Do Reichenbachian Common Cause Systems of Arbitrary Finite Size Exist?
aa r X i v : . [ s t a t . O T ] F e b Do Reichenbachian Common Cause Systems of Arbitrary FiniteSize Exist?
Claudio Mazzola
School of Historical and Philosophical InquiryThe University of QueenslandForgan Smith Building (1), St. Lucia, QLD 4072, [email protected]
Peter Evans
School of Historical and Philosophical InquiryThe University of QueenslandForgan Smith Building (1), St. Lucia, QLD 4072, [email protected]
Abstract
The principle of common cause asserts that positive correlations between causally unrelated eventsought to be explained through the action of some shared causal factors. Reichenbachian commoncause systems are probabilistic structures aimed at accounting for cases where correlations of theaforesaid sort cannot be explained through the action of a single common cause. The existence ofReichenbachian common cause systems of arbitrary finite size for each pair of non-causally correlatedevents was allegedly demonstrated by Hofer-Szabó and Rédei in 2006. This paper shows that theirproof is logically deficient, and we propose an improved proof.
Two random variables are positively correlated just in case their joint probability is greater thanthe product of the respective probabilities. Positive correlation does not imply causation, yet purelychancy positive correlations are hard to come about. So, whenever two random variables appear tobe positively correlated and yet neither of them causes the other, some shared cause is likely to beat work, which could increase the probability of their joint occurrence, and in terms of which theircorrelation could consequently be explained.This, in a nutshell, is the principle of the common cause . Everyday examples of the principleabound: a physician receiving several patients who exhibit the same symptoms would likely attributetheir condition to the same pathogen; a teacher presented with two almost identical research as-signments would presumably infer that they had been copied from the same online source; and thesimultaneous shutting down of all electrical appliances in a building is most likely attributed to afailure in their common power supply. Nor are instances of the principle hard to find in the empiricalsciences. To wit, commonality of phenotypical traits amongst species is typically explained through ome common ancestry; the distribution of iron filings along concentric patterns is explained throughthe action of a central magnetic field; and even the matching shapes of continents are explainedthrough their detachment from a single original land.Reichenbach [7] first gave formal shape to the principle by demanding that instances of non-causalpositive correlations be attributed to the presence of a conjunctive fork . Informally, conjunctive forksare elementary causal structures whereby a common cause increases the probability of two otherwiseindependent effects. Put formally:
Definition 1.
Let (Ω , p ) be a classical probability space with σ -algebra of random events Ω and prob-ability measure p . For any three distinct A, B, C ∈ Ω , the ordered triple h A, B, C i is a conjunctivefork for the pair ( A, B ) if and only if: = p ( C ) = 0 (1.1) p ( A ∧ B | C ) − p ( A | C ) p ( B | C ) = 0 (1.2) p (cid:0) A ∧ B (cid:12)(cid:12) C (cid:1) − p (cid:0) A (cid:12)(cid:12) C (cid:1) p (cid:0) B (cid:12)(cid:12) C (cid:1) = 0 (1.3) p ( A | C ) − p (cid:0) A (cid:12)(cid:12) C (cid:1) > (1.4) p ( B | C ) − p (cid:0) B (cid:12)(cid:12) C (cid:1) > . (1.5)Conjunctive forks are meant to explain positive correlations in two ways. First, whenever a con-junctive fork of the form h A, B, C i exists, the pair ( A, B ) can be easily demonstrated to be positivelycorrelated: Proposition 1.
Let (Ω , p ) be a classical probability space with σ -algebra of random events Ω andprobability measure p . For any three distinct A, B, C ∈ Ω , if h A, B, C i is a conjunctive fork, then: p ( A ∧ B ) − p ( A ) p ( B ) > . (1.6)Second, conditions (1.2)–(1.3) in the definition of a conjunctive fork declare that the correlationin (1.6) should vanish conditional on the occurrence of the common cause. This means that suchcorrelation is purely epiphenomenal, being a mere by-product of said cause.Still, a conjunctive fork may not exist for each pair of causally independent but positively correlatedevents ( A, B ) in the given probability space. The principle of the common cause can be preserved inthe face of this limitation in two ways. On the one hand, one may look for conjunctive forks in a largerprobability space [3]; alternatively, one may attribute the correlation not to a single common cause,but rather to the collective action of a plurality of common causes. In [1] Hofer-Szabó and Rédeipursue the latter strategy, introducing what they called Reichenbachian common cause systems : Definition 2.
