Resolution of Simpson's paradox via the common cause principle
Authors: A. Hovhannisyan
Armen Allahverdyan
Venue: NeurIPS 2025 poster ()
Keywords: probabilistic inference
decision making
Simpson's paradox
causality
common cause
Primary area: probabilistic_methods
TL;DR: The principle of the common cause reformulates Simpson's paradox, and efficiently resolves it when the cause is minimal.

Abstract: Simpson's paradox poses a challenge in probabilistic inference and decision-making. Our study revisits the paradox by re-estimating its frequency with an unbiased data generation process and reaffirms that it is not an artifact of deficient data collection. Thus, it can lead to incorrect recommendations in fields as diverse as statistics, psychology, and artificial intelligence. We show that the paradox can be resolved by assuming a minimal — though not necessarily observed — common cause (or screening) variable for the involved random variables. In our approach, conditioning on this minimal common cause establishes the correct association between events, which coincides with the conditioning (i.e., fine-grained) option of the original Simpson paradox. This resolution applies to both discrete cases of binary variables and continuous settings modeled by Gaussian variables. For a non-minimal common cause, the resolution of the paradox is possible, but detailed knowledge of the common cause is required. Our findings extend traditional understandings of the paradox and offer practical guidance for resolving apparent contradictions in probabilistic inference, ultimately enhancing decision-making processes. This point is illustrated by several examples.

Resolution of Simpson's paradox via the common cause principle
Arshak Hovhannisyan
Alikhanyan National Laboratory, Yerevan, Armenia
Armen Allahverdyan Alikhanyan National Laboratory, Yerevan, Armenia
Abstract
Simpson's paradox poses a challenge probabilistic inference and decision- making. Our study revisits the paradox by re-estimating its frequency with an unbiased data generation process and reaffirms that it is not an artifact of deficient data collection. Thus, it can lead to incorrect recommendations in fields diverse as statistics, psychology, and artificial intelligence. We show that the paradox can be resolved by assuming minimal though not necessarily observed common cause (or screening) variable for the involved random variables. In our approach, conditioning this minimal common cause establishes the correct association between events, which coincides with the conditioning (i.e., fine-grained) option of the original Simpson paradox. This resolution applies to both discrete cases of binary variables and continuous settings modeled by Gaussian variables. For non-minimal common cause, the resolution of the paradox is possible, but de- tailed knowledge of the common cause required. Our findings extend traditional understandings of the paradox and offer practical guidance for resolving apparent contradictions in probabilistic inference, ultimately enhancing decision-man -maiing processes. This point is illustrated by several examples.
Introduction
Simpson's 's paradox was discovered more than a century ago 1 generated a vast literature, and is well-recognized in several fields including, statistics, epidemiology, psychology, social science, [4). This counter-intuitive effect limits the ability to draw conclusions from probabilistic data. The effect is important because it demands more than simply extracting relative frequencies from
data; e.g. it necessitates looking at exchangeability 9 or causality 77 13 14].
The paradox starts with two random variables A and B. Now A = (A1, A2) contains control variable Ag and the target variable Aj. while B is a side random variable that depends on both A₁ and A2- The meaning of A and B is clarified via examples presented below. If there is no information on the outcome of B, the behavior of can be studied on two levels. The first (aggregated) level is that of marginal probabilities p( A a). The second level is finer-grained and is represented by conditional probabilities p(A = al B = b) for all possible values B. Simpson's 's paradox amounts to certain relations between those probabilities; possible, because conclusions drawn from probabilities on different levels contradict each other. Without Simpson's 's paradox, decision-make can proceed at the aggregate level, because looking the fine-grained level is either redundant or inconclusive. Thus, Simpson's 's paradox first and foremost
section 2 for details. It states that decision-making
arshaa howhaankooohanniyynnn aanl.am am allaaherdyyn @ aanl.am
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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involves sxcision-making Moreover, it demonstrates limitations of the sure-thing principle [5]. pillar of traditional decision making 27 A recent review of the sure-thing principle (and its limitations other than Simpson's paradox) can be found in Ref. (28). Limitations of probabilistic decision-making are important for the modern artificial intelligence (probability models, uncertainty estimation, etc).
In section 2 Simpson's paradox is defined in detail, and previous efforts to resolve it in several specific situations are reviewed and criticized. In particular, we show that while certain previous solutions of the paradox assumed the existence of ccausally-sufffcient time-ordered directed acyelic graphs (TODAGs) that describe the 3 variables involved in the paradox, several important examples of the paradox need not support this assumption; see sections 2.2.2 and [ Based on the previous literature, we argue that Simpson's 's paradox is sufficiently frequent when the probabilities of the involved variables are generated from the unbiased (non-informative) distribution, modeled via Dirichlet density; see Appendix [A] Hence this is a genuine decision-making paradox and not an artifact due to inappropriate data gathering.
Our proposal here is to search for the resolution of the paradox by assuming given two dependent random variables A and B there is a random variable C. that makes A and B conditionally independent; i.e., screens out A from B. Examples of Simpson's 's paradox show that such a C' is frequently plausible, though it is normally not observed directly. In particular. C is conceivable the dependence between A and B are not caused by a direct causal influence of B on A. Then the existence of C' postulated by the common cause principle. (If the dependence is due to a causal influence of A on B, Simpson's 's paradox can formally exist, but factually it is absent because the decision is obviously to be taken according to the aggregated level.)
Introducing the screening variable C' allows us to reformulate and extend Simpson's 'ss paradox: its Iwo options- along with many other options refer particular choices of C; see section 3 Now, the paradox seems to be further from being resolved than before. However, we show that when the variables A₁, Ag. B, and C holding the paradox are binary (the minimal set-up of the paradox), the decision-making is to be made according to the fine- mee-grained probabilities. i.e., the paradox is resolved. Such definite relation is impossible for a tertiary (or larger) C: now depending on C all options of Simpson's 's paradox are possible, c.g. the precise control of C can be necessary for decision eecision-making.
Next, turn to Simpson's 's paradox for continuous variables, which was discussed earlier than the discrete formulation []. It holds the main message of the discrete formulation. In addition, it includes the concept of the conditional correlation coefficient (only for Gaussian variables is the random- variable dependence fully explained by the correlation coefficient). The continuous formulation is important because it applies to big data [3 24 29] and because (statistically) it is more frequent than the discrete version (30]. The advantage of continuous Gaussian formulation is that the general description of the paradox under the common cause is feasible; see section 6 For this situation, we show conceptually the same result as for the discrete version: in the minimal (and most widespread) version of the paradox, the very existence of an (unobservable) common cause leads to preferring the fine-grained option of the paradox.
The rest of this paper is organized as follows. Section 2 is short but sufficiently inclusive review of Simpson's 's paradox and its resolutions proposed in the literature 3 It also discusses two basic examples for illustrating different aspects of the paradox; see section 2.2.2 In section reformulate Simpson's 's paradox by assuming that there is a common cause (or screening variable) C' behind the three variables. Now C. need not be observable, since we show that it will be sufficient assume that it exists and (provided that all variables are binary) Simpson's 's paradox is resolved by choosing its fine-grained option. A similar conclusion is reached for Gaussian variables; see section 6 Section 4 considers published data from Ref. [16] on a case of smoking and surviving. This example is not easily treated via the existing methods. Still, we show that the existence of a common cause for this situation is plausible and that Simpson's 's paradox can be studied via our method and leads to a reasonable result. Section 5 treats data on COVID-19, which was suggested in Ref. (3T). We demonstrate that an assumption of plausible common cause points to different conclusions than in Ref. (3I). The last section summarizes our results and their limitations. It also outlines future research directions.
3 Among the issues not addressed in this paper is the explanation of Simpson's 's paradox using counterfactual random variables. This subject is reviewed in 66.
2

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2 Formulation of Simpson's paradox and previous works
2.1 Formulation of the paradox for binary variables and its necessary conditions
To formulate the paradox in its simplest form, assume three binary random variables A = {a ã1 h. A2 {az, )2). , {6, b}. The target event is d,, and we would like to know how it influenced by A2 which occurs at an earlier time than the time of A1: L2 S EAA- This can be done by looking conditional probability. For
p(an (a)) < ((2))
which is equivalent to p(a1) < p(a])ā2), we would conclude that @g enables a1. However, I1 is compatible with
Mailaz, b)
p(ay (äz, b), p(an (ā2, b),
p(alla2. b)
where also occured in an earlier time: tg < tA Examples supporting T are studied below
(sections 2.2.2 4 and (5) and also Appendix C Since 2 hold for each value of B we should
perhaps conclude that dg enables a₁ in contrast to (T). Decision-maker would not know whether to apply (
This is Simpson's paradox. Its equivalent formulation is when all inequalities in
1
are inverted
For Simpson's is paradox
B to hold, is necessary to have one of the following two conditions:
p(an 02, b) < p(a1 02 b) < p(an 22, b) < p(01 (22 b):
(4)
p(a1 02 b) < p2 b) < |2, b) < p(a1 a22 b).
(5)
To find these relations, expand p(a (a2) and p(a (ي2) over the probabilities in [cf. and note that e.g. p(a1la2) is weighted mean of p(a1la2, b) and p(a: 02 b). Given
[6
57)
Simpson's s paradox can be generated via suitable choices of p(bla2) and p(b(ā2); see Appendix
2.2
Attempts resolve the paradox
2.2.1
Replacing prediction with retrodiction
Over time, several resolutions to the paradox have been proposed. Barigelli and Scozzafava 110 11 proposed to replace [ by
p(a2)p(a (a2) < p(az)p(arl@a)
(6)
ie. to interchange A₁ and As in (I. Then it is casy to see that its inversion under additional conditioning over B is impossible. While (T) stands for prediction - i.c. aiming at a2 (and not at ā₂) will more likely produce ã (than a1) the proposal by Ref. [10 11] looks for retrodiction. Though retrodicting (in contrast to predicting) does not suffer from Simpson's 's paradox, retrodicting and predicting different things, and cannot generally be substituted for each other.
Rudas also sought to change the criterion (T) so that does not allow inversion after additional conditioning over B, but still has several reasonable features [32] The proposal is to employ ((22)(((a (a2) - p(ā (az)] < p(ā²) ( [p(a [gg) p-pppjj_a_p [ instead of
Notice the conceptual
relation of this with the previous proposal (6)
An unnatural point of both these proposals is that they depend on the ratio p(a2) e.g. for the Example 1 mentioned below this means that if the treatment was applied more, it has better chances to be accepted. This drawback acknowledged in [32]
2.2.2 Exchangeability and causality
According to Lindley and Novick, the paradox may be resolved by going beyond probabilistic consid- erations (as we do below well) and by employing the notion of exchangeability or causality (9; see
*We leave aside the following pertinent problem; see [19] for details. If probabilities are extracted from finite populations, the more conditioned version 2 is less reliable. because it is extracted from smaller population. For all probability provbiiity-providing populations will be sufficiently large.

