let (Ω , p ) be a classic a l pr obability spa c e with σ -alg ebra of rando m ev ent s Ω and pr obability measure p . Moreover, let A, B ∈ Ω satisfy (17) and let the set { a i , b i , c i , d i } n i =1 of n ≥ 2 real num b es satisfy conditions (35)-(38) and (32)-(33). Fina lly , let (44)-(46) b e in place. Given the afores aid h y pothesis , (28)-(30) can b e dir ectly obta ined from (39)-(41) thanks to the theorem of total probability , a nd vice-versa. Therefore, we only need to show that (31) obtains if a nd only if (42)-(43) do . T o this pur po s e, let us first obser ve that, as a further c o nsequence of the theo rem o f to ta l probability , the following equality holds: d n = [ d n − a n b n ] + " a − n − 1 X k = i a k c k # " b − n − 1 X k =1 b − b k # " 1 − n − 1 X k =1 c k # 2 . (47) Thanks to (4 7) it is then immediate to verify that (42)-(43) ar e simultaneously satisfied if (31) is. Con- versely , let us suppo se that (42)-(43) are the case. Then (42) and (47) will jointly imply that d n − a n b n = ε, (48) which , together with (43), straighfor wardly implies (31), as required. Endo wed with the ab ov e r esult, we ar e now in a p osition to deter mine the necessa ry conditions so that, in general, HR-admissible num be r s { a i , bi , c i , d i } n i =1 could exist for δ ( A, B ) and n ≥ 2 . Quite in ter estingly: Lemma 4. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω , n o H R-admissible n umb ers exist for δ ( A, B ) if C orr e ( A, B ) + p ( A ) p ( B ) ≤ 0 . (49) Pr o of. Let (Ω , p ) be a classic a l pr obability spa c e with σ -alg ebra of rando m ev ent s Ω and pr obability measure p , and let A, B ∈ Ω be arbitrarily chosen. Lemma 4 will b e established by con tra po sition, so let 10 us assume that, for so me n ≥ 2 , a set { a i , b i , c i , d i } n i =1 of HR-admissible num b ers do es exist for δ ( A, B ) . Moreov er, let us a ssume iden tities (44)-(4 6). T o prov e our lemma, t wo preliminary steps will b e requir ed. First, we shall prov e that some a j , b k ∈ { a i , b i , c i , d i } n i =1 exist such that a − a j > 0 (50) b − b k > 0 . (51) Next, on that basis, we shall demonstrate that at least so me such a j , a k ∈ { a i , b i , c i , d i } n i =1 exist, for whic h j = k . T o estabish the first claim, let us b egin by noticing that, as a plain conse q uence of (28)-(29) and (30): 0 = a − a = n X i =1 ac i − n X i =1 a i c i = n X i =1 c i ( a − a i ) (52) 0 = b − b = n X i =1 bc i − n X i =1 b i c i = n X i =1 c i ( b − b i ) . (53) On the other hand, (33) dema nds that c i > 0 for a ll i = 1 , ..., n , while (34) implies that a j = a and b k = b can be sa tisfied by at most one term a j and one ter m b k for j, k = 1 , ..., n ≥ 2 . The a b ove equa lities therefore imply that a − a i be p ositive fo r some v alues of i and negative for others, while similarly b − b i be pos itiv e for some v alues of i and negative fo r others. This is enough to pro ve (50) and (51 ), as des ired. T o prov e our second auxiliary result, let us first rela bel all nu m ber s in { a i , b i , c i , d i } n i =1 so that a 1 < ... < a k < a ≤ a k +1 < ... < a n . (54) This in turn implies that a i − a j < 0 i = 1 , ..., k ; j = k + 1 , ..., n. (55) Now, let us pr o c eed b y reductio, and let us assume that b i − b > 0 i = 1 , ..., k . (56) Then, according to the result pr eviously established, some