Let (Ω , p ) be a classical probability space with σ -algebra of random events Ω and prob-ability measure p . For any A, B ∈ Ω , a Reichenbachian common cause system (RCCS) of size n ≥ for the pair A, B is a partition { C i } ni =1 of Ω such that: p ( C i ) = 0 ( i = 1 , ..., n ) (1.7) p ( A ∧ B | C i ) − p ( A | C i ) p ( B | C i ) = 0 ( i = 1 , ..., n ) (1.8) [ p ( A | C i ) − p ( A | C j )][ p ( B | C i ) − p ( B | C j )] > , ..., n = i = j = 1 , ..., n ) . (1.9) RCCS s are claimed to generalise the notion of a conjunctive fork in two respects. On the one hand,conditions (1.7), (1.8) and (1.9) are intended to generalise constraints (1.1), (1.2)–(1.3) and (1.4)–(1.5) For alternative formalisations of the principle see for instance [8] and [6]. For a comprehensive review of the currentstatus of the principle in probabilistic causal modelling, see [4]. espectively. On the other hand, Hofer-Szabó and Rédei show that every RCCS determines a positivecorrelation between the terms of the corresponding pair, thereby emulating the explanatory functionof conjunctive forks. But do
RCCS s actually exist for each positively correlated pair of causallyindependent events?Hofer-Szabó and Rédei claim that they do [2]. More precisely, they offer a proof to the effect that,for each non-strictly positively correlated pair of events ( A, B ) in a classical probability space (Ω , p ) and any integer number n ≥ , some extension of (Ω , p ) can be constructed whereby a RCCS of size n can be found for said pair. Their proof, however, is invalid. The aim of this article is to show whyit is, and to replace it with an amended one. Each of these goals will be pursued respectively in thefollowing two sections. Hofer-Szabó and Rédei’s proof is articulated into two major steps, which can be briefly summarisedas follows:
Step 1 : Hofer-Szabó and Rédei notice that conditions (1.7)–(1.9) and the axioms of classicalprobability theory jointly constrain the possible values that can be assigned to probabilities p ( A | C ) ,...., p ( A | C n ) , p ( B | C ) , ...., p ( B | C n ) , p ( C ) , ...., p ( C n ) , so that { C i } ni =1 could be a RCCS of size n for the pair ( A, B ) in a probability space (Ω , p ) . They thereby demonstrate that some set of n realnumbers satisfying such constraints can always be found, for each possible value of n ≥ and anycorrelated pair ( A, B ) . Step 2 : On this basis, Hofer-Szabó and Rédei show how, for each such n and ( A, B ) , an extension( Ω ′ , p ′ ) of (Ω , p ) can be constructed so that Ω includes a partition { C i } ni =1 and so that the values of p ( A | C ) , ...., p ( A | C n ) , p ( B | C ) , ...., p ( B | C n ) , p ( C ) , ...., p ( C n ) meet the constraints mentionedabove. This, they conclude, suffices to prove that some extension of the assumed probability spacecan in general be found, which includes a RCCS of size n for a given pair.Step 1 is where the mistake with Hofer-Szabó and Rédei’s demonstration lies, so we shall specificallyfocus on it. To facilitate our analysis, it will be convenient to further subdivide Step 1 into three sub-components. Step 1a : The proof begins by stating the set of conditions that a set { a i , b i , c i } nn =1 of n realnumbers must satisfy so that { C i } ni =1 be a RCCS of size n for the pair ( A, B ) in probability space Whether
RCCS s are an accurate generalisation of conjunctive forks is controversial. See [5] and [9] for two opposingviews on this matter. Ω , p ) . These conditions are: p ( A ) = n X i =1 a i c i (2.1) p ( B ) = n X i =1 b i c i (2.2) p ( A ∧ B ) = n X i =1 a i b i c i (2.3) n X i =1 c i (2.4) < [ a i − a j ][ b i − b j ] (1 , ..., n = i = j = 1 , ..., n ) (2.5) ≤ a i , b i ≤ i = 1 , ..., n ) (2.6) < c i < i = 1 , ..., n ) , (2.7)under identifications p ( C i ) = c i ( i = 1 , ..., n ) (2.8) p ( A | C i ) = a i ( i = 1 , ..., n ) (2.9) p ( B | C i ) = b i ( i = 1 , ..., n ) . (2.10)Hofer-Szabó and Rédei call a set of n numbers satisfying the above constraints