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33] for recent discussion on causality and exchangeability. Within that proposal, the data generally provides only propensities, and one needs additional assumptions of sample homogeneity (exchange- ability) for equating propensities with probabilities even for large sample size. Exchangeability and the closely related notion of ergodicity remain influential in the current analysis of statistical problems exemplified by Simpson's 's paradox [34]. Lindley and Novick studied the following two examples that support Simpson's 's paradox (more examples are discussed in sections 4 |
and Appendix C
Example 1. Medical treatment [9] A {a1, } (the target variable) is the recovery rate of medical patients: d1 = recovery, ā1 = no recovery. AA {az, ā2) refers to a specific medical treatment: d2 = treatment, 2₂ = no treatment. B = {b. b} is the sex of patients: b = male, b = female. The times which the random variables A1, Ag and B refer clearly hold tB t,a2 < tt,-
Example 2. Plant yield [9] A₁ = { 01 (the target variable) is the yield of single plant: a1 high, @i low. A2 = {az, 22) } refers to the variety (color) of the plant: 0g dark,
02 = light. B = (b, b) refers to the height of the plant: b = tall, = low. The times hold
A, < <tg- < tAT
Lindley and Novick proposed that assumptions on exchangeability lead to preferring I for Example 2 and 2 3 for Example 1 (9]. They also proposed that the same results can be Tound by using causality instead of exchangeability [9]. The same proposal was made earlier by Cartwright in the context of abstract causality (7 8). Pearl elaborated this proposal assuming that the above examples can be represented via time-ordered direct acyclic graphs (TODAG) [13 14), where an arrow > represents the influence of earlier variable to the later one; Fig. for details. If we follow this assumption, then given the time constraints for the examples each of them can be related to a unique TODAG:
Example 1: B + A + A + B,
(7)
Example 2: A2 + B - A1 t A2
(8)
In 7 the suggestion is to condition over B [hence using 2 3 if B influences both A₁ and Ae 9
13
This is because conditioning over the cause reduces spurious dependencies. This reasoning was generalized as the back-door criterion [13]. In contrast, it is advised to use ( in (8) since B
an effect of A2, but still cause of A (9 13 [4]. The intuition of this suggestion is seen in
the extreme case when B screens A, and A, from each other, i.e. A, B and A2 form Markov chain. Then the conditional probability p(A1|A,, B) = p( ((Ai will not depend on Ag begging the
original question in IT. Thus, for the two examples considered in (9) Refs. [13 14 recommendations. The basis of these recommendations was criticized in
make similar
Refs. 13 14]
imply that Simpson's paradox for 7 g can be solved via do-caleulus. This is
only partially correct: only (7) is solved with do-calculus. Indeed, the do-calculus in (7) defines p(a, (do(A))
Σg p(b)p(a) AA2 b). Now p(a, [do(A2 = az)) > p(a, [do(A, a2)). amounts to
the fine-gran geeinrri version 3 of the paradox, which agrees with the conclusion of (9. However, for (8)
we have p(a [do (A2)) = p(a1 (A2), and is not clear what prevents us from going back to the original
formulation of Simpson's paradox, i.e., comparing p(a (A, = a2)) < p(a1 (on]d Ag = az)), with
p(ai |do A2 = u2), b) > [/ai/d((As (2 ā2), b) and p(an |do(A2 = a2), b) > p(ai do a2). b).
Let us now argue that realistically, Example and Example 2 need not support TODAGs 7 respectively. In fact, both arrows B + Ag and B - A₁ in Example I are generally questionable: sex need not influence the selection of the treatment, B A A2 (unless the data was collected in that specific way), and many treatments are sex-i rex-indifferent, i.c. B A Ai. For Example it is more natural to assume that B does not causally influence A. In such situation, the common cause principle proposes that there is an unobserved random variable C, which is common cause for A and B [35 36]; see section 3 Similar reservations apply to Example 2: now Ag - B is perhaps argued on the basis of color (A²) being more directly related to the genotype of the plant, while the height (B) is a phenotypical feature. First, color-genotype and
relations need not hold for all plants Second (and more importantly), it is more natural to assume that the plant genotype influences both its color and height than that the color influences height. Hence the genotype can be a common cause for A and B. Implications of such common cause scenarios are studied below.

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B
A2
B
A
Figure Directed acyclic graphs between random variables in discussing Simpson's paradox. The first and second graphs were studied in Refs. [13 14] see B). The third or fourth graphs are basic assumptions of this work; see (.. In the first graph, B influences A and A2, but B is not the common cause in the strict sense, because there is an influence from A2 to A1. A similar interpretation applies to the second graph. We emphasize that the joint probability p( A1, A2, B) for the first and second graphs has the same form, i.e. such graphs are extra constructions employed for interpretation of data. In contrast, the third and fourth graph imply a definite (but the same for both graphs) limitation on the joint probability p(A1, A2, B, C), which is expressed by
A2) B and C innolved
3
Common cause principle and reformulation of Simpson's paradox
3.1
Common cause and screening
The common cause for A = (A1; A2) and B means that there exists random variable C'
{cc] 35
36
p(A1, A2, B|C) = 1 p(c)p((A,,
(9)
p(A1, A2, B) = B)= P(c A2lc) p(c) > 0
(10)
where 9 holds for all values assumed by AI, A2, B and C, and where 10) follows from 9
5 The
same applies if C' causes A and screens A from B. These two scenarios are shown Fig.
as (resp.) the third and fourth graphs. Sections 4 S and Appendix provide several examples
C
of causing (or screening) variable C' in the context of Simpson's 's paradox. p(A1, A2, B) in (10
can be considered as a matrix, where Ag) [B] enumerates rows [columns]. Now there is
minimal value of |C| (the number of realizations of O)) such that (I0) holds (37]. This minimal
number called the positive rank (rank, (p)) of the matrix (1,, A2, B)
There are methods
for its estimation
[38] e.g., it holds rank [p] min |A1] |A2). B where [ is the number of
realization of B. Representations 10) are not unique, even for fixed |C| (38)
The common cause principle was proposed to explain probabilistic dependencies [35] 36). It later found important applications in data science, where approximate relations similar to (9) are applied to effective data compression Neg-negative matrix factorization, Probabilistic Latent Dirichltt indexing, etc) (39 40). Pearson and Yule expressed early ideas about the common causes that explain certain dependencies; see [41] for a historical review. Involving a common cause in Simpson's 's paradox means that we do not consider this paradox referring causally sufficient situation; for recent
discussions on causal (in (in)sufficiency see [42 43).
Note from that C' gets rid of the conditional dependence on B in A2 C).. Thus, sensible way of looking at the association between a₁ and a2 is to check the sign of
p(ay a2, C) p(a1 T22 C.) for each value of C.
(11)
To support the usage of the common cause C for ceciisionmmakinng we note that ( has an important
implication in the context of (T) (This implication generalizes the argument given in (36].) Assume that p(a2, b, c) > 0 for all values c of C. Note from (9) that there exists an event such that p(ai (a2, b) < p(ai la2, b, = p(a (a2, and an event c' such that p(ai (a2, b) > p(a (az,lal,,, b, e') p(ai (02, e¹). Hence, conditioning over facilitates (hinders) the association between ai and d2. then conditioning over e (c') ) is not worse in this facilitation (hindering)
'There are formulations of the common cause principle that look for holding for certain events only and not for random variables 35 36) We do not focus them.
'To deduce the first relation assume that p(a Jáz, b) < p(as d2, b, c) = p(as (az, c) for all c, multiply both parts by p(az, b, c) > 0, sum over and get contradiction p(as, az b) < p(as, 0z b). Likewise for the second relation.
5

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After the above reformulation, Simpson's 's paradox seems even less resolvable since C is not observed. Indeed, there are common causes that reproduce I those that reproduce 2 3). but there are many other possibilities. Common causes that are close to B (C ~ B) imply option while R leads to ( These conclusions are based on the fact that (9) holds exactly for B and C = A. Thus, Simpson's 's paradox is not a choice between two options
of the paradox,
and 1 it is
choice between many options given by different common causes C.
Finally, two remarks about the applicability of [ First, if C' is a common cause for both A = (A1, A2) and B, the times of these variables naturally hold to < min[t Ag. tg). When C screens A from B, it holds tB < tt < mmintt,, t In certain applications of [11. it will suffice to have even a weaker condition to tA
3 we do not need p( (2)) i.e. only p A2) is needed for connecting with 2 Indeed, A2 does not necessarily need be random variable, but can simply be a label describing the situation. Now the same holds for (IT): once (9) is written as
Second, we note that for applying 1 2
p(A,, B|A2, C) p(Ai|A,, C)C()
(12)
we need only p(C]A2) to pass from
12 to quantities involved in 2 3 i.e. p(A2) is not needed.
3.2 A common cause (or screening variable) resolves Simpson's 's paradox for binary variables
The following theorem shows a definite statement for all binary causes. The message of the theorem that once we know that C' is binary, then the correct decision is 2
Theorem 1: If A1, A2, B and C = fc. o} are binary, and provided that and are valid, all causes C hold
p(a1 c) > p(a1 (م2, c),
p(a1 c) > (0,
C)2
(13)
i.e all C' holding 9 predict the same sign of association between @1 and 02 as
This theorem
is proved in Appendix B The main idea is that to prove (13), we need to invert
The resolution of Simpson's paradox offered by Theorem is consistent with the do-calculus; see 13) for a review. To show this, consider from Fig. 1 two TODAGs that support the causal structure
of Thoremm 1:
B A2 + Ar, B B A2 AI,
C+ C+A.-
(14)
(15)
For both these TODAGs we have
p(ai/do(A2 a2)) Ec plai a2, C)p(C).
(16)
If (as stated in our theorem 1) p(ai la e) > p(az|āz, c) and pla1 é) > p(a1|02, 2) then we get قم2,
[do(A2 oz)) > p(a1 dd = a2)). Hence, if the influence of A on a1 is decided via
do-conditioning over A2, then the conclusion agrees with the option 2 3 of Simpson's paradox.
An important aspect of theorem 1, is that once we can motivate one of the above TODAGs (14 15)
with binary C, then additional data-gathering is necessary for resolving the paradox, i.e., do not need to know P(A1, A2| A2), etc.
Note the difference between TODAGs probability p(A1, A2, B). In contrast, 14 require a specific probabilistic feature
and
are consistent with any joint
3.3 Non-binary causes
Let us assume that we have Simpson's 's paradox I 2 3 and also the common cause condition 9
However, C (c1, C2, C2,C3) 03} is now a tertiary random variable. turns out that now all three options of Simpson's 's paradox become possible: there are common causes C' that support (
p(ai |a2, C) < p(ailaaa [ū2, C),, for C= (2,, ca).
(17)
There are also common causes C' which support 2 Eventually, there are tertiary common causes C (c1, 02 03) for which p(a, laz, c.) p(a: 02 (x) has different signs for different values of supported. Hence, already for the tertiary cause, one needs
i 1, 2, 3; i.c., neither (I) nor