b j ∈ { b i } n i = k +1 ⊂ { a i , b i , c i , d i } n i =1 should exist such that b − b j > 0 . (57) How ever, in that ca se b i − b j > 0 i = 1 , ..., k (58) 11 would ensue. T ogether with (55), this would imply [ a i − a j ][ b i − b j ] < 0 i = 1 , ..., k (59) consequently contradicting (34). By reductio, this shows that (5 0)-(51) must be satisfied for some a j , b k ∈ { a i , b i , c i , d i } n i =1 where j = k . Let us now come to the main pa rt of our pro of. Thanks to the results so established, we can now safely claim that, for any set { a i , b i , c i , d i } n i =1 of HR-admissible num b ers, some a i , b i ∈ { a i , b i , c i , d i } n i =1 are alwa ys to be found such that ab − a i b i > 0 . (60) T ogether with (31), this implies that ab + ε > a i b i + ε = d i > 0 , (61) contradicting (49 ). By co ntrap osition, this means that whenever (49) is satisfied, no set { a i , b i , c i , d i } n i =1 of real num b ers can s atify (31) given the other co nditions for a HR-admissible s et fo r δ ( A, B ) . Hence, no HR-admissible set can exist for δ ( A, B ) . Let us now mov e to the sufficient condition for the existence of HR-admissible nu m ber s { a i , b i , c i , d i } n i =1 for δ ( A, B ) a nd n ≥ 2 . Remark ably , this turns out to b e the same as the necessary condition: Lemma 5. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω sati sfying (17), a set { a i , b i , c i , d i } n i =1 of HR-admissible numb ers for δ ( A, B ) exists for e ach n ≥ 2 if C orr e ( A, B ) + p ( A ) p ( B ) > 0 . (62) Pr o of. Let (Ω , p ) be a classic a l pr obability spa c e with σ -alg ebra of rando m ev ent s Ω and pr obability measure p . Mo reov er , let A, B ∈ Ω satisfy (17) and (62). T o start with, let us observe that (43) is in fa c t a system of n − 1 equations, namely one for ea ch v alue of i = 1 , ..., n − 1 . Therefore, (39)-(43) jointly comprise a system of 4 +( n − 1 ) = n +3 equations in 4 n v ar iables. This means tha t ea ch HR-admissible set for δ ( A, B ) is determined by a set of 4 n − ( n + 3 ) = 3 n − 3 parameters, fo r every n ≥ 2 . T o establish the existence o f such set, we acco rdingly need to pr ov e that such para meters exist. T o this purpose, let num b ers a , b and ε be understo o d as p er (44)-(46). Proo f will pro ceed b y induction on the cardinality of n . Let us b egin by assuming n = 2 as our inductiv e basis. This has the effect of transforming (39)-(43), 12 resp ectively , into: a 2 = a − c 1 a 1 1 − c 1 (63) b 2 = b − c 1 b 1 1 − c 1 (64) c 2 = 1 − c 1 (65) d 2 = ε + [ a − a 1 c 1 ][ b − b 1 c 1 ] [1 − c 1 ] 2 (66) d 1 − a 1 b 1 = ε. (67) Since a , b and ε are known by h yp othesis, choosing num b ers c 1 , a 1 and b 1 will therefore suffice to fix the v alues of all 4 n = 8 v ariables in the system. Let us acco rdingly c o nstrain c 1 so that: c i → 0 . ( 68) Owing to this, (63)-(66) immediately pro duce a 2 → a (69) b 2 → b (70) c 2 → 1 (71) d 2 → ε + ab, (72) while on the other hand (17) directly r e quires that 1 > a, b > 0 , (73) as it would be easy to verify . T aken together, this ensures that 1 > a 2 , b 2 > 0 (74) 1 > c 1 , c 2 > 0 , (75) while (62) and (17) imply that 1 > d 2 > 0 . (76) T o determine the remaining num b ers, w e further need to set a 1 and b 1 . In this ca se, our choice will dep end 13 on the v alue of ε , as follows: ε ≥ 0    a > a 1 > 0 