admissible for thepair ( A, B ) . Step 1b : Next, Hofer-Szabó and Rédei point out that equalities (2.1)–(2.4) effectively restrictthe number of independent parameters in each admissible set { a i , b i , c i } nn =1 to n − . Specifically,they maintain, numbers c n − , c n , a n , and b n can be expressed as functions of the remaining n − parameters as follows: c n − = γ + n − X j,k =1 c j c k [ a j − a k ][ b j − b k ] − n − X k =1 c k [ a − a k ][ b − b k ] + γ n − X k =1 c k [ a − a n − ][ b − b n − ] + γ + n − X k =1 [ a n − − a k ][ b n − − b k ] (2.11) c n = 1 − n − X i =1 c i (2.12) a n = a + n − X k =1 c k [ a n − − a k ] − a n − − n − X k =1 c k + a n − (2.13) b n = b + n − X k =1 c k [ b n − − b k ] − b n − − n − X k =1 c k + b n − (2.14) here, for notational convenience, we set: a = p ( A ) (2.15) b = p ( B ) (2.16) γ = p ( A ∧ B ) − p ( A ) p ( B ) . (2.17) Step 1c : Finally, thanks to the above equivalences, it is demonstrated by induction on n that aset { a i , b i , c i } ni =1 of admissible numbers for ( A, B ) exists for each n ≥ . More specifically it is arguedthat, for any such n , some set of n − parameters { a i , a n − , b i , b n − , c i } n − i =1 can always be chosenso that, in virtue of (2.5)–(2.7) and (2.11)–(2.14), the resulting set { a i , b i , c i } ni =1 be admissible for ( A, B ) .Leaving aside some minor and relatively harmless mathematical glitches, the main problem withthis part of the proof is logical. Hofer-Szabó and Rédei’s argument in fact rests on the presuppositionthat if numbers { a i , b i , c i } nn =1 meet the conditions (2.1)–(2.7) defining an admissible set for ( A, B ) , then a partition satisfying identifications (2.8)–(2.10) will thereby qualify as a RCCS for the same pair(i.e., will satisfy (1.7)–(1.9)): quite evidently, if that were not the case, demonstrating the existenceof admissible numbers could not prove anything about the existence of RCCS s. Hofer-Szabó andRédei even make this assumption explicit when declaring that ‘admissible numbers have been chosenprecisely so that [(1.7)–(1.9)] are satisfied’ [2, p. 755]. Put in other words, they submit that given(2.8)–(2.10), conditions (2.1)–(2.7) jointly imply (1.7)–(1.9). Nevertheless, and here lies the logicalerror with their proof, this is demonstrably false.To verify this, let us hereafter grant (2.8)–(2.10). This stipulation immediately turns (2.6) intoa direct consequence of the axioms of probability theory, whereas (2.4) essentially demands that { C i } ni =1 be a partition of the assumed probability space, as indeed it is required from a RCCS .Having established this, (2.1) and (2.2) are then easily obtained from the theorem of total probability.This shows that all of the aforesaid conditions are actually independent from (1.7)–(1.9), so the realjob must be done by the remaining three equalities. Now (2.7) does effectively imply (1.7), while(2.5) is openly equivalent to (1.9). By exclusion, it follows that (2.1)–(2.7) jointly imply (1.7)–(1.9),as Hofer-Szabó and Rédei’s proof requires, if and only if (2.3) logically implies (1.8). However, this isnot so.To illustrate, let
X, Y ∈ Ω be arbitrarily chosen from (Ω , p ) , and let { Z i } ni =1 ⊆ Ω be a partition ofthat space. The theorem of total probability then produces the following, general formula: p ( X ∧ Y ) = n X i = i p ( X | Z i ) p ( Y | Z i ) p ( Z i ) + n X i = i p ( Z i ) [ p ( X ∧ Y | Z i ) − p ( X | Z i ) p ( Y | Z i )] . (2.18)Now let A = X , B = Y and Z i = C i for i = 1 , ..., n . To get (2.3) from (2.18), it is then necessary andsufficient that: n X i =1 p ( Z i ) [ p ( X ∧ Y | Z i ) − p ( X | Z i ) p ( Y | Z i )] = 0 . (2.19)Now, if (1.8) is true, then surely (2.19) is the case and (2.3) follows as a result. Therefore, (1.8)is clearly a sufficient condition for (2.3), while conversely (2.3) is certainly a necessary condition for(1.8). Still, it is equally clear that (1.8) being the case is not the sole circumstance whereby (2.3) canbe so obtained. Because