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prior information on the common cause to decide on the solution of Simpson's 's paradox. Alternatively, we can infer this unknown cause via one of the methods proposed recently for obtaining the most plausible common cause [44 45 It is not excluded that such inference methods will provide further information on the solution of Simpson's paradox.
Note that (17) is counter-example to an opinion that the structure of the TODAG as such can determine which option ① or (2 of Simpson's paradox applies. Indeed, [17 and Theorem 1
support the same TODAGs (4, 15 but they lead to different options of the paradox.
4 Example: smoking and surviving
In section 2.2.2 we discussed two examples studied in the literature and argued that they be also interpreted via the common cause principle. In the present case, the standard approaches do not seem apply, but the common cause can still be motivated. This example on survival of smokers versus nonsmokers is taken from Ref. [16]. Its technical details are discussed in Appendix [ Binary A represents the survival in a group of women as determined by two surveys taken 20 years apart:
A {a1, a1 } {died, alive},
A2 = {az, ] = {smoker, nousmoker).
(18)
B {b, 6) {age 18 64, age 65 74),
(19)
where p(b) 0.1334, and where and 6 denote age-groups. According to the data of (16). Simpson's 's paradox reads [see Appendix D.I for several technical clarifications]:
p(a)) 0.2214 < p(a (2)) 0.2485,
(20)
plarla2, b) > p(a1 |22 b),
p(01la2, b) p(a1 و2 b).
(21)
Note that B here influences A₁: the age of a person is predictor of his/her survival. There are few people who quit or started smoking, so causal influences from B A2 can be ignored (16). We can assume that influences from smoking to age are absent. Then this example is intermediate between two situations considered in (7/9 13). Recall that when B influenced Ag, these references advised to decide via the fine- -ha-irdi option of the paradox, while for the case of the inverse influence (from Ag
to B) they recommend to employ the gaaniee version; see 7 g and Fig. [
Hence, we should expand on the above situation to achieve a workable model. We can assume that A2 and B are influenced by common cause. Genetic factors influence individual's 's age and tendency to smoke. Originally proposed by Fisher (46). this hypothesis was later substantiated in several studies; see Refs. [4 48 for reviews. Note that this refers to genetics of the smoking behavior itself, and not to health problems that can be caused by smoking plus genetic factors. Several sets of studies that contributed genetic determinants of smoking behavior are as follows. (i) Children of smoking parents tend to smoke. (ii) Smoking behavior of adopted kids correlates stronger with that of their biological parents. (iã) Monozygotic (genetically identical) twins correlate in their smoking behavior much stronger than heterozygotic twins. Smoking behavior includes both the acquisition and maintenance of smoking. Monozygotic twins show correlations in both these aspects.
As a preliminary hypothesis, we suggest that genetic factors are the common cause of both smoking and age. To apply Theorem we introduce genetic variable C =(risk smoking, no risk to smoking). TODAG (14) can describe our simplified model. Now we need to consider genes, which can have influence nicotine addiction and show evidence of pleiotropy. i.c. they can influence more than one aspect of health and survival [49]. CHRNAS is pleiotropic gene that encodes subunits of the nicotinic acetylcholine receptor, which is important in neural signaling and nicotine addiction. The receptor can influence various aspects of smoking behavior: nicotine binding and response, reward pathways, craving intensity, smoking cessation success rates, etc; see Ref. (50) for a review. CHRNAS has two alleles G (guanine) and A (adenine), which differ by single nucleotide. Now A is the risk allele, which is associated with increased smoking. G is the non-risk allele (50). A the dominant allele with respect to G, and we treat CHRNA5 as binary genetic variable; see Appendix D for clarifications. Thus, Theorem applies and we conclude consistently with other studies that smoking is not beneficial for survival.
7

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Example: COVID-19, Italy versus China
Here the COVID-19 death rates are compared in Italy and China [31 51]. According to the data, aggregated death rates in Italy are higher than in China, but in each age group, the death rates are higher in China, More precisely,
A1 {ai, ā1 = {died, alive),
A2 {az, a2) {China, Italy),
(22)
B = {b, b} = {age 60 - 79, age 80+},
(23)
where p(a1) is the death rate out of COVID-19, ( is found from the number of positively tested people in each age group, p(b) = 0.1012, and where p(b(a2) 0.1017 and p(blāz) 0.3141.
According to the data of 31J, Simpson's 's paradox reads
p(a1 (22) = 0.0608 < p(a1(ā2) = 0.0760,
(24)
p(an Jaz, b) = 0.0507 > p(a1 (22 b) 0.04900,
p(ai/a b) 0.150 p(ai (ā2. b) 0.135.
(25)
The authors of (31] proposed that this situation is described by TODAG A2 + A + A2; ef. (8). Then the conclusion from [9
[3] will be that the aggregated version of Simpson's paradox works, i.c. Italy did worse than China. The authors of Ref. [31] reached the same conclusion.
When applying the common cause set-up from section 3.1 we can look at 12 because A2 is better described as label (avoiding dealing with the probability of country). Hence, from the viewpoint of (12), we need a common cause that supplements A₂ and acts on both A₁ and B. We propose that the quality healthcare system can be the common cause C' here. In particular, a more affordable healthcare system may cause a higher proportion of older people in the country's 's society, Indeed, for 2019, Italy had a larger percentage of people aged above 65 than China: 24.05 versus 12.06 %. On the other hand, the healthcare system will influence death rates in all age groups. If C is binary, then our conclusion from Theorem is opposite to that of [3I] China did worse than Italy.
Simpson's paradox and common cause principle for Gaussian variables
6.1 Formulation of Simpson's paradox for continuous variables
Simpson's paradox is uncovered earlier for continuous variables than for the discrete case (T. Researching the continuous variable paradox and identifying it in big datasets is currently active
research field 23
24
29
52
54
The association between continuous variables A, = } and A₂ {a2} can be based on
reasonable definition of correlation coefficient [I 31) We focus on Gaussian variables, because this
definition is unique for them and amounts conditional variance. These variables are also important the context of machine learning (e.g. linear regressions) [55].
Hence the formulation of Simpson's 's paradox given B = {b} reads instead of I
23
24
30
a faa,, a2]aa [aja[ai,, a2[b] < for all b,
(26)
ala, 02] = ((a) (a1)) (a2))).
(27)
a (a1, 02 b) ((a) (a1))) (a2))))., (a)s / da a p(a(b), a
(28)
where (a)s and a o[a1, d2 [b] are the conditional mean and covariance; (a) and o[a1, az) are the mean and covariance; p(a(b) is the conditional probability density of A A= {a)
The message of (26) is that the usual and conditional covariance have different signs, i.e., they predict different types of associations between A₁ and Ag. For instance, a[a), u2] > 0 means correlation, while a (01, d2 (b] implies anti-correlation. Note a subtle difference between this formulation of Simpson's 's paradox and that presented in section 2.2 In (26 27p the formulation is symmetric with respect to Aj and A2.
8

---

6.2 General solution for Gaussian variables
For fuller generality, we shall assume that = {a}, and {b} are Gaussian column vectors with a number of components (i.e., dimensionality) nAA and ng. respectively. We also define
= (a, y y is a number, yy a matrix,
(29)
where ' means transposition. We assume that a Gaussian nx dimeensioon variable X= = { is the common cause variable for and B:
P Ø) = (2 n2/2 (det (det(2)) e
(30)
mx /2 ()) --1==s=*
P(x) a (2t)
(31)
A
(y) CCx,
(32)
0 B
(yy - (3) )=)) = Q
(33)
where the common cause feature of X = {w) is ensured by the block-diagonal structure of the covariance matrix Q: A and are (resp.) covariance matrices for and B. (30). (n + ng) x nx matrix that ensures the coupling between (A, B) and X. For simplicity and without loss of generality we assumed that (x) and hence (y) in (30). We get from (30) after arranging similar terms (and omitting normalization):
P(a:) P(ylar) (x) =1=T y "v--v '2l 1/ 1-2 -c lc
(34)
V -=
(35)
Employing (87 from Appendix F we obtain:
- -0o=-i"0--
(36)
P(y) x e 1v" (@+csc")-1)
(37)
(yy") Q + CSC
(38)
We now recall
29
33
introduce the block- diagonaa form for CSCT and find
-A+J KT
KC
Q + CSC
B+C
(39)
K(B + - :.)
(2 (C+
(40)
where (40) is deduced via formulas from Appendix In 40 need only the upper-left block, so
that all other blocks are omitted. Collecting pertinent expressions from (2 38 along with (38)
33). we obtain
(4y (y) (y )a)a = Q (a) - (a (aT))))) 1))b = A + x)
(41)
(42)
6.3
The minimal set-up of Simpson's 's paradox: 3 scalar variables + scalar cause
For this simplest situation, y (ai, 02 is 3-dimensional vector, A is a 2 2 matrix, C is 3 x matrix, while S and B are positive scalars. Now 38 read:
((1 (a1))) (o2)b)) A1s CnnCrse,
0 0< < B C2 11
(43)
((a: -
(a))a)) A12
(44)
(a,az) = A12 + Cn
(45)
Now consider scenario of Simpson's 's paradox, where
(a)a2) Ai2 + S21 > 0 and
(46)
((a) - (a1))) (o2)))) A + C
(47)

---

Due to 0 < < 1, these two inequalities demand A12 < 0. Likewise, (a1a2) A12 + C21 < 0 and A12 C1iC2i
0 demand A12 > 0. is seen that under Simpson's paradox for this minimal situation, the sign of {(a1 (a1) (a² {02)))))> coincides with the sign of ((a) (a1)))(a2 (a2)6))b. We are thus led to the following:
Theorem 2: In the minimal situation (53 with the (minimal) common cause, the continuous Simpson's 's paradox (26) is resolved in the sense that the decision on the sign of correlations should proceed according to the fine-grained option: ((a) - (a1))) (02)b)}a; see 27)
For non-minimal common causes, all possibilities of the paradox can be realized; see Appendix G
7 Conclusion
We addressed Simpson's 's paradox: the problem of setting up an association between two events aj a2 given the lurking variable B. This decision-making paradox provides two plausible but opposite suggestions for the same situation; see ① and (2 3). Either the first option is correct, the second
option correct, or none of them is correct.
We focus on cases when there is a common cause C' for B and A = A2) (which combines a1, 02 and their complements). A Alternatively, C screens out A from B; ef. Fig. 1. These cases include those in which there is no causal influence from A to B, as well as from B to A. Hence, the dependency between A and B are be explained via the common cause C, which is statement of the common cause principle G5| 36). Now the association between a and a2 is to be decided by looking at p(a, (a2, c) for various values of C. This task is normally difficult given the fact that C' is frequently not fully known and is not observed. However, provided that AI, A2. B and C' are binary, p(a1 [02, c) shows the same association as the option 2 3 of Simpson's paradox. In this sense, Simpson's 's paradox is resolved in the binary situation, provided that the situation allows a binary cause or a binary screening variable. The same conclusion on resolving Simpson's 's paradox was reached for Gaussian variables in the minimal situation. Several examples can illustrate the plausibility of a minimal C.
Our solution of Simpson's 's paradox is not a generalization of the existing solution, since it employs a different idea. As we argued in section 2.2.2] the only unambiguous solution proposed far refers to the directed acyclic graph 7 We also provided a counter-example against an opinion that the solution of Simpson's 's paradox can be decided based on the directed acyclic graph structure only.
We provide the first resolution of Simpson's 's paradox for Gaussian variables. This scenario of the paradox differs from the discrete in at least one essential aspect (it is symmetric), and was historically known carlier than the discrete version, and is more frequent in practice. This scenario of the paradox cannot be analyzed via standard directed acyclic graphs, and has to be worked out directly.
Our results have several limitations, but (we believe) these can be overcome with further research. (i) We limited ourselves to results that hold for all (minimal) common causes. For many applications, this is too stringent: if the common cause known to exist, but is not observed directly, then it may be sufficient to infer it e.g. via the (generalized) maximum likelihood (45) or the minimal entropy method [44). This may provide pertinent information on the real common cause and the structure of Simpson's 's paradox. (ii) We insisted on precise common cause. The screening relation (T)) is also useful, when does hold approximately, but the support of C is relatively small. Such an approximate relation (10) provides data compression via feature detection, which is the main message of unsupervised methods such as Non-negative Matrix factorization and Probabilistic Latent Dirichlet indexing (39 40). The impact of such approximate, but efficient causes on probabilistic reasoning is an interesting research subject that we plan to explore in the future. (ûi) All examples we presented observationally incomplete: plausible cause had to be inferred, but it was not shown to exist or function from the real data.
Acknowledgments and Disclosure of Funding
This work was supported by the HESC of Armenia under Grants 1F030, 21AG- 1C038 and
10