b > b 1 > 0 (77) ε < 0    1 > a 1 > 0 1 > b 1 > 0 (78) Either option is allow ed b y (73), and either will ensur e that 1 > d i > 0 , (79) as it would b e straightforward to c heck with the a id of (43), (17) and (62). Thanks to Lemma (3), this is enough to establish that some set { a i , b i , c i , d i } n i =1 of qua si-admissible num b ers e x ist f or δ ( A, B ) if n = 2 . T o further show that suc h set is HR-admissible for δ ( A, B ) , w e only need to observe that (34) can b e obtained from bo th (77) and (78), owing to (28)-(30). Let us now assume, as our inductiv e hypo thesis, that some set { a i , b i , c i , d i } m i =1 be HR-a dmissible for δ ( A, B ) , where n = m > 2 . T o pr ov e that a HR-a dmissible set for δ ( A, B ) also exists if n = m + 1 , let us consider the set { a j , bj, c j , a m − 1 , b m − 1 } m − 2 j =1 ⊂ { a i , b i , c i , d i } m i =1 , and let us ch o ose num b ers a ′ m , b ′ m , c ′ m − 1 , c ′ m such that: a j > a ′ m > 0 ( j = 1 , ..., m − 1) (80) b j > b ′ m > 0 ( j = 1 , ..., m − 1) (81) c m , c m − 1 > c ′ m − 1 > 0 (82) 1 > c ′ m > 0 . (83) Given (39)-(43), the set { a j , bj, c j , a m − 1 , b m − 1 } m − 2 j =1 ∪  a ′ m , b ′ m , c ′ m − 1 , c ′ m  of 3( m − 1) + 3 = 3( m + 1) − 3 = 3 n − 3 parameters will then suffice to determine 4( m + 1) num b ers:  a j , b j , c j , d j , a m − 1 , b m − 1 , c ′ m − 1 , d m − 1 , a ′ m , b ′ m , c ′ m , d ′ m , a m +1 , b m +1 , c m +1 , d m +1  m − 2 j =1 . Because num b ers  a j , b j , c j , d j , a m − 1 , b m − 1 , c ′ m − 1 , d m − 1 , a ′ m , b ′ m , c ′ m ,  m − 1 j =1 satisfy conditions (31)-(33) and (34) b y hypo thesis, all we need to show is that s aid c onstraints be also sa tisfied by the r emaining nu m ber s { d ′ m , a m +1 , b m +1 , c m +1 , d m +1 } . T o this purpose, let us first no tice that (32) must b e true of d ′ m b y virtue 14 of (31) and (80)-(81). Next, tha nk s to (39)-(42), it will b e sufficient to suppose that a ′ m → 0 (84) b ′ m → 0 (85) c ′ m → c m − 1 − c ′ m − 1 (86) to obtain a m +1 → a m (87) b m +1 → b m (88) c m +1 → c m (89) d m +1 → d m (90) which we a lready know, by our inductiv e hypothesis, to satisfy (31)-(33). Mo reov er, (80) and (81) , along with the inductive a s sumption whereb y [ a m − a i ][ b m − b i ] > 0 ( i = 1 , ..., m − 1) , (91) ensures that [ a m +1 − a i ][ b m +1 − b i ] > 0 ( i = 1 , ..., m ) , (92) which together w ith o ur inductive hypothesis suffices to establish (34). Due to Le mma 3 a nd Definition 8, the set of 4( m + 1 ) num b ers s o determined is therefore HR-admissible for δ ( A, B ) . Lemma 5 is thus demonstrated b y induction. Before getting to the end of our existential proo f, w e need o ne more definition: Definition 9. L et (Ω , p ) and (Ω ′ , p ′ ) b e classic al pr ob ability sp ac es with σ -algebr as of r andom events Ω and Ω ′ and with pr ob ability me asur es p and p ′ , r esp e ctively. Then (Ω ′ , p ′ ) is c al le d an extension of (Ω , p ) if and only if ther e exists an inje ct ive lattic e homomorphism h : Ω → Ω ′ , pr eserving c omplementation, such that p ′ ( h ( X )) = p ( X ) for al l X ∈ Ω . (93) The results of our demonstra tion can th us b e crys ta llized in to the following prop o sition: Prop osition 4. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom event s Ω and pr ob abili ty me asur e p . F or any A, B ∈ Ω satisfying (17) and any n ≥ 2 , an extension (Ω ′ , p ′ ) of (Ω , p ) including a GHR-RCCS of size n for δ ( A, B ) exists if and only if A and B satisfy (62). Pr o of. The only-if clause immediately follows from Lemma 2 and Lemma 4. Pro of of the if-clause, instead, is structura lly similar to Step 2 of Hofer-Szab ó and Rédei’s pro of for the existence of HR-RCCSs of 15 arbitrary finite size in [8], although Hofer- Sza bó and Rédei’s conditions (60 )-(63) will ha ve to be replaced b y: r 1 i = c i d i p ( A ∧ B ) (94) r 2 i = c i a i − c i d i p  A ∧ B  (95) r 3 i = c i b i − c i d i p  A ∧ B  (96) r 4 i = c i − c i a i − c i b i + c i d i p  A ∧ B  , (97) which , owin g to (31), actually reduce to the afor esaid conditions for ε = 0 . 4 Generalised Reic hen bac hian Common Cause Systems Revisited There are tw o aspe cts in whic h HR-RCCSs ma y not b e considered fully satisfactory generalisations of conjunctiv e co mmon cause s . The first asp ect is that they can admit of elemen ts that ar e probabilistically independent of one or b oth ev ent s from the corresp onding cor r elated pair . This is at o dds with the in tuition that p ositive ca us es should c eteris p aribus increase the probability of their effects, and that negative caus es should c eteris p aribus decrease their probability . The second aspect is that they rule out the possibility that t wo distinct causes could equally alter the probability of one, or b oth, of their effects. On the face of it, there is s imply no reaso n why a s ystems of common causes should b e so constr ained. T o ov erco me these limitations, in [10] I prop osed a revisitatio n of HR-RCCSs, along the following lines: Definition 1 0. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω , a Reic hen bachian Common Cause System* (M-RCCS) of size n ≥ 2 for C orr ( A, B ) is a p artition { C i } n i =1 of Ω such that: p ( C i ) 6 = 0 ( i = 1 , ..., n ) (98) C orr ( A, B | C i ) = 0 ( i = 1 , ..., n ) (99)  p ( A | C i ) − p ( A )  p ( B | C i ) − p ( B )  > 0 ( i = 1 , ..., n ) . (100) Whether M- RCCSs are r eally to b e preferr ed to HR-RCCSs is controv ers ia l [20]. How ever, this is no place to settle that issue. Rather, in this section we shall limit our selves to offer an alternative extension of the generalised common cause mo del, by tak ing M-R CCSs as a basis. Let us accor ding ly define: Definition 1 1. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω , a Gener alised Reichenb achian Common Cause System* (GM-RCCS) of size n ≥ 2 for C orr ( A, B ) is a p artition { C i } n i =1 of Ω such that: p ( C i ) 6 = 0 ( i = 1 , ..., n ) (101) C orr ( A, B | C i ) = C or r e ( A, B ) ( i = 1 , ..., n ) (102)  p ( A | C i ) − p ( A )  p ( B | C i ) − p ( B )  > 0 ( i = 1 , ..., n ) . (103) 16 Just as with GHR-RCCSs, it can b e shown that GM-RCCSs inv ariably pro duce a po sitiv e deviation betw een the observed co rrelation o f their effects a nd their exp ected cor relation. T o this e nd, let us first in tro duce the following lemma: Lemma 6. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . L et A, B ∈ Ω and let { C i } i ∈ I b e a p artition of Ω satisfying c onditions (101)- (102). Then: δ ( A, B ) = X i ∈ I p ( C i ) [ p ( A | C i ) − p ( A ) ][ p ( B | C i ) − p ( B )] . (104) Pr o of. Let (Ω , p ) be a classic a l pr obability spa c e with σ -alg ebra of rando m ev ent s Ω and pr obability measure p . Let A, B ∈ Ω and let { C i } i ∈ I be a par tition of Ω for which (1 01) holds. The theorem of total probability thereby implies that C orr ( A, B ) = n X i =1 p ( C i ) [ p ( A | C i ) − p ( A )][ p ( B | C i ) − p ( A )] + n X i =1 p ( C i ) C orr ( A, B | C i ) . (105) Let us now suppose that (102) b e satisfied, to o. Then, owing to the fact that n X i =1 p ( C i ) = 1 , (106) few elemen tar y calculations w o uld transform the ab ov e equality in to: C orr ( A, B ) − C or r e ( A, B ) = n X i =1 p ( C i ) [ p ( A | C i ) − p ( A )][ p ( B | C i ) − p ( A )] , (107) which acc o rding to (10) is just a restatement o f (104). Based on the ab ov e lemma, it would then be easy to demonstra te the following prop os ition: Prop osition 5. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom event s Ω and pr ob abili ty me asur e p . F or any A, B ∈ Ω and any { C i } i ∈ I ⊆ Ω , if { C i } i ∈ I is a GM-RCCS of size n ≥ 2 for δ ( A, B ) , then (17) obtains. GM-RCCSs acco rdingly per form a similar explanator y function as GHR-RCCSs. Quite in terestingly , moreov e r , for every t wo ev en ts A and B in a clas sical probabilit y space such that δ ( A, B ) > 0 a nd every n ≥ 2 , a GM-RCCSs of size n for δ ( A, B ) exists in some extension of the given probability space if and only if a GHR-RCCS does. Demonstrating this will be our next ob jectiv e. 4.1 Existence of GM-R CCSs The existential pr o of we s hall elab orate in this section will follow the broad lines of the one develop ed in §3.1. J ust as with GHR-RCCSs, we shall fir st determine the necessary and sufficien t co nditions the v alues of p ( A | C 1 ) , ...., p ( A | C n ) , p ( B | C 1 ) , ...., p ( B | C n ) , p ( A ∧ B | C 1 ) , ...., p ( A ∧ B | C n ) , and p ( C 1 ) , ...., p ( C n ) ought to satisfy s o that { C i } n i =1 be a GM-RCCS of size n ≥ 2 for δ ( A, B ) . Subsequently , we 17 shall determine the necessary and s ufficien t co nditions for the existence of suc h num b ers, and on tha t basis we s hall finally establish the necessa r y and sufficien t co nditions for the existence of a n extension of the given pro babilit y spa ce, where a GM-R CCS of size n for δ ( A, B ) could be found. Quite eviden tly , GM-RCCSs differ from GHR-RCCSs o nly in that they substitute condition (23) with (103). Consequently , in order to complete the first step of our pr o of, w e only need to replace Definition 8 with the following o ne: Definition 1 2. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω and any n ≥ 2 , a set { a i , b i , c i , d i } n i =1 of r e al numb ers is c al le d M-admissible for δ ( A, B ) if and only if it is quasi-admissible for δ ( A, B ) and it further satisfies [ a i − p ( A ) ][ b i − p ( B )] > 0 ( i = 1 , ..., n ) . (108) Just as be fo r e, the adequacy of the ab ov e definition is easily established: Lemma 7. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω and any { C i } n i =1 ⊆ Ω wher e n ≥ 2 , t he set { C i } n i =1 is a GM-RCCS of size n for δ ( A, B ) if and only if ther e exists a set { a i , b i , c i , d i } n i =1 of M-admissible numb ers for δ ( A, B ) for which identities (35)-(38) ar e t rue. M-admissible num b ers are quasi-admissible by definition. This fact allows us to build on our previous discussion, to easily prov e the following result: Lemma 8. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω , n o M-admissible numb ers exist for δ ( A, B ) if C orr e ( A, B ) + p ( A ) p ( B ) ≤ 0 . (49) Pr o of. Lemma 8 is demonstrated in a similar wa y as Le mma 4. Let us acco rdingly (Ω , p ) b e a cla ssical probability space with σ -alg ebra of random even ts Ω a nd probability measur e p , let A, B ∈ Ω b e arbitr a rily chosen so as to satisfy (17), and let us further as s ume that for some n ≥ 2 , a set { a i , b i , c i , d i } n i =1 of M- admissible nu m ber s exists for δ ( A, B ) . Mo reov er, let iden tities (35)-(38) a nd(44)-(46) be in place. Showing that so me a i , b i ∈ { a i , b i , c i , d i } n i =1 exist satisfying a − a i > 0 (109) b − b i > 0 (110) in this case would only require so me elemen tary calculations, as the ab ov e inequalit y directly follow from (108) along with (28) and (29). The remainder of the pro of would then pro ceed in exactly the same way as the analogous pro of for Lemma 4. Condition (62) is thus necessa ry for the existence of M-admissible n um b ers for δ ( A, B ) , for any t wo even ts A and B satisfying (17) and any n ≥ 2 . Moreov er, as with GHR-R CCSs, it is also sufficient: 18 Lemma 9. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom events Ω and pr ob ability me asur e p . F or any A, B ∈ Ω satisfying (17), a set { a i , b i , c i , d i } n i =1 of M-admissible numb ers for δ ( A, B ) exists for e ach n ≥ 2 if C orr e ( A, B ) + p ( A ) p ( B ) > 0 . (62) Pr o of. Let (Ω , p ) be a classic a l pr obability spa c e with σ -alg ebra of rando m ev ent s Ω and pr obability measure p . Mo reov er , let A, B ∈ Ω satisfy (17) and (62). Pro of will pro ceed by induction o n n . Let n = 2 according ly be our inductiv e basis . Because for n = 2 conditions (34) and (108) b ecome equiv alent , this case was actually covered in the inductive pro o f for Lemma 5. Next, a s our inductive h ypo thesis, let n = m and let { a i , b i , c i , d i } m i =1 be M-admissible fo r δ ( A, B ) . O n this basis, let us now pro ceed to the last step of our inductive proo f, and let n = m + ( r − 1) , where r ≥ 2 . Let us first choos e so me c k ∈ { a i , b i , c i , d i } m i =1 . Then, (30) and (33) ensure that it is po ssible to find a set n c j k o r j =1 of r ≥ 2 identical real n um ber s, lying inside the interv al (0 , 1) , such that r X j =1 c j k = r X j =1 c k r = c k . (111) F urthermore, it is trivially p ossible to find three sets n a j k o r j =1 , n b j k o r j =1 and n d j k o r j =1 of r ≥ 2 identical real n umbers s a tisfying a j k = a k j = 1 , ..., r (112) b j k = b k j = 1 , ..., r (113) d j k = d k j = 1 , ..., r. (114) Given (44)-(45), our inductive h ypo thesis then implies: a = m X i =1 a i c i = m X k 6 = i =1 a i c i + a k c k = m X k 6 = i =1 a i c i + a k c k r r = m X k 6 = i =1 a i c i + r X j =1 a j k c j k = m + r X k 6 = i =1 a i c i (115) b = m X i =1 b i c i = m X k 6 = i =1 b i c i + b k c k = m X k 6 = i =1 b i c i + b k c k r r = m X k 6 = i =1 b i c i + r X j =1 b j k c j k = m + r X k 6 = i =1 b i c i (116) 1 = m X i =1 c i = m X k 6 = i =1 c i + c k = m X k 6 = i =1 c i + r X j =1 c j k = m + r X k 6 = i =1 c i (117) ε = d k − a k b k = d j k − a j k b j k j = 1 , ..., r . (118) This guara nt e es that the set { a i , b i , c i , d i } m k 6 = i =1 ∪ { a j , b j , c j , d j } r j =1 of 4 ( m + r − 1) num b ers so obtained satisfies (28)-(31). F urthermore, (32)-(33) and (10 8) are clearly satisfied owing to our inductive hypothesis and to the wa y n umbers { a j , b j , c j , d j } r j =1 were chosen. This is enough t o pr ov e that { a i , b i , c i , d i } m k 6 = i =1 ∪ { a j , b j , c j , d j } r j =1 is M-admissible for δ ( A, B ) , therefore concluding our inductive proo f. 19 Our existen tia l pro o f is now vir tually complete. Let us just add one final touch: Prop osition 6. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom event s Ω and pr ob abili ty me asur e p . F or any A, B ∈ Ω satisfying (17) and any n ≥ 2 , an extension (Ω ′ , p ′ ) of (Ω , p ) including a GM-RC CS of size n for δ ( A, B ) exists if and only if A and B satisfy (62). Pr o of. Pro of is in all similar to the pro of for Prop osition 4, mutatis mutandis. T wo final remarks may b e added at this at this p oint. First, Pro po sition 4 and Prop osition 6 b oth rectify the results anno unced in [11], where it was implicit ly assumed th at expec ted co rrelations be greater than or equa l to zero, leading to the erroneo us cla im that a genera lised common cause sho uld exist, in so me extension o f the initial pro babilit y space, for every p ositive deviation. Seco nd, GHR-R CCSs and GM- R CCSs for a given deviation may not co exist in the same proba bilit y space. Nev erthless, Prop osition 4 and 6 join tly guara n tee the following result: Prop osition 7. L et (Ω , p ) b e a classic al pr ob ability sp ac e with σ -algebr a of r andom event s Ω and pr ob abili ty me asur e p . F or any A, B ∈ Ω satisfying (17), an extension (Ω ′ , p ′ ) of (Ω , p ) including a GHR-RCCS of size n ≥ 2 for δ ( A, B ) ex ists if and only if ther e is some extension (Ω ′′ , p ′′ ) of (Ω , p ) including a GM-RCCS of size m ≥ 2 for δ ( A, B ) . This means that, irresp ective o f the different probabilistric pro p erties of GHR-RCCSs and GM-RCCSs, neither mo del can explain more o r different devia tions than the other. The t wo mo dels ar e thu s to b e assessed based not o n what they ca n explain, but how. The question a s to whether M-RCCSs s hould b e preferred to HR-RCCSs, th erefore, is carr ied o ver to th eir gener alised coun terpa rts. 4.2 Conclusion The principle o f the co mmo n c a use decrees that impro bable coincidences be put down to the action o f some common cause . The s tandard interpretation of the principle takes this as a req uirement that p ositive correla tio ns b et ween causally unrelated even ts be remov ed by conditioning o n so me conjunctiv e common cause. The interpretation here promoted, and encapsulated in the extended principle of the common cause, urges by contrast that common causes b e called fo r in o rder to explain p ositive dev iations b etw een the estimated co rrelation of tw o even ts and their exp ected co r relation. This pap er has outlined tw o distinct pr obabilistic mo dels for systems of co mmo n causes that incorp orate the ex tended intepretation of the principle. GHR-RCCSs have b e e n elab ora ted by combining the generalis e d common cause mo del with HR-RCCSs. GM-R CCSs, instead, hav e b een obtained by integrating generalised common causes with M-RCCSs. The necessary and sufficient conditions for the exis tence of finite systems of either kind hav e been determined. 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