each term of the form p ( X ∧ Y | Z i ) − p ( X | Z i ) p ( Y | Z i ) can perfectly wellbe positive or negative, (2.19) can in fact be satisfied even in case not all such terms are equal to zero. his is easily checked by putting, for instance, p ( C ) = 12 p ( C ) = 12 p ( A | C ) = 14 p ( A | C ) = 18 p ( B | C ) = 13 p ( B | C ) = 16 p ( A ∧ B | C ) = 124 p ( A ∧ B | C ) = 116 where n = 2 ; (2.19) is satisfied but (1.8) is violated.This proves that (1.8) is not a necessary condition for (2.3), so the latter does not logically imply theformer. Contrary to what Hofer-Szabó and Rédei declare, admissible numbers have not been chosenso that (1.7)–(1.9) are satisfied; so proving the existence of admissible numbers cannot in any waycontribute to demonstrating the existence of RCCS s. Hofer-Szabó and Rédei’s proof is consequentlyinvalid.Before proceeding further, it will be worth stopping briefly to consider a possible rejoinder. Lookingback at Hofer-Szabó and Rédei’s proof, a chain of two subsequent inferences can be seen at work. First,(2.1)–(2.4) are inferred from (2.5)–(2.7), (2.11)–(2.14); and subsequently, (1.7)–(1.9) are allegedlyinferred from (2.1)–(2.7). We have shown that the latter inferences in the chain are invalid, but whatif (1.7)–(1.9) could be directly obtained from (2.5)–(2.7), without the mediation of (2.1)–(2.4)? If thatwere the case, then the crux of Hofer-Szabó and Rédei’s proof could be salvaged after all.A few elementary calculations would show that, in effect, (2.12), (2.13) and (2.14) are respectivelyequivalent to (2.4), (2.1) and (2.2). Our problem therefore reduces to establishing whether (2.11) doesreally imply (1.8). On the face of it, this is not quite clear. Hofer-Szabó and Rédei seem to treat(2.11) as equivalent to (2.3), but this might be only because they think, erroneously, that the latterbe equivalent to (1.8). Indeed, they do not directly obtain (2.11) from (2.3), but they derive it insteadfrom this other formula: p ( X ∧ Y ) − p ( X ) p ( Y ) = 12 n X i,j =1 p ( Z i ) p ( Z j ) [ p ( X | Z i ) − p ( X | Z j )][ p ( Y | Z i ) − p ( Y | Z j )] . (2.20)This equation can be easily demonstrated to hold for every pair of random events X and Y in aclassical probability space and every partition { Z i } in =1 of that space for which (1.8) is true. Thus, ifHofer-Szabó and Rédei’s calculations are correct, (1.8) is at least a sufficient condition for (2.11). Butdoes (2.20) show that (2.11) in turn implies (1.8)?To answer this question we need to look at an even further formula, namely: p ( X ∧ Y ) − p ( X ) p ( Y ) = 12 n X i,j =1 p ( Z i ) p ( Z j ) [ p ( X | Z i ) − p ( X | Z j )][ p ( Y | Z i ) − p ( Y | Z j )] +12 n X i =1 p ( Z i ) [ p ( X ∧ Y | Z i ) − p ( X | C i ) p ( Y | C i )] . (2.21)This equality follows from the theorem of total probability alone, and it is true of all X and Y andall partitions { Z i } in =1 of a classical probability space. Formula (2.20) is clearly a special case of this Notice that the logical direction of the proof is opposite to the direction in which it is presented. ore general equality, and it obtains, once again, if and only if: n X i =1 p ( Z i ) [ p ( X ∧ Y | Z i ) − p ( X | Z i ) p ( Y | Z i )] = 0 . (2.19)On the other hand, it should be clear by now that (1.8) is merely a sufficient condition for (2.19).Put in other words, it is possible that (2.19), and hence (2.20), be satisfied without (1.8) being true.Consequently, it is possible that (2.11) be satisfied whilst (1.8) is not, so the rejoinder presently underexamination would be unsuccessful. Unfortunately, Hofer-Szabó and Rédei’s proof cannot be salvagedin this way. Our proof, as it will become apparent, will closely follow the structure of Hofer-Szabó and Rédei’sone, and all differences will ultimately depend on the different form of the mathematical constraintsinvolved. For this reason, it will be helpful to articulate it into four sub-steps, each one correspondingto one step from the original proof.