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A How frequent is Simpson's paradox: an estimate based on the non-informative Dirichlet density
To estimate the frequency of Simpson's paradox under fair data-gathering, we can try to generate the probabilities in 3 randomly in an unbiased way, and calculate the frequency of holding the
paradox 30 56] The best and widely accepted candidate for an unbiased density of probabilities is the Dirichlet density, which is widely employed in statistics and machine learning 37 58). The Dirichlet probability density for n probabilities (91..... Qn) reads:
T k Onk
Inl01, (n)
II
81
- 1),
(48)
r Mawl
x=1
k=1
T1 II D(a..... (ola,, (n)) =
(49)
where og the parameters of the Dirichlet density, 8(x) is the delta-function, and [[a]
So d is the Fuler's r-function. Since Dig (n.) is non-zero only for qk 2 and 2k=1 9h I, the continuous variables themselves have the meaning of probabilities.
Many standard prior densities for probabilities are contained in (48) c.g., homogeneous (ar1 ...On 1), Haldane's 's (a1 ...Op
0), Jeffreys (a3 ,.,0 1/2). For estimating the frequency of Simpson's 's paradox, Ref. [66] employed homogeneous and Jeffreys prior.
For modeling non-infonmee con-informative Dirichlet density we find it natural to take
Q1 .., 1/n.
(50)
The homogeneity feature, 01 =..o in 500 is natural for an unbiased density. The factor 4 in (0 makes an intuitive sense, since (x Or, become homogeneous (non-informative) probabilitess. Eq. (50) arises when we assume that the distribution of random probabilities is independent of whether they were generated directly from (48) with n components, alternatively from (48) with
nm components 0₃ = = asm and then marginalized. This requirement indeed leads to 55). as can be checked with the following feature of (48):
1./ dg'' dg', 8(gn-1 dn - s'm)
Diqi.... 4n- 1 d' aq, ..., an)
D(g,,..., Qu 2, In [11, On an-1 + an).
(51)
The message of (51) is that aggregating over two probabilities leads to the same Dirichlet density with the sum of the corresponding weights 0x and an.
We estimated the frequency of Simpson's 's paradox assuming that 8 probabilities p(A1, A2, B) in generated from 48 50) with 8 (binary situation). This amounts checking two
relations (they amount to 1 and its reversal)
(p(())2)) p(a1laz))lp(e (2, p(a1 222 b)] < 0,
(52)
pp(ar|02, b) p(a, āā b)][p(a) b) p(a, 02 b)] 0.
Our numerical result is that the frequency of two incqualities in 52 is 4.29% + 0.001%. For this precision it was sufficient generate N 10 samples from 48 50 with 8. This result compares favorably with x 1.66% obtained for a = ...atg = 1 (homogencous prior), and x 2.67%
obtained for α₃ ...Ag 0.5 (Jeffreys prior) [56). It is seen that the frequency of Simpson's 's paradox a decreasing function of a1 = = 0g = a [56).
Roughly, the above result ~ 4.29% means that in every 1000 instances of 3 binary variables, 42 instances will show Simpson's 's paradox. This number is reassuring: not very large meaning that the standard Decision-making based on the marginal probabilities in (1) will frequently be reasonable. But it is also not very small, showing that Simpson's paradox is generic and has its range of applicability.
14

---

B Proof of Theorem 1
The main idea of proving T33) is inverting (
p(a [0010220)) c)
p(bl)ppla l02. ()p(())2) (p(b)e) 1)p(a) a2 ()p(8)(2)
(53)
p(b(e)p(bla)) + (p(blc) 1)p(6(az)
p(l11l02, b) ((0l))((//2)(p((111 b) p(a1 l02, b)]
(54)
1 p(p(b p(blaz)
p(cla2) p(blc)p(blaz) + (p(ble) - 1)p(b/a2)
p(blc) + p(blc)
p(blc) + p(bla2) -1 p(blc) + p(b]c) -1
(55)
where unknown quantities p(a] (42, c) and p(cl02) are represented via known ones (i.e. p(A,, A2, B)) and free parameters y( BCC).. Eqs. (54 55) hold upon changing a2 by d2 and are deduced in Appendix
via specific notations that should be useful when dealing with (9) for a non-binary C.
The rest of the proof is algebraic but non-trivial. It also works out and employs constraints Simpson's 's paradox itself. Expanding both sides of (T).
64 on
p(an ((2) p(an l02, ()p(((a2) + p(an la,, b)p(b)a)):
(56)
p(a (ā2) = p(ailaz, b)p(bla2) + p(ailãz. b)p(bla2)
(57)
and using there 2 we subtract the sides of from each other and find:
p(e1 [ b) + p(b a2) il a1 a22 b) p(a1 02 b)] p(a, وā b) + p(b)p a2) p(a ق22 b) p(a, 22 b)]. We return to 22 and note that we can assume without loosing generality
(58)
p(a1 (a2, b) p(ailaz, b).
(59)
Eqs. (56 57 imply that for the validity of 3 3 59
it is necessary to have p(an (á2, b) > p(an (az b).
which together with ( 3 59) revert to Now
56
57) read
p(a1a22 b) + [1 p b az a2) b) - p(p a2, b))
(60)
< p(a1 22 b) + [ - -1 ((b))))(((( b) p(a1 (22 b)),
p(a aa, b) - p(b (p - pla b))
< p(a (22, b) (bb(a)) 02, p(an قي2 b)),
(61)
where (60) and 61 are equivalent. Eqs. 60 imply
[1 (blaz) laz, b) - pla1laaa.101 b)) <
[1 p(bla=]((p(aaaa) b) -
b)),
(62)
p(bla2)(p(al a2 b) p(a1 a22 b)) > (pbla)) (z b) p(a, (az b)).
(63)
As checked directly, Eqs.
62 63 lead to
p(blaz) > p(blūz).
(64)
Now we return to 55 and assume there p(blc) + p bb -
< O, which leads to
p(blc) + p(bjp)))))))))))))))) < from 55). Writing down from 555 the formula for p(clāz) and mak- ing the same assumption we get p(b[c) + p(blāz) - 1 < 0. Now look at (54) and its analog obtained via 02 -> Ø2 and use there these two results together with 63 64) and (4) to deduce the first inequality in [3) under assumption p(blc) + p(blc) - < 0. It should be obvious that the second
inequality 13 holds under the same assumption since we nowhere used any specific feature of e
compared to
15

---

For p(ble) + p(bla2) - > 0 we need to use instead of 54
another form of (53
p(a [0, c) - p(a1
[ - p((
b) - p(a1 02 b)]
(65)
p(b[c) p(b(a2)
The rest is similar to the above: we proceed via 62 64 the analog of (65) obtained via 01 -> 02.
and and deduce 13 from 55
65
and
C More examples of Simpson's paradox
We collected several examples of the paradox that are scattered the literature. We discuss them employing notations in equations (1-3) of the main text emphasizing (whenever relevant) the existence of the common cause (or screening) variable C.
Example 5. Snow tires provide cars with better traction in snowy and icy road conditions. However, nationally in the US, cars fitted with snow tires are more likely to have accidents in snowy and icy accident), ), Ag {ag changed tires, 02
conditions [59]. A {oi accident, á1 not changed]. B = {states]. Here the choice of the state (warm or cold) has a direct causal link to accidents in winter conditions. Now snow tires tend to be fitted to cars only in snowy winter months and in states with colder weather. Cars in warmer months and in states with warmer weather are much less likely to have accidents in snowy and icy conditions. Plausibly, there is random variable, C== (good weather conditions, bad weather conditions). which causes A, and screens A from B: (A(CB) p(A(C). The times are distributed as t tc < t < tAA
Example 6. This example emerged from discussing our own experience with hospitals. We need to choose between two hospitals 1 and 2: A {aı recovered,
not recovered), A₂ {az = hospital 1, 02 = hospital 2). B = (first half year, second half year). C' {types of illness]. Here we do not expect direct causal influence from B to A, if (as assume) the hospitals do not treat seasonal illnesses. We expect that C causes A, and screens from B.
Note that the data from which the probabilities for Simpson's 's paradox are calculated is the number of patients N (A1, Ag, B) that came the hospital. Simpson's 's paradox does not occur if within each season the hospitals accept an equal number of patients: Σ>, A, .N (AI, A2, B) does not depend on the value of B. This creates a conceptual possibility for judging between the hospitals. This is however not realistic, because imposing on these hospitals an equal number patients can disturb their usual (normal) functioning.
Example Simpson's 's paradox is realized when comparing scores of professional athletes, e.g. the batting averages of baseball players [60]. Here A₁ refers to a score of an athlete in game, e.g. A high score, low score, A2 denotes concrete athletes, while B is the time-period (e.g. playing season). The causing variable C' can refer to the psychological and physical state of an athlete that influences his/her game success, and the number of games he/she participated each season.
D Elaborations on smoking and surviving
1. Our interest in this subject started from learning about the works by R. Fisher, who proposed that least part of the association between smoking and survival may be due to genetic common causes. Then we noted that qualitative genetists clarified his statements in 1980s, but they are nearly forgotten in modern genetics, where genetic determinants of smoking are still studied actively [49].
The data presented in Ref. [16] considers three random variables A, =(ded,, alive), Ag = (smokingg non-smoking), ). and B -yyunger, older). To this we added unobserved genetic variable: C = (rrik to smoking, no risk to smoking), which roughly corresponds to the gene CHRNA5 described below.
A fairly general TODAG for this situation is
A B < C - A AI, A2 B, C A1;
(66)
e.g. because once C can influence B, then potentially also A₂ can have a direct influence on B. At the present stage of our knowledge on pertinent genetic and age-dependent factor influencing smoking, this TODAG is not manageable. So we had to simplify it drastically.
16

---

First, we erased the link A₁ < B. because the physical age by itself does not influence survival. The physiological age correlates well with the physical age for some people, which can already affect their survival rate. However, the physiological age in this experiment was not recorded or controlled.
Once we assumed A1 of B, then from the viewpoint of Simpson's paradox, it was already natural to assume A2 A B as well, because the link A + B does not influence p[AA] (d((2)) (Postulating the direct influences A + B are more or less akin predetermining the influences of smoking.) At
any rate, we emphasize that both A + B and A2 A B are essential assumptions of the model. We
these assumptions we end up from 66 with the following TODAG [cf. [40]]
(67) remains to explain in which sense the gene can influence the age. For example, if an allele of a gene (see below) can be a common cause of both smoking and (independently) smoking generated smoking-generated deceases, then aged (but still healthy) people can be those which did not have this allele.
AI, C A1.
2. The classical Mendelian genetics assumed (and in many instanced verified) that as far the influence on the phenotype is concerned, one can restrict a gene to a binary variable: recessive and dominant alleles of the gene. Each organism has two genes (one from mother and another one from fäther), and now the three pairs recessiveddomnnat, dominant-recessive type of influence to the phenotype, while the version recessive-recessin to another influence.
dominant-dominant amount to one
In modern genetics, many exclusion from this classical binary-gene law are known. For example, the gene of the blood type has 3 alleles (A,B, and O). Here A and B are dominant with respect to O, but together they are co-dominant and hence there are blood groups: AO, BO, AB, 00.
To understand whether the genes controlling the smoking behavior can be modeled as binary (i.e. dominant and recessive), we need to consider concrete genes, which according current genetics research have serious effects on nicotine addiction and show evidence of pleiotropy, i.e., they can influence more than one aspect of health and survival: see [49]
CHRNA5 is a gene that encodes subunits of the nicotinic acetylcholine receptor, which is important in neural signaling and nicotine addiction. The receptor can influence various aspects of smocking behavior: nicotine binding and response, reward pathways, craving intensity, smoking cessation success rates, etc; see Ref. [50] for review.
CHRNA5 has two alleles and A. They are denoted by (guanine) and (adenine), because the alleles differ by single nucleotide. Now A is the risk allele, which is associated with increased smoking. G is the non-risk allele [50). Now A the dominant allele with respect to G, and CHRNA5 can be said to be binary with the following reservation: there is dose-effect and the pair AA turns out to be more risky than AG (in contrast to GG, which is risk free). Within our crude model we neglect this difference and treat CHRNAS as binary.
D.1 Technical details on the example from section of the main text
This example is taken from Ref. [16]. Its concise version was discussed in section 5 Here we provide more details on how the data was presented and how we analyzed it. In this case, binary A₁ represents the survival of a woman as determined by two surveys taken 20 years apart: A₁ = {died, alive). The binary A₂ reads A₂ = [smoker, nonsmoker], j. while B = {Bi,- .... B₁ means the age group of the person recorded in the first survey. The B1 now includes women between the ages of 18 and 24. Likewise, Bz, B3, Ba. B5. BB refer to (resp.) ages (25 - 34), (35- 44). (45- - 54), (55 64). (65 74). The corresponding probabilities read:
p(B)) = 0.0946, p(B2) = 0.2272, p(Bz) = 0.1859, p(B)) = 0.1681,
p(Bs) 0.1908, p(Bb) 0.1334.
(68)
There is also the seventh age group that included people who were 75+ at the time of the first survey. We shall, however, disregard this group, since the data is pathological: nobody from this group survived till the second survey. It turns out that the aggregated data (1) of the main text hints that smoking is beneficial for survival:
A2) _ P(A, Bx)
(69)
p(A, -died|Ag smoking) = 0.2214 < p(A, = A2 = nonsmoking) = 0. 0.2485.
(70)
17

