Identification and Estimation of the Bi-Directional MR with Some Invalid Instruments
Authors: Feng Xie
Zhen Yao
Lin Xie
Yan Zeng
Zhi Geng
Venue: NeurIPS 2024 oral (2024)
Keywords: Mendelian Randomization
Instrumental Variable
Causal Effect
Testability
Primary area: causal_inference

Abstract: We consider the challenging problem of estimating causal effects from purely observational data in the bi-directional Mendelian randomization (MR), where some invalid instruments, as well as unmeasured confounding, usually exist. 
To address this problem, most existing methods attempt to find proper valid instrumental variables (IVs) for the target causal effect by expert knowledge or by assuming that the causal model is a one-directional MR model. 
As such, in this paper, we first theoretically investigate the identification of the bi-directional MR from observational data. In particular, we provide necessary and sufficient conditions under which valid IV sets are correctly identified such that the bi-directional MR model is identifiable, including the causal directions of a pair of phenotypes (i.e., the treatment and outcome).
Moreover, based on the identification theory, we develop a cluster fusion-like method to discover valid IV sets and estimate the causal effects of interest.
We theoretically demonstrate the correctness of the proposed algorithm.
Experimental results show the effectiveness of our method for estimating causal effects in both one-directional and bi-directional MR models.

Identification and Estimation of the Bi-Directional MR with Some Invalid Instruments
Feng Xie', Zhen Yao', Lin Xie', Yan Zeng',,Aa Geng'
Beijing Technology and Business University
Abstract
We consider the challenging problem of estimating causal effects from purely ob- servational data in the bi-directional Mendelian randomization (MR), where some invalid instruments, well as unmeasured confounding, usually exist. To address this problem, most existing methods attempt to find proper valid instrumental vari- ables (IVs) for the target causal effect by expert knowledge or by assuming that the causal model is one-directional MR model. As such, in this paper, we first theo- retically investigate the identification of the bi-directional MR from observational data. In particular, we provide necessary and sufficient conditions under which valid IV sets are correctly identified such that the bi-directional MR model is iden- tifiable, including the causal directions of a pair of phenotypes (i.e., the treatment and outcome). Moreover, based on the identification theory, we develop a clus- ter fusion-like method to discover valid IV sets and estimate the causal effects of interest. We theoretically demonstrate the correctness of the proposed algorithm. Experimental results show the effectiveness of our method for estimating causal effects both one-directional and bi-directional MR models.
1 Introduction
Mendelian randomization (MR) is powerful method for estimating the causal effect of a potentially modifiable risk factor (treatment) on disease (outcome) in observational studies, which has been used across a wide variety of contexts, such as clinic [Mokry et al., 2015. Carreras- lorres ef all 2017 sociocconomics [C Cuullaar-Pardddact
2016, Brumpton all 2020J and drug targets |Ference all 201, Li et all 2020]. Most cxisting MR methods assume a one-directional causal relationship. However, bi-directional relationships are ubiquitous in real-life scenarios, for example, obesity and vitamin D status (Vimaleswares el all 2013), body mass index (BMI) and type 2 diabetes (T2D), etc. Understanding and analyzing bi-directional relationships can provide deeper insights into complex systems, leading to more effective interventions and solutions.
Within causal inference framework, MR can be implemented (IVs) analysis which genetic V serve as IVs for the modifiable risk factor. Specifically, as shown in Figure M. suppose obesity (X) and Vitamin D status (Y) are the treatment and outcome of interest, respectively, and lifestyle factors (U) are the unmeasured confounders between them. One use genetic variants G as IVs to explore the causal effect of X on Y in observational data [Vr maleswaran et all 2013 if G are valid IVs, i.e., G satisfy the following three assumptions [Wright Bowden and Turkington, 1990, [Heráan and Robins, 2006]:
form of instrumental variables
1928. Goldberger 1972
A1. [Relevance] The genetic variants G are associated with the exposure X;
A2. [Exclusion Restriction] The genetic variants G have no direct pathway to the outcome Y:
"Corresponding author
"The genetic variants that act as IVs in MR studies are Single Nucleotide Polymorphisms (SNPs).