Step 1a* : The error with Hofer-Szabó and Rédei’s proof, as we saw, rests in their definition ofadmissible numbers. More specifically, it lies in the fact that (2.3) fails to fully capture condition(1.8), for which it is a necessary but not sufficient condition. The first step in rectifying Hofer-Szabóand Rédei’s proof, therefore, will consist in modifying their characterization of admissible numbers.Let us accordingly lay down the following definition:
Definition 3.
Let (Ω , p ) be a classical probability space with σ -algebra of random events Ω and prob-ability measure p . For any A, B ∈ Ω satisfying (1.6) and any n ≥ , the set { a i , b i , c i , d i } ni =1 is called admissible* for ( A, B ) if and only if the following conditions hold: p ( A ) = n X i =1 a i c i (3.1) p ( B ) = n X i =1 b i c i (3.2) n X i =1 c i (3.3) d i − a i b i ( i = 1 , ..., n ) (3.4) < [ a i − a j ][ b i − b j ] (1 , ..., n = i = j = 1 , ..., n ) (3.5) < a i , b i , d i < i = 1 , ..., n ) (3.6) < c i < i = 1 , ..., n ) . (3.7)Notice the above definition is largely similar to the one informally proposed by Hofer-Szabó andRédei, save for the fact that (2.3) is replaced by the stronger condition (3.4), and for the fact thatadmissible* sets contain n more numbers { d i } ni =1 than admissible sets, for each finite value of n . Thelatter change was indeed made necessary in order to include (3.4).Before we proceed any further, we ought to be sure that the above definition does not suffer from he same shortcomings as the one that it is intended to replace. This is established through thefollowing lemma – proof of which is elementary, and which has been consequently omitted: Lemma 1.
Let (Ω , p ) be a classical probability space with σ -algebra of random events Ω and probabilitymeasure p . For any A, B ∈ Ω satisfying (1.6) and any { C i } ni =1 ⊆ Ω where n ≥ , the set { C i } ni =1 isa RCCS of size n for ( A, B ) if and only if there exists a set { a i , b i , c i , d i } ni =1 of admissible* numbersfor ( A, B ) such that: p ( C i ) = c i ( i = 1 , ..., n ) (3.8) p ( A | C i ) = a i ( i = 1 , ..., n ) (3.9) p ( B | C i ) = b i ( i = 1 , ..., n ) (3.10) p ( A ∧ B | C i ) = d i ( i = 1 , ..., n ) . (3.11) Step 1b* : Now that Definition 3 is securely in place, we can continue with our demonstration. Fol-lowing Hofer-Szabó and Rédei, our next step will be to restrict the number of independent parametersneeded to identify an admissible* set. This will be achieved by means of our second lemma:
Lemma 2.
Let (Ω , p ) be a classical probability space with σ -algebra of random events Ω and probabilitymeasure p . Let A, B ∈ Ω satisfy (1.6), let set { C i } ni =1 ⊆ Ω with n ≥ , and let { a i , b i , c i , d i } ni =1 bedefined so as to meet conditions (3.8)–(3.11). Finally, let (3.5)–(3.7) obtain. Then, { a i , b i , c i , d i } ni =1 is admissible* for ( A, B ) if and only if : a n = a − n − X k =1 c k a k − n − X k =1 c k (3.12) b n = b − n − X k =1 c k b k − n − X k =1 c k (3.13) c n = 1 − n − X k =1 c k (3.14) d n = " a − n − X k =1 a k c k b − n − X k =1 b k c k − n − X k =1 c k (3.15) d k − a k b k ( k = 1 , ..., n − (3.16) where a := p ( A ) (3.17) b := p ( B ) . (3.18) Proof.