---

This conclusion is partially reversed, once the age group B is introduced:
p(A1 died A₂ smoking, B&) > p(A1 = died A₂ = nonsmoking, Bx), k 1,3,,,,,5,6
(71)
p(4 died A2 smoking, B2) < p( died A2 nonsmoking, Bz).
(72)
We need to coarse-grain the above data to formulate the Simpson paradox clearly. Now
B = {b, b). b = B1 U B2 U B3 U B4 U Bs,
b = B6:
(73)
p(A1 died] A2 smoking. b) = 0.1820 > p(A died/ A2 nonsmoking, b) 006,
(74)
p(11 died] A2 smoking, b) = 0.8056 > p( A1 died|A2 nonsmoking, b) 0.7829.
(75)
This leads the formulation Simpson's 's paradox discussed in section V of the main text. Eq. is the only coarse- grriinig that leads to the paradox.
The authors of Ref. (16) provide the following heuristic explanation for the prediction difference between (70) and (71): they noted that aged people from the survey are mostly not smokers and most would have died out of natural reasons. This is the statistical explanation of the Simpson paradox.
This explanation is not especially convincing because of (68 72): it is seen that B3 is the most probable group, for which (70) and agree.
E
Matrix notations for inverting the common cause equation
Here develop matrix notations for inverting the common cause equation:
A2, = E2(A, C)((ICC)
(76)
where the summation goes over all values of C. We work for the case when the variables A1, A2, B and C' are binary, though the matrix notations we introduce below are useful more generally.
Eq. (76) can be written in matrix form
ii1
(1 12
(112)
(ik1)
(77)
ik-2
(2
(ih:2)
where il: = 11, 21, 22 and the following notations were introduced
[11I] = p(ai, a/a1,a2,00 b), [121] = p(aj, 02 b),....
(111) p(a1, 02 c), (121) = p(a1, 02 .).....
(78)
(1|1) = p(bcc), (211) p(blc)....,
D= = (2 + (1/1)- 1
Inversion of the Eq. 77 gives
(ik1)
(2/2)
(112)
(ik-2) D (2(1)
(111)
[1-22
(79)
Eq. 79
implies
21
(1k1) (112)
(2(2)
- -1122) (1)1)
[ [1 2
(80)
D - (2
where analogously to 78
we introduced the following notations:
[i|u1] ((1]a22 b), [1|21] p(a1, ã2, b)....
(1[11) = p(ailag, <), (1/21) = p(a, lūz, c)....
(81)
[i]t] = p(bla2), [2[1] p p(bla2)...... ).........
(1|1} = p(cla2), {2 = p(
18

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The matrix relation (80) results in
(2 D
[ (2 [2
(1))))(1
(82)
Using
82
79
we get relations employed in the main text:
(i)k1) = (22)) (2P0))))1
(83)
{1]k} (2222) + 3. ((212) 1) (
(84)
(2)k2) (AD-DER114 M
(85)
(10--1)
[2]k} (((1)) (111]])))))
(86)
F Certain matrix relations
There is useful formula for matrix inversion
(Z + - -V
(87)
Eq. 88 is derived via two auxiliary formulas. First note that
(88)
which follows from (V= Next, note moving U according to (88)
U (Q==
(89)
which leads
(1+ -
(90)
To deduce 87 from 90). we manipulate Z and W in respectively LHS and RHS of (87 and hence transform (87) to the form (9)), but with the following replacements: U Z- and WV.
Eq. 187
leads a generalized Sylvester formula:
det Z + UWV] det[Z] det[W] det[W VZ-
(91)
The ordinary Sylvester formula for determinants reads
dct n KN M LM N] det[IM M KM M).
(92)
where IN is the N x N unit matrix, KN M is N M matrix etc. Eq. (9) follows from the fact that (for M 2 N) LM x KN a has the same eigenvalues as KN at Las (plus M - zero
eigenvalues for M N > 0).
Inverting block matrix goes via
Au A21
A12
S-1 App A
-5-1 And A21S A12
(93)
Ap2
Ap1
S An
A12 A21.
(94)
where dimensions of A11, A12, A21 and A22 are, respectively, M M, MN M, (N M) (N- M). and where S is the Schur-complement of the block matrix over its upper diagonal part. Eq. (93) straightforward to prove.
A11 A21
A12
det
det[An - A12 A2i] [22])
(95)
A22
19

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G
Common cause with higher dimensionality for continuous variables
Let's discuss the scenario where the number of components of common cause is two. Recall equation (55) of the main text and note that now C is a x 2 matrix and S is 2 x 2 matrix. For CSCT we have
TerCi + 12
21 + V2C22
Vi Cai + C2C32
CSC
C2 + U2C22
ui C2 + u2C22
Mi uiCai + ugC32
(96)
Ca + V2C32
u1C31 + u2C32
CaCai + C2
where
11 C11$11 + C12821,
0/2 C11$12 + C12822,
(97)
U1 = C11 + C22821,
U2 = C21 C22822.
(98)
k C31811 C32321,
22 C31$12 + C32522.
(99)
We need to keep track of element of the matrices, since
((a1 (a2) - (a2))))o = C)
(100)
((a: (a1) (a2)a)) = A12,
(101)
(a1a2) (A+ 1112
(102)
and for which we have
(A - J) A12 + U1C21 + DzC22,
(103)
(KCBB (C 12
(ViCs1 + D2Cse) U2C32).
(104)
B C1 + h2Cs2
Now, we consider the simplest case for a common cause
0
$ =
(105)
$
The equations simplify to ((a1 - (a1)))(a2 - (a2))))
A12 + sCC11 + C12C22)
(106)
s²
(Cn1Ca1 + (12Cs2) + C22C32X107)
+ s + 2
(a1az)
A12 + sC1n1 C12C22).
(108)
By setting Cai = 0 and considering
B, we get
((a1 - (a1))) (a2)6)lo
A1 + sC11C21,
(109)
(alaz)
= A12 + s(C11 + C12C22).
(110)
Obviously, inequalities (aΙaz) > 0 and ((11-(a1))) (az (22) 0 their inverted alternatives), do not determine the sign of A12, hereby the sign of ((a1 - (a)) (a2)=)) Thus, for a common cause with two components we already see that can support both fine-erained and coarse-grained options.
20