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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A3. [Randomness] The genetic variants G are uncorrelated with unmeasured confounders U. Figure 11 illustrates these three assumptions for ge- netic variant G that is a valid IV relative to the causal relationship X + Y. Given valid IV G in the lin- ear model, one can estimate the causal effect of risk factor X on the outcome Y of interest consistently by using two-Stage Least Squares (TSLS) [Burgess et all 2017]- However, in most MR studies, the ge- netic variants may include some invalid IVs due to horizontal pleiotropy- where some genetic variants may have pleiotropic effects on both the treatment X and the outcome Y via different biological path- ways (Davey Smith and Ebrahum, MIXK, el 2008]. How to select valid IV's from such genetic vari- ants becomes the major issue in the MR study. In general, valid IVs need to be chosen based on domain knowledge or expert experience [Kichmond ef all 2017, Martens all, 2006, Zhao et all 2123.. However, it is usually difficult to select appropriate valid IVs without precise prior knowl- edge of genetic variants, and an invalid IV may cause biased estimation of the effect of X on Y [Sanderson et all, 2022, Richmond and Smith, 2022]. Therefore, it is desirable to investigate ways of selecting genetic variants as valid IVs solely from observed variables.
Lifestyle Factors
SNP. (A5) TAI)
Vitamin Status
Ohecity (A2)
Graphical illustration of a valid IV model, where dashed lines indicate the absence of arrows. G is a valid IV relative to the causal relationship X + Y.
Figure 1:
Many contributions have been made to handle MR studies with some invalid IVs under specific as- sumptions. Existing methods can be roughly divided into four strategies. The first strategy assumes the InSIDE assumption, which states that the genetic variants' pleiotropic effects on the outcome Y uncorrelated with their effects on the exposure X, and requires that the number of IVs increases with the sample size [Bowden et all 2016 The second strategy is based on the Majority rule as- sumption, which assumes that over 50% of the candidate IVs are valid [Kang et ald 201G, Howden et all. 2016. Windmenjer et all 2019, Hantord et ali, 2021J. The third strategy assumes the Plurality rule assumption, which means that the number of valid IVs is larger than any number of invalid IVs with the same ratio estimator limit [Guo et all 2018. Windmeijer et all, 2021]. The last strategy assumes the Valid IVs assumption, which requires that the number of valid IVs is greater than or equal to 2, and the Rank-faitffulees assumption, which states that the probability distribution P rank-faithhfe to a directed acyclic graph G if every rank constraint on a sub-covariance matrix that holds in P is entailed by every linear structural model with respect to G [Silva and Shimiuu, ααι eta 2022 Cheng el all RUPK) Although these methods have been used in various fields, they assumed that the one-directional causal relationship between X and Y is known and may fail to handle scenarios where bi- bi-directional relationships are present.
Recently, Darrous et all [2021]] proposed latent heritable confounder MR method to estimate bi-
directional causal effects with hierarchical prior on the true parameters. However, their method is sensitive to prior misspecification. Later, Liand Ya [2022] introduced a focusing framework to test bi-directional causal effects under the noise distribution of the effect estimates assumption, which is exciting. However, this assumption is strictly parameterized (See Condition 1 in [LLand Ye, 2022] for more details). Moreover, they focus on the two-sample MR model that assumes homogeneity across samples, whereas we consider the one-sample MR model. See Appendix A for more details.
In this paper, we aim to investigate the identifiability of the one-sample bi-directional MR model and to provide an estimation method. Specifically, we make the following contributions:
1. We present sufficient and necessary conditions for the identifiability of the bi-directional MR model, enabling both valid IV sets for each direction and the causal effects of interests to be correctly identified.
2. We propose a practical and effective cluster fusion-like algorithm for unbiased estimation. This algorithm correctly identifies which IVs are valid, determines the causal directions which the valid IV sets belong, and estimates the bi-directional causal effects.
3. Extensive experiments demonstrate the efficacy of the proposed algorithms in MR models.
2 Model Definition and IV Estimator
2.1 Bi-Directional MR Causal Models
Suppose we have n independent samples (X,, Y., G.) from the distribution of (X, Y. G), where X R and Y e IR is pair of phenotypes of interest, and G (G1, E 12 denotes

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measured genetic variants, which may include invalid IVs. Analogous to Li and Ye [2022], for each sample, indexed by the phenotype X, is taken a linear function of the phenotype Y, genetic variants G. and an error term ex,, while phenotype Y, is modeled as a linear function of the phenotype X, genetic variants G, and an error term ey;- Without loss of generality, we assume that all variables have a zero mean (otherwise can be centered). Specifically, the generating process of the data is given below:
X - + T + == XBx-x++++ + GTyy +
where - is the causal effect of Y on X and - is that of X on Y. 7X ((xx,1,-... Tx..)" and yy- = (,,, )Y are the direct effects of genetic variants G on X and Y. respectively. Note that xx and σy are error terms that correlated due unmeasured confounders U between X and Y. Besides, there may exist a single genetic variant G, E G with both 7x,, of O and yy, A O, which implies that G3 also has direct pathway the outcome, violating Assumption A2 [Exclusion Restriction]. Like Li and Yd [2022]], we here restrict the independence between any two genetic variants in G (i.e., genetic variants are randomized). In Section EL we discuss the situation where this restriction is violated. Unlike Li Ye [2022]], we allow genetic variants to cause unmeasured confounders (partially violating Assumption A3 [Randonness|).