Let (Ω , p ) be a classical probability space with σ -algebra of random events Ω and probabilitymeasure p . Moreover, let A, B ∈ Ω be positively correlated and let the set { a i , b i , c i , d i } ni =1 of n ≥ eal numbers satisfy conditions (3.8)–(3.11) and (3.5)–(3.7). Finally, let identities (3.17) and (3.18)be in place.To begin with let us observe that, owing to the assumptions just stated and owing to the theoremof total probability, (3.1)–(3.3) are in fact equivalent to (3.12)–(3.14). Our proof thus reduces toshowing that (3.4) obtains if and only if (3.15)–(3.16) do. This will be done with the help of thefollowing formula d n = [ d n − a n b n ] + " a − n − X k = i a k c k b − n − X k =1 b − b k − n − X k =1 c k , (3.19)which, it must be noticed, can be obtained from the theorem of total probability alone. This is to saythat (3.19) holds quite generally, and that it does not depend on any specific assumptions other thanthe obvious request that " − n − X k =1 c k = 0 . (3.20)Clearly, in our case this is guaranteed by (3.7) and (3.3).Thanks to the above equality, proving necessity becomes elementary. If (3.4) holds, then (3.16)follows a fortiori, whereas to obtain (3.15) we only need to observe that (3.4) makes the first addendumin (3.19) equal to zero. Let us now turn to sufficiency, so let (3.15) and (3.16) be in place. Once again,the first addendum in (3.19) becomes equal to zero, this time thanks to (3.15). This result, along with(3.16), suffices to establish (3.4). Step 1c* : Just like conditions (2.11)–(2.14) did for admissible sets in the original proof, equations(3.12)–(3.16) effectively reduce the number of independent parameters whose existence is to be provedin order to establish the existence of admissible* sets of arbitrary finite cardinality. Let us observe,in particular, that (3.16) is in reality a system of n − equations, namely one for each value of i = 1 , ..., n − . Therefore, (3.12)–(3.16) jointly comprise n −
1) = n + 3 equations in n variables. This means that, for each n ≥ , each admissible* set for ( A, B ) is determined by a set of n − ( n + 3) = 3 n − parameters, whose existence we are now going to establish by induction. Lemma 3.
Let (Ω , p ) be a classical probability space with σ -algebra of random events Ω and probabilitymeasure p . For any A, B ∈ Ω satisfying (1.6), a set { a i , b i , c i , d i } ni =1 of admissible* numbers for ( A, B ) exists for each n ≥ .Proof. Let (Ω , p ) be a classical probability space with σ -algebra of random events Ω and probabilitymeasure p . Moreover, let A, B ∈ Ω satisfy (1.6), and let us further assume, for ease of expression,identities (3.17)–(3.18). Proof will proceed by induction on n .First, let n = 2 be our inductive basis: in this case, equations (3.12)–(3.16) reduce to a = a − c a − c (3.21) b = b − c b − c (3.22) c = 1 − c (3.23) d = [ a − a c ][ b − b c ][1 − c ] (3.24) d = a b . (3.25) ecause a and b and are given, choosing numbers c , a and b will therefore suffice to fix the values ofall n = 8 variables in the system. Now let us observe that, as a direct consequence of (1.6), numbers a and b must lie strictly between one and zero: > a, b > . (3.26)Since we are interested simply in the existence of some admissible* set, we are free to assume that c → (3.27) > a > a > (3.28) > b > b > (3.29)in order to ensure that < a , a , b , b , d , d < (3.30) < c , c < (3.31) < [ a − a ][ b − b ] , (3.32)which is straightforward to check. The set { a i , b i , c i , d i } i =1 so defined accordingly satisfies (3.5)–(3.7),in addition to (3.12)–(3.16). By Lemma 2, it is therefore admissible* for ( A, B ) .Next, let n = m > and let { a i , b i , c i , d i } mi =1 be admissible* for ( A, B ) as our inductive hypothesis.To prove that an admissible* set for ( A, B ) exists if n = m + 1 , let us consider the following subset ofthe admissible* set for n = m { a j , b j , c j , d j } m − j =1 ⊂ { a i , b i , c i , d i } mi =1 , which we take by assumption to constitute a set of m − parameters. Let us then choose numbers a ′ m , b ′ m , c ′ m such that a j > a ′ m > j = 1 , ..., m − (3.33) b j > b ′ m > j = 1 , ..., m − (3.34) c j > c ′ m > j = 1 , ..., m − . (3.35)Given (3.12)–(3.16), the set { a j , b j , c j , d j } m − j =1 ∪ (cid:8) a ′ m , b ′ m , c ′ m (cid:9) of m −