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Reviews:
[{"review_id": "rq9hr8Zit0", "reviewer": "Reviewer_y8Yk", "summary": "This paper studies Simpson’s paradox from a causal perspective and applies the result to decision-making with examples. The work first formalizes Simpson’s paradox. Consider a target event $A_1$ and an action $A_2$ performed before $A_1$, along with a context variable $B$ that also precedes $A_1$ (all variables are binary). The marginal distribution $p(a_1 \\mid a_2) < p(a_1 \\mid \\bar{a}_2)$ can exhibit a different property than the conditional distribution $p(a_1 \\mid a_2, b) > p(a_1 \\mid \\bar{a}_2, b)$, making it unclear whether to apply the action $A_2 = a_2$ or $A_2 = a_2'$. \n\nNext, the paper reviews earlier attempts to resolve Simpson’s paradox and discusses their limitations. It then proposes a new approach: assume there exists a binary variable $C$ that is a cause of $A_1$ and $A_2$, and that the causal structure among $A_1$, $A_2$, $B$, and $C$ satisfies the condition that no additional information about $A_1$ and $A_2$ can be obtained from $B$ once we condition on $C$. Under this assumption, we have $p(a_1 \\mid a_2, c) > p(a_1 \\mid \\bar{a}_2, c)$ and $p(a_1 \\mid a_2, \\bar{c}) > p(a_1 \\mid \\bar{a}_2, \\bar{c})$, which aligns with the conditional distribution given $B$, and we should choose the action $A_2 = a_2'$. The result is further extended to the case of Gaussian variables.", "questions": "**Q1**. Is the term $P(s)$ missing in Equation (11)?", "limitations": "Please refer to **W1** and **W2**.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "# Strength \n\n**S1.** The paper is well written. It provides a wide range of illustrative examples to support its claims and theorems, helping to ground the theoretical content in concrete scenarios. The logical flow is smooth and easy to follow.\n\n**S2.** The theoretical results are presented with commendable mathematical rigor.\n\n**S3.** The theoretical results are interesting in math.\n\n# Weakness\n**W1.** \nA key concern lies in the use of Theorem 1 for decision-making. Theorem 1 states that conditioning on $B$ yields the same sign of association between $A_1$ and $A_2$ as conditioning on $C$, i.e., \n\n$$\np(a_1 \\mid a_2, B) > p(a_1 \\mid \\bar{a}_2, B) \\quad \\text{then} \\quad p(a_1 \\mid a_2, C) > p(a_1 \\mid \\bar{a}_2, C).\n$$\n\nHowever, it is questionable whether this implies that taking action $A_2 = a_2$ is better than taking action $A_2 = \\bar{a}_2$. \n\nIn causal inference, observational distributions reflect correlations, not causal effects. Decision-making should be based on causal effects, not merely on associations. As shown in \\cite{pearl2009causality} (referenced as [13] in the paper), causal effects are defined in terms of the interventional distribution $p(a_1 \\mid \\text{do}(A_2 = a_2))$. It is entirely possible for the observational association to be positive while the causal effect is negative. For instance, in the presence of unobserved confounding between $A_1$ and $A_2$ (as in the bow arc graph in \\cite{pearl2009causality}), it could happen that\n$$\np(a_1 \\mid \\text{do}(a_2)) < p(a_1 \\mid \\text{do}(\\bar{a}_2)),\n$$\neven though\n$$\np(a_1 \\mid a_2, C) > p(a_1 \\mid \\bar{a}_2, C).\n$$\nTherefore, the connection between Theorem 1 and decision-making should be more explicitly clarified, possibly by incorporating additional assumptions or causal identifiability arguments.\n\n**W2.** \nAnother limitation of Theorem 1 lies in its strong assumptions. Although $C$ may be unobserved, the theorem relies on two difficult-to-verify assumptions: the conditional independence condition (Eq.~(10)) and the assumption that $C$ is binary. Since $C$ is unobserved, neither assumption can be verified from data, making the theorem challenging to apply in practice easily.\n\nThe paper acknowledges this limitation, stating:  \n“All examples we presented are observationally incomplete: a plausible cause had to be inferred, but it was not shown to exist or function from the real data.” (lines 337--338)\n\nGiven the considerable gap between the assumptions of the theorem and real-world data, and the lack of discussion on when or why these assumptions might be satisfied, I believe the practical contribution of Theorem 1 is limited.", "was_revised": true, "final_justification": "The paper presents a novel and sound solution to Simpson’s paradox under certain assumed graphical structures. The theoretical results offer interesting insights. However, I find the binary C assumption, as well as the graph over B and C, to be somewhat unnatural and difficult to validate. Therefore, I recommend a borderline accept.", "rebuttal": "We thank the reviewer for the evaluation of our manuscript and for\nrequesting important clarifications. The reviewer noted two weaknesses **W1**\nand **W2**, and asked one question **Q1**. Below we respond to them.\n**W1** and **Q1** can be addressed fully. **W2** cannot be\naddressed fully without serious additional research. Still, in the\ncontext of **W2** we presented a new argument in support of the\npublishability of our contribution. \n\nResponse to **Q1**. Yes, indeed, $P(c)$ is missing in Equation (11).\n\nResponse to **W1**. Our result is fully consistent with the\ndo-calculus and with the proposal of the reviewer. To show this,\nconsider two DAGs that support the causal structure of our theorems;\nplease see Fig.~1 of the manuscript. (Please note that Fig. 1 wrote\nthese two DAGs together in a condensed form, and also did not specify\nthe causal direction $A_2\\to A_1$.)\n\nThe first DAG from Fig. 1 is \n$$\nB\\leftarrow C\\to A_2\\to A_1, \\qquad C\\to A_1. \n$$\nThe second is \n$$\nB\\to C\\to A_2\\to A_1, \\qquad C\\to A_1. \n$$\nFor both these DAGs\nwe have\n$$\np(a_1|{\\rm do} (A_2=a_2) )=\\sum_{C=c,\\bar{c}} p(a_1|a_2,C) p(C).\n$$\nHence, if (as stated in our theorem 1)\n$$\np(a_1|a_2,c)>p(a_1|\\bar{a}_2,c), \\quad p(a_1|a_2,\\bar{c})>p(a_1|\\bar{a}_2,\\bar{c}),\n$$\nthen we get (as desired)\n$$\np(a_1|{\\rm do} (A_2=a_2))>p(a_1|{\\rm do} (A_2=\\bar{a}_2)).\n$$\n\nResponse to **W2** consists of two parts.  In part **1** we argue\nthat also the existing solutions of Simpson's paradox require conditions\non data, and that our results compare well with the existing benchmarks.\nIn part **2**, we discuss an important aspect of our theorems, which went\nunnoticed in the above criticism of the reviewer. For\n\n**1.** We agree that our results on resolving Simpson's\nparadox demand motivating one of the above DAGs. However, the \nviable resolution of the paradox (proposed by Lindley and Novick, and\nextended and formalized by Pearl; please see Refs.~[9,13] of our\nmanuscript), also demands motivating certain DAGs.  This is the general\npoint of Simpson's paradox: it is resolved only by going beyond its\noriginal, purely probabilistic formulation. More precisely, Simpson's\nparadox was so far unambiguously solved only for one DAG. To this we add\ntwo more DAGs outlined above. Hence we suggest that our contribution in\nresolving Simpson's is at least comparable to the existing\n``benchmark''. \n\nLet us provide details of our opinion. \nPearl in Ref.~[13] studied two DAGs. The first one is $A_1\\leftarrow B\\to A_2\\to A_1$. For this DAG we agree that \n$$\np(a_1|{\\rm do} (A_2=a_2))>p(a_1|{\\rm do} (A_2=\\bar{a}_2))\n$$\nresolves the paradox. The resolution amounts to the fine-grained option of the paradox. This is expected: within \n$A_1\\leftarrow B\\to A_2\\to A_1$, $B$ influences both $A_1$ and $A_2$; hence it is natural to condition over $B$.\n\nPearl also considers another DAG: $A_1\\leftarrow A_2\\to B\\to A_1$. For this situation \n$$\np(a_1|{\\rm do} (A_2))=p(a_1|A_2),\n$$\nand it is not clear what prevents us from going back to the original formulation of Simpson's paradox, i.e. what prevents us from \ncomparing \n$$\np(a_1|{\\rm do} (A_2=a_2))<p(a_1|{\\rm do} (A_2=\\bar{a}_2)),\n$$\n\nwith \n\n$$p(a_1|{\\rm do} (A_2=a_2),b)>p(a_1|{\\rm do} (A_2=\\bar{a}_2),b),$$ \n$$ p(a_1|{\\rm do} (A_2=a_2),\\bar{b})>p(a_1|{\\rm do} (A_2=\\bar{a}_2),\\bar{b}).$$\n\nWe understand that Lindley and Novick (despite of refraining to some extent from the causality language), \nand Pearl presented heuristic arguments for resolving the paradox also for the situation of the second DAG\n$A_1\\leftarrow A_2\\to B\\to A_1$. But still the formal level of resolution for the two DAGs is quite different, as was\nemphasized also by Armistead (our Ref.~[17]). \n\n(Please note that the third DAG $B\\leftarrow A_2\\to A_1\\to B$ is\nnot studied, since we do not consider Simpson's paradox\nin the scenario where the variable $B$ is the effect of the involved variables $A_2$ and\n$A_1$.)\n\nTo summarize, we see that so far Simpson's paradox was unambiguously\nsolved only for one (out of two) DAGs. \n\n**2.** We would like to emphasize an important aspect of our theorems,\nwhich went unnoticed in the above criticism of the reviewer. For theorem\n1, once we can motivate one of the above DAG with binary $C$, then no\nadditional data-gathering is necessary for resolving the paradox, i.e.,\nwe do not need to know $P(A_1,A_2|C)$,  etc. Likewise, for theorem\n2, once we assume the minimal Gaussian set-up, the paradox is resolved\nwithout knowing further details. \n\nIf in the set-up of theorem 1, $C$ is not binary (but one of the above\nDAGs applies), then the resolution of the paradox still applies in the\nsense that one should compare $p(a_1|a_2,C)$ with $p(a_1|\\bar{a}_2,C)$,\nbut to understand the message of this resolution, one now needs the\ndetails of the common cause $C$. The same holds for theorem 2."}, {"review_id": "yF0lLsXuMl", "reviewer": "Reviewer_J6Fc", "summary": "This paper claims to resolve Simpson's paradox through a new assumption of a common cause, but I think this is an insufficient description of their contributions.\n\nInstead, the paper proposes a formulation of Simpson's paradox with a specific graphical structure, in which some $C$ is a common cause between the treatment, outcome, and a confounder.\n\nIn this setting, they claim that conditioning on the minimum common cause $B$ resolves the paradox.", "questions": "I am still having a difficult time understanding the criticism with the exchangeability resolution of Simpson's paradox. Is the problem just that TODAG's cannot express hidden confounding? Can't this be resolved using an ADMG and M-separation? There are still backdoor adjustments for ADMGs. If I am missing something here, I am happy to improve my score. However, as it stands, I must recommend rejection.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The paper is lacking the precision needed to really evaluate its results. As it stands, I am worried about correctness.\nFor example, they claim that some $C$ is a common cause for $A, B$. Since they then say that $C$ is unobserved, we are working with the following ADMG:\n\n$A_1 \\rightarrow A_2$\n\n$A_1 \\leftrightarrow A_2$\n\n$A_1 \\leftrightarrow B$\n\n$A_2 \\leftrightarrow B$\n\nConditioning on $B$ here should open up an active path between $A_1$ and $A_2$, so their claim seems to be incorrect with respect to Pearl's resolution. Their criticism of Pearl's resolution of the paradox seems to be that they believe the canonical setup $A_1 \\rightarrow A_2$ with $A_1 \\leftarrow B \\rightarrow A_2$ does not describe all settings. But, of course, Pearl's resolution is more general and has to do with exchangeability. They do acknowledge this point, so there is a chance I am not understanding their problem with the exchangeability resolution. Conditioning on $B$ in their setting clearly does not ensure exchangeability.\n\nWhether or not the paper is correct, the paper does not ``resolve'' Simpson's paradox in general. It simply studies the paradox in a specific setting.\n\nMinor comment:\nLine 140: It is a little confusing how you referred to Equations (1) and (2-3) so much later in the text. I think it might be easier to just refer to the aggregate direction vs the conditional one. Or use “Eq.” or “Equation.”", "was_revised": true, "final_justification": "The authors clarified my misunderstanding about the goals of their paper. I think it is valuable to expand our understanding of the settings under which Simpson's paradox occurs. However, they do not provide a general resolution of the paradox, so I find the results somewhat incremental. This makes this a borderline accept.", "rebuttal": "We thank the reviewer for evaluating our manuscript. We are encouraged by\nthe statement that the reviewer will be happy to improve the score, if\nwe clarify our statements. We try to do so below. \n\nWe identified 4 questions in the reviewer's report.\n\n**Q1** What precisely is shown in the present work?\n\nTo answer this question it is easier to look at the following two DAGs;\nplease see Fig.~1 for a condensed version of them. \nThe first DAG is\n$B\\leftarrow C\\to A_2\\to A_1$ and $C\\to A_1$. The second DAG is $B\\to\nC\\to A_2\\to A_1$ and $C\\to A_1$. \n\nNow for the minimal set-up of binary variables Simpson's paradox leads to\n$$\np(a_1|a_2,c)>p(a_1|\\bar{a}_2,c), \\quad p(a_1|a_2,\\bar{c})>p(a_1|\\bar{a}_2,\\bar{c}),\n$$\nwhich in its turn produces \n$$\np(a_1|{\\rm do} (A_2=a_2))>p(a_1|{\\rm do} (A_2=\\bar{a}_2)).\n$$\nAltogether, it means Simpson's paradox is resolved in a specific way.\n\n**Q2** What precisely is our criticism of the Simpson's paradox solution presented by Pearl?\n\nIn fact, our criticism consists of two separate points. The first one\nwas presented in the manuscript, and will be reminded below. The second\none is an expansion and formalization of one of criticisms presented by\nArmistead (our Ref.~[17]). We only very briefly mentioned it in the manuscript, \nso below we will provide its full details.\n\nA major part of Pearl's resolution (which expands on the previous work\nby Lindley and Novick; our Ref.~[13]) is that examples presented Lindley\nand Novick are mapped into two DAGs: $A_1\\leftarrow B\\to A_2\\to A_1$ and\n$A_1\\leftarrow A_2\\to B\\to A_1$. However, looking at those specific\nexamples we found qualitative arguments for not restricting ourselves\nwith those two DAGs only. For example, we motivate that the common cause\nsituation might instead be a more plausible description of (at least)\none of those examples. Such criticisms are not unexpected: any realistic\nexample is (infinitely) complicated, and might admit different models. \n\nThe second criticism is as follows. For the DAG $A_1\\leftarrow A_2\\to B\\to A_1$, Pearl resolves the \nSimpson's paradox via $p(a_1|{\\rm do} (A_2=a_2))$. However, for this specific DAG we have\n$$\np(a_1|{\\rm do} (A_2=a_2))=p(a_1|A_2=\\bar{a}_2),\n$$\nand it is not clear what prevents us from going back to the original formulation of Simpson's paradox, i.e. what prevents us from \ncomparing \n$$\np(a_1|{\\rm do} (A_2=a_2))<p(a_1|{\\rm do} (A_2=\\bar{a}_2)),\n$$\nwith \n$$\np(a_1|{\\rm do} (A_2=a_2),b)>p(a_1|{\\rm do} (A_2=\\bar{a}_2),b), \\quad \np(a_1|{\\rm do} (A_2=a_2),\\bar{b})>p(a_1|{\\rm do} (A_2=\\bar{a}_2),\\bar{b}).\n$$\nWe understand that Lindley and Novick (despite of refraining to some extent from the causality language), \nand Pearl presented plausible arguments for resolving the paradox (for this DAG) via $p(a_1|{\\rm do}( A_2=a_2))=p(a_1|A_2={a}_2)$.\nBut still the level of resolution for the two DAGs is quite different, as was\nemphasized also by Armistead (our Ref.~[17]). \n\n**Q3** How Pearl's solution relates to exchangeability? \n\nTo our understanding the exchangeability was proposed by Lindley and\nNovick as a partial substitute of the causality. Pearl (in his published\npapers and books) did not endorce the exchangeability viewpoint. This is\nalso apparent from the review written by Lindley on the book by Pearl;\nplease see D.V. Lindley, \"Seeing and doing: the concept of causation\",\nInternational Statistical Review, 70, 191-214 (2002).  After Lindley and\nNovick, the exchangeability viewpoint was extensively developed e.g. in\nO. Saarela, D.A. Stephens, and E.E. Moodie, \"The role of exchangeability\nin causal inference\", Statistical Science. 38, 369-385 (2023). This\npaper reviews the history of the probabilistic causal inference, and\nalso explicitly supports the above opinion on Pearl and exchangeability. \n\nTo our understanding, Pearl never formulated a general solution to\nSimpson's paradox. Following his general ideas, one can take $p(A_1|{\\rm\ndo} (A_2))$ as a possible general solution. As we stated explicitly when\nanswering the first question, this solution agrees with our main\nresult. \n\n**Q4** How do our results relate (and/or can be interpreted) via ADMGs. \n\nConsider again the two DAGs studied in our manuscript:  \n$B\\leftarrow C\\to A_2\\to A_1$ and $C\\to A_1$; $B\\to\nC\\to A_2\\to A_1$ and $C\\to A_1$. \nEach of these DAGs leads to the following ADMG: $A_2\\leftrightarrow A_1$, \n$A_2\\leftrightarrow B$, $A_1\\leftrightarrow B$, and $A_2\\to A_1$. (Please note that the description of this ADMG by the reviewer contains a misprint.) Likewise, if one starts with this ADMG and assumes that there is only one hidden variable $C$, then one ends with one of the above DAGs. We see that $B$ is not a collider in the DAGs.\nWe do not employ this ADMG in our consideration because the concept of m-separation here is not as useful, as working directly with the above two DAGs; please see our Fig.1."}, {"review_id": "MzaL6rBmlG", "reviewer": "Reviewer_GFht", "summary": "This paper addresses the long-standing problem of Simpson’s paradox, where aggregated and disaggregated data lead to contradictory decisions. The authors propose resolving the paradox by assuming the existence of a common cause or screening variable, conditioned on which the relevant variables become independent. By conditioning on this (possibly unobserved) variable, the paradox can be resolved in two settings: (i) binary treatment, binary outcome and binary common cause, and a *minimal* setting with Gaussian variables. They show that if the common cause can take more than two values in the discrete case, it may not be possible to resolve the paradox unless the common cause is observed.", "questions": "1. Do you have practical guidelines for when it is reasonable to assume the existence of a common cause variable $C$, and when such an assumption might be implausible?