Following Hausman
Then let Eq. (II can be reorganized to avoid recursive formulations as follows:
[183]] we assume that 8x- * 1.
1/(1 - Bx
x -G + GT + -xx
Y = xG + GT + +exxx-y + E
If By = 1, the causal effects 3x-Y and By - are not identifiable, even in the presence of valid IV.
Goal. In this paper, we aim to address whether the bi-directional MR model in (q.()) is identifi- able; that is, whether we can estimate the causal effects Bx-- and ,y given infinite data, even without prior knowledge about which candidate IVs are valid or invalid.
2.2 IV Estimator
In this section, we briefly describe the classical IV estimator, the two-stage least squares (TSLS) estimator, which is capable of consistently estimating the causal effects 3x and By x of interest in the above bi- di-directional MR causal model of (q.(I) when the valid IVs are known [Wooldndge 2011 We denote by G - the set consisting of all valid IVs in G relative to X Y. and G
the set that consists of at least one invalid IV in G relative to X + Y. Similar notations are applied to the inverse causal relationship
X, i.e., G - and G -Y-
Proposition (Two Stage Least Square (TSLS) Estimator). Assume the system is a linear bi- directional causal model 7 For given causal relationship X + Y in the system, the causal effect of X on Y can be identified by
= - X Bx-
where P (G (GS--)) GĀ- is the projection matrix.
Remark 1. Note that, when utilizing the TSLS estimator, the causal effect is biased if the candidate set G includes invalid IVs. For instance, consider the causal relationship X < Y. given an invalid IV set G C--Y the causal effect of X on Y is given by
8x = [x-xxx]x] X = +
X XTP(GTTY +ey),
Boia
where P- (G (GX G
In the remaining of the paper, we denote such a procedure by _x-Y- = TSLS(X, G.-Y)). Similarly, for causal relationship Y + X, one can also have the corresponding causal effect of Y on X given the set of IVs, i.e., By = TSLS(Y, X, GY-X) For details about the proof of Proposition II and Remark MI please refer to Appendix G and G7
3

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Identifiability of Bi-Directional MR Model
In this section, we first show by a simple example that the valid and invalid IV sets can impose different constraints. Then, we formulate these constraints and present the necessary and sufficient conditions that render the model identifiable.
3.1 A Motivating Example
Figure 0 gives illustration of our basic idea. Suppose there are 2 valid IVs, ie., G1 and G3. and invalid IVs, Le., G2. G4 and Gs. relative to the causal relationship X - Y. For the inverse + X, identical conclu- sions be derived. Assume the generating process of X and Y follows Eql where specifically we set all variables to have zero
G₁
G2
G3
and unit variance and
G.
X
G5Tx,5 and Y - =X +
Gs
GsTY,5 + where 3x-Y = +x = 0.6, 7x,1
Figure An illustrative example where valid and invalid IV sets induce distinct constraints, where G is valid (G2, G4, G5)T T is invalid due to pathways G2 + Y, G4 + Y and G3 + Y.
7x.2 7x.3 7x,4 = 1.8, 7x.5 1.2,
0.5, TY,1 = 0.6, 7v.3 = 0.3.
(G1. Gs) set, while G
Interestingly, we have the following observations:
Observation Given valid IV set where there are at least two IVs, e.g., G (G1, G3)T, we have:
corr(Y Xwias), G1) O,
corr(Y- - Xwia G3) = =0,
where corr(-) denotes the Pearson's 's correlation coefficient between two random variables, and W[G.) TSLS(X, Y, (G:)) with i E [1,3}.
Observation 2. However, given an invalid IV set, egg. GĀ-Y = -G2, G4, Gs) we have: corr(Y XW. G2) * 0, corr(Y - G.) G4) + 0, corr(Y Xwi Go) + 0, where W(G,,G,) TS,, (G,, G21) with + j and i.j E f2, 4,.