1) + 3 = 3( m + 1) − n − parameters will then suffice to determine m + 1) numbers: (cid:8) a j , b j , c j , d j , a ′ m , b ′ m , c ′ m , d ′ m , a m +1 , b m +1 , c m +1 , d m +1 (cid:9) m − j =1 . Because conditions (3.5)–(3.7) hold by assumption for numbers { a j , b j , c j , d i , a ′ m , b ′ m , c ′ m , } m − j =1 , all weneed to show is that said constraints be also satisfied by { a m +1 , b m +1 , c m +1 , d ′ m , d m +1 } . To thispurpose, let us first notice that (3.6) must be true of d m by virtue of (3.16) and (3.33)–(3.34). Next,thanks to (3.12)–(3.15), it will be sufficient to suppose that c ′ m → (3.36) o obtain a m +1 → a m (3.37) b m +1 → b m (3.38) c m +1 → c m (3.39) d m +1 → d m , (3.40)which we already know, by our inductive hypothesis, to satisfy (3.5)–(3.7). Due once again to Lemma2, the set of m + 1) numbers so determined is therefore admissible* for ( A, B ) . This completes ourinductive proof. Step 2* : Now that we are in possession of n admissible* numbers for each arbitrary integer valueof n ≥ and every correlated pair ( A, B ) , the remainder of our proof will simply mimic the logicalmachinery set up by Hofer-Szabó and Rédei in the latter part of their proof. That is, admissible*numbers will hereafter accomplish the task which admissible numbers were supposed to perform inthe original proof. To establish the existence of RCCSs of arbitrary finite size, we accordingly needonly one more definition: Definition 4.
Let (Ω , p ) and (Ω ′ , p ′ ) be classical probability spaces with σ -algebras of random events Ω and Ω ′ and with probability measures p and p ′ , respectively. Then (Ω ′ , p ′ ) is called an extension of (Ω , p ) if and only if there exists an injective lattice homomorphism h : Ω → Ω ′ , preserving comple-mentation, such that p ′ ( h ( X )) = p ( X ) for all X ∈ Ω . (3.41)Thanks to this, we can now finally state the result we have been after: Proposition 2.
Let (Ω , p ) be a classical probability space with σ -algebra of random events Ω andprobability measure p . For any A, B ∈ Ω satisfying (1.6) and any n ≥ , there is some extension (Ω ′ , p ′ ) of (Ω , p ) whereby a RCCS of size n exists for ( A, B ) .Proof. The proof here is in all respects similar to Step 2 in [2], the sole difference being that sets ofadmissible numbers should be replaced with sets of admissible* ones. This, in particular, will requirereplacing Hofer-Szabó and Rédei’s equations (60)–(63) with: r i = c i d i p ( A ∧ B ) (3.42) r i = c i a i − c i d i p (cid:0) A ∧ B (cid:1) (3.43) r i = c i b i − c i d i p (cid:0) A ∧ B (cid:1) (3.44) r i = c i − c i a i − c i b i + c i d i p (cid:0) A ∧ B (cid:1) . (3.45)Owing to (3.4), however, Hofer-Szabó and Rédei’s equations can be immediately recovered.Remarkably, Proposition 2 applies to every pair of positively correlated events, be they strictlycorrelated or not. The analogous result announced by Hofer-Szabó and Rédei, on the contrary, con-cerned non-strictly correlated pairs only. The proof presented in this section, therefore, improves ontheir demonstration in two ways: it firstly rectifies the proof, but it also generalises the proof. Conclusion
Reichenbachian common cause systems have been developed as a generalisation of the conjunctivefork model, to account for cases whereby the observed correlation between two causally independentevents cannot be explained through the action of a single common cause. The existence of such systemsin suitable extensions of the assumed probability space was allegedly demonstrated by Hofer-Szabóand Rédei. This paper has shown that their proof is logically deficient: the admissibility condition(2.3) is necessary, but not sufficient, for the screening-off condition (1.8). Accordingly, we proposean alternative admissibility condition (3.4), which provides a more straightforward representationof the screening-off condition, and we demonstrate that the resulting set of admissibility conditionsovercomes their logical error.
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