\n\n2. Related to the first question, in cases where the common cause is unobserved, how should one reason about or estimate its cardinality?\n\n3. In the example of smoking and survival, could you expand on how your method would resolve the paradox? Specifically, given that genetic factors are unlikely to be binary, how would your approach handle this casE?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n1.Although the Simpson's paradox is a relatively old and well-studied problem, this paper provides a novel perspective on it.\n\n2. The concrete real-world examples that were given were helpful and improved my understaning.\n\n3. The paper is well-structured and has a clear presentation.\n\n**Weaknesses**\n\nPerhaps the main drawbacks of the paper are the following two:\n\n1. Since the common cause is often unobserved, the criterion of Eq. (10) is not testable -- or at least if there are ways to falsify it, the authors do not mention anything in that regard. \n\n2. The offered resolution only works if the common cause can take up to 2 values. This is also connected to the first weakness above, in the sense that an assumption must be made about an unobserved variable, which is not testable. \n\nThe authors acknowledge the second weakness, and discuss the limitation that if $C$ takes more than two values, the paradox may not be resolvable.\n\n**Minor Comments**\n\n- Equation (6) does not necessarily hold. I think the only point that the authors wanted to make was that $A_2$ and $B$ must be correlated, which is a true claim, but the argument for it does not have to go through Equation (6), for which counterexamples can be easily constructed.\n\n- There are two typos that I think are important to fix: one is Equation (11), where either the right hand side must be multiplied with $p(c)$, or the first term should be the joint probability $p(A_1,A_2,c)$ rather than the conditional. The second one is on the first line of the proof of Theorem 1, where equations 51 and 52 are the results of inverting equation (11) rather than (10). Since the derivations appear much later in section E, this part was quite difficult to understand with the typo.", "was_revised": true, "final_justification": "The authors answered most of my questions satisfactorily. Only one of my concerns remains, despite which I'd like to retain my positive assessment of the paper. I believe this paper introduces a nice perspective on Simpson's Paradox.", "rebuttal": "We thank the reviewer for the clear and encouraging evaluation of our\nmanuscript. We fully agree with all comments concerning misprints and\nunclarities. \n\nResponse to weakness 1.\nWhen answering the question of the reviewer below, we shall explain in\ndetail mechanisms of producing a common cause situation, even if it is\nnot observed directly.  Here we would like to emphasize that there is a\ngeneral consensus that resolving Simpson's paradox implies going beyond\nthe three-variable data within which the paradox was formulated.  This\n``looking beyond data'' is present in all {\\it viable} solutions of\nSimpson's presented so far. \n\nBy viable we mean the following. As reviewed in our manuscript,\nsolutions to Simpson's paradox proposed so far separate into two groups.\nWithin the first group (please our section 2.2.1) people proposed new\nways of applying probability theory formulas. The proposals were to a\nlarge extent {\\it ad hoc}, i.e. specially designed to solve the paradox.\nThey were not tested more generally, and were not shown to lead to a\nfully coherent resolution schemes. For example, interchanging prediction\nwith retrodiction (as proposed by Barigelli and Scozzafava) is certainly\nnot a self-consistent thing to do and can lead to various errors when\napplied systematically. \n\nA viable solution was presented by Lindley and Novick and later on\nclarified and extended by Pearl; please see Refs.~[9,13] of our\nmanuscript. It focuses on the three random variables $A_1$, $A_2$ and\n$B$, and demands that their mutual behavior is governed by a certain DAG\n(directed acyclic graph). There is no solution, before the structure of\nthe DAG is clarified. \n\nNow the discussion we provided in section 2.2.2 illustrates that finding\nand motivating DAG is not easier than finding and motivating a common\ncause. \n\nResponse to weakness 2. In this context, \nwe would like to emphasize an important aspect of our theorems. For theorem\n1, once we can motivate one of the above DAG with binary $C$, then no\nadditional data-gathering is necessary for resolving the paradox, i.e.,\nwe do not need to know $P(A_1,A_2|C)$, {\\it etc}. Likewise, for theorem\n2, once we assume the minimal Gaussian set-up, the paradox is resolved\nwithout knowing further details. \n\nIf in the set-up of theorem 1, $C$ is not binary (but the common cause\nreasoning applies), then the resolution of the paradox is there in the\nsense that one should compare $p(a_1|a_2,C)$ with $p(a_1|\\bar{a}_2,C)$.\nHowever, to understand the message of this resolution, one now needs the\ndetails of the common cause $C$. The same holds for theorem 2. \n\nResponse to question 1. The assumption of common cause between\n$A=(A_1,A_2)$ and $B$ is reasonable if direct\ncausal links between $A$ and $B$ are absent, i.e., when we suspect that the\ndependency between them is not causation. This situation is especially\nimportant for Simpson's paradox, because if $B$ causes $A_2$ and $A_1$,\nthen it is intuitively expected that one should prefer the fine-grained\nversion of Simpson's paradox (i.e. the version which conditions over\n$B$). This in fact was shown by Lindley and Novick, and Pearl within a\nDAG $A_1\\leftarrow B\\to A_2\\to A_1$. \n\nA particular (but important) mechanism of producing a common cause is when $A$ and $B$\nare functions of $C$:\n$$\nA=f(C,\\lambda), \\qquad B=g(C,\\mu),\n$$\nwhere $\\lambda$ and $\\mu$ are independent random variables: $p(\\lambda,\\mu)=p(\\mu)p(\\lambda)$. \nNow the common cause follows from\n$$\np(a,b|c)=\\sum_{\\lambda,\\mu} \\delta(a-f(c,\\lambda))\\delta(b-g(c,\\mu))p(\\mu) p(\\lambda)=p(a|c)p(b|c).\n$$\n\nAnother important derivation of the common cause proceeds via the\nmaximum entropy principle, when the probability $p(a,b,c)$ is recovered\nfrom maximizing the entropy $-\\sum_{a,b,c}p(a,b,c)\\ln p(a,b,c)$,\nassuming that $p(a,c)$ and $p(b|c)$ are known and are to be imposed as\nconstraints in the maximization. \n\nResponse to question 2. The estimation of the cardinality of the\ncommon cause is an important problem in linear algebra,\nmatrix factorization theory, {\\it etc}; please see J. E. Cohen and U. G.\nRothblum, “Nonnegative ranks, decompositions, and factorizations of\nnonnegative matrices,” Linear Algebra and its Applications, 190,\n149–168, (1993). \nN. Gillis, Nonnegative Matrix Factorization. SIAM, 2021. \n\n\nIf the joint probability $p(a,b)$ of $A$ and $B$ is regarded as a matrix with non-negative elements, then common cause representations can be seen as factorizations of\n$p(a,b)$ into two matrices with non-negative elements:\n$$\np(a,b)=\\sum_{c=1}^{|C|} x_{ac}y_{cb}, \\qquad x_{ac}\\geq 0, \\quad y_{cb}\\geq 0,\n$$\nwhere the minimal value of $|C|$ (the minimal cardinality of the cause) is called the non-negative rank of the matrix $p(a,b)$.\nThere exist efficient methods of estimating it; please see N. Gillis, Nonnegative Matrix Factorization. SIAM, 2021. \n\nResponse to question 3.\n\n1. Our interest in this subject started from learning about the\nworks by R. Fisher, who proposed that at least a part of the association\nbetween smoking and survival may be due to genetic common causes.\nThen we noted that qualitative genetists clarified his statements in\n1980s, but they are nearly forgotten in modern genetics, where genetic\ndeterminants of smoking are still studied actively; please see e.g., The\nTobacco and Genetics Consortium, \"Genome-wide meta-analyses identify\nmultiple loci associated with smoking behavior\", Nat. Genet. 42, 441-447\n(2010). \n\nThe data presented in Ref.~[16] considers three random\nvariables $A_1=$\\{died, alive\\}, $A_2=$\\{smoking, non-smoking\\}, and\n$B=$\\{younger, older\\}. To this, we added an unobserved genetic variable:\n$C=$\\{risk to smoking, no risk to smoking\\}, which roughly corresponds\nto the gene CHRNA5 described below. The real DAG that refers to these 4\nvariables may be quite complicated. In particular, it may contain direct\ncausal influences $A_2\\to B$ in addition to more obvious relations $C\\to\nB$ and $C\\to A_2$. However, we also noted that postulating the direct\ninfluences $A_2\\to B$ is more or less akin to predetermining the\ninfluences of smoking. Hence, we employed a simpler DAG $A_1\\leftarrow C\\to\nA_2\\to A_1$, $C\\to B$. \nThough the actual situation is probably much more complex, we thought it might be helpful to note that the simple DAG is already able to explain the data in a satisfactory way. \n\nMore formally, a fairly general DAG for this situation would be\n$A_1\\leftarrow B\\leftarrow C\\to A_2\\to A_1$ and $A_2\\to B$, e.g. because\nonce $C$ can influence $B$, then potentially also $A_2$ can have a\ndirect influence on $B$. At the present stage of our knowledge on pertinent\ngenetic and age-dependent factors influencing smoking, this\nDAG is not manageable. So we had to simplify it drastically.  \n\nFirst, we erased the link $A_1\\leftarrow B$, because the physical age by itself does not influence survival. The physiological age correlates well with the physical age for some people, which can already affect their survival rate. However, the physiological age in this experiment was not recorded or controlled. \n\nOnce we assumed $A_1\\not\\leftarrow B$, then from the viewpoint of\nSimpson's paradox, it was already natural to assume $A_2\\not\\to B$ as\nwell, because the link $A_2\\to B$ does not influence $p(A_1|{\\rm\ndo}(A_2))$. At any rate, we emphasize that both $A_1\\not\\leftarrow B$\nand $A_2\\not\\to B$ are essential assumptions of the model. \n\nIt remains to explain in which sense the gene can influence the age. For example, if an allele of a gene (see below) can be a common cause of both smoking and (independently) smoking-generated deceases, then aged (but still healthy) people can be those who did not have this allele. \n\n2. The classical Mendelian genetics assumed (and in many instances\nverified) that as far as the influence on the phenotype is concerned, one\ncan restrict a gene to a binary variable: recessive\nand dominant alleles of the gene. Each organism has two genes (one from\nmother and another one from father), and now the three pairs -\nrecessive-dominant, dominant-recessive, dominant-dominant - amount to\none type of influence on the phenotype, while the version\nrecessive-recessive to another influence. \n\nIn modern genetics, many exclusions from this classical binary-gene law\nare known. For example, the gene of the blood type has 3 alleles (A,B,\nand O). Here, A and B are dominant with respect to O, but together they\nare co-dominant, and hence there are 4 blood groups: AO, BO, AB, OO. \n\nTo understand whether the genes controlling smoking behavior can be\nmodeled as binary (i.e., dominant and recessive), we need to consider\nconcrete genes, which, according to current genetics research, have\nserious effects on nicotine addiction and show evidence of pleiotropy,\ni.e., they can influence more than one aspect of health and survival;\nplease see The Tobacco and Genetics\nConsortium, \"Genome-wide meta-analyses identify multiple loci associated\nwith smoking behavior\", Nat Genet. 42, 441-447 (2010). \n\nCHRNA5 is a gene that encodes subunits of the nicotinic acetylcholine\nreceptor, which is important in neural signaling and nicotine addiction.  The\nreceptor can influence various aspects of smoking behavior: nicotine\nbinding and response, reward pathways, craving intensity, smoking\ncessation success rates, {\\it etc}; please see for a review: G. Lassi,\net al. \"The CHRNA5–A3–B4 gene cluster and smoking: from discovery to\ntherapeutics.\" Trends in neurosciences 39, 851-861 (2016). \n\nCHRNA5 has two alleles G and A.  They are denoted by G\n(guanine) and A (adenine), because the alleles differ by a single\nnucleotide.  Now, A is the risk allele, which is associated with\nincreased smoking. G is the non-risk allele. Now A is the dominant\nallele with respect to G, and CHRNA5 can be said to be binary with the\nfollowing reservation: there is a dose-effect, and the pair AA turns out to be\nmore risky than AG (in contrast to GG, which is risk-free). Within our\ncrude model, we neglect this difference and treat CHRNA5 as binary."}, {"review_id": "CqPN4lbqRA", "reviewer": "Reviewer_Ffck", "summary": "This paper revisits Simpson’s paradox and shows that it can be systematically resolved by assuming the existence of a minimal (possibly unobserved) common cause variable that renders the involved variables conditionally independent. The authors demonstrate that, in both binary and Gaussian settings, conditioning on such a common cause recovers consistent associations and resolves the paradox in its minimal form. They support their claim with examples from real-world data (e.g., smoking and COVID-19 cases) and provide theoretical results showing that decision-making based on fine-grained (conditioned) probabilities is valid when a minimal common cause exists.", "questions": "In the smoking and survival example, could the authors clarify the following:\n\n- Why can the gene ($C$) be assumed to influence age ($B$) (Lines 234–243), while smoking behavior ($A_2$) is assumed to have no effect on age ($B$)?\n\n- Why is it reasonable to model the gene ($C$) as a binary variable?", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "- The paper is well-written and fluent.\n- It includes many examples that make the paper accessible.\n-  The assumption of the existence of a common cause variable $C$ is quite reasonable, and the perspective of resolving Simpson's paradox through this common cause is novel and interesting.\n- However, some of the additional assumptions about the common cause may not be realistic in real-world data, even in the examples discussed by the authors. Please see the questions for details.", "was_revised": true, "final_justification": "The author’s rebuttal has clarified my concern", "rebuttal": "We thank the reviewer for evaluating our manuscript and asking two pertinent questions. Below we respond to them.\n\nReviewer: Why can the gene $C$ be assumed to influence age $B$ (Lines 234-243), while smoking behavior $A_2$\nis assumed to have no effect on age $B$?\n\nResponse: A fairly general DAG for this situation would be\n$A_1\\leftarrow B\\leftarrow C\\to A_2\\to A_1$ and $A_2\\to B$, e.g. because\nonce $C$ can influence $B$, then potentially also $A_2$ can have a\ndirect influence on $B$.  At the present stage of our knowledge on pertinent\ngenetic and age-dependent factor influencing smoking, this\nDAG is not manageable. So we had to simplify it drastically.  \n\nFirst, we erased the link $A_1\\leftarrow B$, because the physical age by itself does not influence survival. It is true that for some people, the physical age correlates well with the physiological age, which can already alter survival. However, the physiological age in this experiment was not recorded or controlled. \n\nOnce we assumed $A_1\\not\\leftarrow B$, then from the viewpoint of\nSimpson's paradox it was already natural to assume $A_2\\not\\to B$ as\nwell, because the link $A_2\\to B$ does not influence $p(A_1|{\\rm\ndo}(A_2))$. At any rate, we emphasize that both $A_1\\not\\leftarrow B$\nand $A_2\\not\\to B$ are essential assumptions of the model. \n\nIt remains to explain in which sense the gene can influence the age.\nFor example, if an allele of a gene (see below) can be a common cause of\nboth smoking and (independently) smoking-generated deceases, then aged (but healthy) \npeople can be those which did not have this allele. \n\nReviewer:\nWhy is it reasonable to model the gene $C$ as a binary variable?\n\nResponse: A given gene can have many alleles (i.e. versions), but the\nclassical Mendelian genetics assumed (and in many instanced verified)\nthat as far the influence on the phenotype is concerned, one can\nrestrict the influence of a gene to the binary variable: recessive and\ndominant alleles. Each organism has two genes (one from mother and\nanother one from father), and now the three pairs - recessive-dominant,\ndominant-recessive, dominant-dominant - amount to one type of influence\nto the phenotype, while the version recessive-recessive to another\ninfluence. \n\nIn modern genetics, many exclusion from this classical binary-gene law\nare known. For example, the gene of the blood type has 3 alleles (A,B,\nand O). Here A and B are dominant with respect to O, but together they\nare co-dominant and hence there are 4 blood groups: AO, BO, AB, OO. \n\nTo understand whether the genes controlling the smoking behavior can be\nmodeled as binary (i.e. dominant and recessive), we need to consider\nconcrete genes, which according to current genetics research have\nserious effects on nicotine addiction and show evidence of pleiotropy,\ni.e., they can influence more than one aspect of health and survival;\nplease see The Tobacco and Genetics\nConsortium, \"Genome-wide meta-analyses identify multiple loci associated\nwith smoking behavior\", Nat Genet. 42, 441-447 (2010). \n\nCHRNA5 is a gene that encodes subunits of the nicotinic acetylcholine\nreceptor, which is important in neural signaling and nicotine addiction.  The\nreceptor can influence various aspects of smocking behavior: nicotine\nbinding and response, reward pathways, craving intensity, smoking\ncessation success rates, {\\it etc}; please see for a review: G. Lassi,\net al. \"The CHRNA5–A3–B4 gene cluster and smoking: from discovery to\ntherapeutics.\" Trends in neurosciences 39, 851-861 (2016). \n\nCHRNA5 has two alleles G and A.  They are denoted by G\n(guanine) and A (adenine), because the alleles differ by single\nnucleotide.  Now A is the risk allele, which is associated with\nincreased smoking. G is the non-risk allele. Now A is the dominant\nallele with respect to G, and CHRNA5 can be said to be binary with the\nfollowing reservation: there a dose-effect and the pair AA turns out to be\nmore risky than AG (in contrast to GG, which is risk free). Within our\ncrude model we neglect this difference and treat CHRNA5 as binary."}]