Based on these two observations. we can see that the valid and invalid IV sets entail clearly different constraints regarding the correlation coefficients,, which inspires tion analysis. For details about the observations, please refer to Appendix C
further derive our identifica-
3.2 Main Identification Results
We begin with the following definition and the main assumptions. Definition 1.
Residual) Let G be subset of candidate genetic variants. A pseudo-residual of (X,Y) relative to is defined as
PR(XYIG) == - -XWG.
(4)
where we TY,
This "pseudo-residual" is applied to measure the uncorrelated relationship with some genetic vari- ants. Note that concepts "pseudo-residual" have been developed address different tasks [Drioon and Rtchardsoni 20AA, Chen et all 2017, Cai et all, 2019, Xie el all 2020], but our formalization is different from theirs in terms of the parameter wag. To the best of our knowledge, it has not been realized that the uncorrelated property involving such pseudo-residuals reflects the validity of the IV's in the bi-directional MR causal model (see Propositions
B)).
As analyzed by Chu all [(лл]]], within the framework of linear models, a variable being a valid IV imposes constraints the joint marginal distribution of the observed variables. In other words, there is no available test to determine whether a variable a valid IV without making further assumptions. Therefore, introduce the following assumption.
Assumption (Valid IV Set). For given causal relationship (if the relationship exists), there exists a valid set that consists of at least two valid IVs. For example, for the causal relationship X - Y. G +Y 2 2.
Pleasse note that these constraints be tested easily by observed data.

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Assumption 01 is the same as in [Silva and Shimizu, 201 one-directional MR model. Notice that this assumption much milder than the Majority rule as- sumption [Kang et al,, 2016 Bowden et all 2016, Windmeijer et all 2019, Hartford et all, 2021]] and Plurality rule assumption [Guo all 2018, Windmeijer et all, 2021]: Assumption does not rely on the total number of genetic variants, while Majority rule and Plurality rule assumptions do.
Cheng et al, 202 if the model is
We now show that under Assumption I. one can identify the class of invalid IV sets.
Proposition 2 (Identifying Invalid IV Sets). Let G {GI..... Gp3. 2 2 be a subset of candidate genetic variants G. Suppose that Assumption I and the size of participants + oo hold. If there exists a G3 G such that,
coIr(PR(x,y 18(G,) G0)+0, #0,
(5)
then G is an invalid IV set for any one of the causal relationships.
Proposition is valuable when given a set of genetic variants but lack prior knowledge of the validity of IV sets. By testing specific correlations in Eq,(S), we conclude that the current IV set is invalid.
Example 1. Consider the example in Figure A Suppose G = (G1, G2), according to Proposition have ( G1) # O. Thus G is invalid IV set.
One may naturally raise the following ques- tion: Can all the invalid IV sets be identified by Proposition ?? In other words, can we identify all the valid IV sets? Unfortunately, Assump- tion iss insufficient condition for identifying the sets valid IVs. See an illustrative exam- ple below.
G₂
G1
U
G1
G1
(a)
Example 2 (Counterexample). Consider the ample in Figure & Suppose G = [G1, G2). In subgraph (a), for G1 G, according Propo- sition 2 we obtain
Figure 3: An illustrative example that valid and invalid IV may induce the same constraints in (5.(5) of Proposition R. where {G1, G2 } is the set of valid IVs in (a), while invalid in (b). It implies Assumption M0s is not sufficient to find valid IV sets.
CorX(PR(X, -1)=
(6)
Similarly. for G2 E G, we have
corr (ox(PR(XX G2) =
In subgraph (b), for G1 E G, we check the condition of Proposition D and obtain
(or(PR 7y,1"X,2 X
(8)
8Y 7x,2
Similarly, for G2 e G, we have
cor(PR(xx1((.))) G2) Tr,7XX11 7Y,17XX2
(9)
8Y 7X,1
We know that (Gi, G2} is invalid IV set in this case. However, if the proportion of effects of Gi on X and is equal to the proportion of the effects G2 on and Y, i.e., 7x.1 = 71,7x,2 we have
corr CorT(PRRX.Y (G2)), =,
(P (G.)) G2) 0,
which is consistent with the conclusion of subgraph (a). These observations imply that one can not identify valid IV sets under only Assumption 0
To give sufficient conditions for identifying valid IV sets, we first introduce the following assump- tion.
Assumption (Generic Identifiability. For given MR causal model, parameters 7 and live in set of Lebesgue measure non-zero.