Author rebuttal:

Meta-review: **(a) Summary**  
The paper addresses Simpson’s paradox by assuming a minimal common cause, presenting the first resolution for Gaussian variables and extending known DAG-based cases. The work combines rigorous theorems with illustrative examples and situates itself relative to classical results by Lindley, Novick, and Pearl.

**(b) Strengths**  
- Novel treatment of Simpson’s paradox for Gaussian settings.  
- Clear exposition supported by examples.  
- Theoretical results are rigorous and connect to established causal inference literature.  
- Author rebuttal successfully clarified key misunderstandings.

**(c) Weaknesses**  
- Relies on strong assumptions, such as an unobserved binary common cause that may be difficult to justify in practice.  
- Contributions extend benchmarks but do not provide a fully general solution.  
- Some modeling simplifications may limit applicability.

**(d) Key reasons for decision**  
Reviewers initially had mixed views, with concerns about correctness and assumptions. After rebuttal and discussion, reviewers **y8Yk** and **J6Fc** revised their ratings upward from reject/borderline reject to borderline accept, citing clarified contributions. Reviewers **GFht** and **Ffck** remained supportive throughout. Overall, the Gaussian case and DAG extensions were recognized as meaningful advances, even if not general.

**(e) Discussion and changes**  
The discussion phase was productive: authors clarified alignment with do-calculus, addressed ADMG and exchangeability questions, and responded to practical concerns about modeling assumptions. While limitations remain, the reviewers’ final justifications reflect that these are acceptable trade-offs for a contribution of this scope.

**Decision: Accept (Poster).**  
Concise justification: The paper provides rigorous and novel extensions to Simpson’s paradox, especially for Gaussian settings, and reviewers’ concerns were largely resolved in discussion.
Decision: Accept (poster)