5

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In other words, Assumption D specifically implies that given a general subset of candidate genetic variants G, there exists a G3 G that satisfies Ea,es/a, B,cov(G,, + "U,
myo - (v. (7x.y + + O, where 3; & is the estimated coefficient be- tween G and X. For instance, Example Q. such an assumption simplifies to 7x,17Y: A TY, 7x,2- Whenever 7x,17y2 = 7x, parameters y and / would fall into set of Lebesgue measure zero, rendering non-identifiable of the model. It is worth noting that in real life, although we cannot ob- tain the ground truths of causal effects between variables, especially when latent confounders involved, Assumption Q is generally satisfied. This is because the set of conditions to violate it occupies very small portion of the entire space.
With Assumption C. we now derive sufficient conditions for generic identifiability in Proposition Proposition 3 (Identifying Valid IV Sets). Let G {GI..... Gp} P 2 2 be a subset of candidate genetic varianty G. Suppose that Assumptions 0 and B and the size of participants n. hold. If each G3 E G satisfies
CoIT(PR(X,y (SA,)) -3)=0,
(10)
then G is proper valid IV for one of the causal relationships, ie., G or G G
Proposition E indicates that, under Assumptions I and D we are able to identify subset that consists of valid IVs for one of the relationships, from the candidate genetic variants G. Here, it should be noted that in bi-directional MR model (with both 9x-Y * O and BY o 0), one cannot determine whether the identified IV set is related to the causal relationship X Y or Y - X. To address this, we here introduce the following conditions that allow the estimated valid IV sets to determine the direction.
Assumption 3. In a bi-directional MR model, for a given causal relationship X + Y. & Var(X) < Var(Y). Similarly, for given causal relationship Y X, B2_x Var(Y) I Var(X).
Assumption 3 is very natural condition that one expects to hold for the unique identifiability of valid IVs for the bi-directional model. This assumption is often reasonable in MR models as discussed in Xue and Pan [2020]] One may refer to Pages 4~5 in Xue and Pan [p0] for more details.
Proposition 4 (Identifying Direction of Causal Influences). Suppose Assumption holds. Given valid IV set denoted as G. if each Gj E G satisfies the condition |corr(G, Y) then G is a valid IV set for the causal relationship X - Y, i.e., G C G condition is not satisfied, G is valid IV set for the causal relationship Y + X, i.e., G C G -X
/////oorGr.... (o, < 1, X +Y Conversely, if this
Proposition 4 indicates that, under Assumption 3, it is possible to ascertain the directional orientation of the identified IV sets. Thus, with Propositions 2-- we have sufficient and necessary conditions under which the bi-directional MR model is fully identifiable, formalized as follows:
Theorem (Identifiability of Bi-directional MR model). Suppose that Assumptions W -~~ E and the size of participants n - oa hold, the Bi-directional MR model is fully identifuble, which includes the identification of valid IV sets, the causal relationships the idenisiid sets relate to, the causal directions between X and Y, and the causal effects of X on Y as well as Y on X.
4 PReBiM Algorithm
In this section, we leverage the above theoretical results and propose a data-driven algorithm, PRe- BiM Sseudd-Residuall contains two key steps, finding valid IV sets (Step I), and inferring causal direction and estimat- ing the causal effects given the identified valid IV sets (Step II). The algorithm is designed with the following rules, which are proved correct: (1) valid IV sets can be correctly discovered given the measured genetic variants (Section H... and (2) the causal direction and causal effects can be uniquely identified (Section The entire process is summarized in Algorithm MI
Bi-directional MR), for estimating the causal effects of interest. It
Algorithm 1 PReBiM
Input: A dataset of measured genetic variants G = (GI,..., Gy) two phenotypes X and Y, significance level and parameter W, maximum number of IVs to consider.
1: Valid IV sets + VandValiddVSettsGG, X. Y, a, W);
> Step I b Step II
2: (8x-y,,8y- + InferCauss/DDreettofffeeeXXXXXX Y. V):
Output: 3x and 4x. the causal effects of X on Y and Y on X, respectively.

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4.1 Step I: Finding Valid IV Sets
Propositions and & have paved the way to distinguish the invalid and valid sets, respectively, This enables us to use a cluster fusion-like method to discover the valid IV sets To improve the algorithm's s effectiveness with a finite sample size, especially when the pool of candidate genetic variants G is substantial, we introduce parameter W, representing the maximum number of instru- mental variables (IVs) to consider. This approach aims to circumvent the necessity of identifying the largest IV set within the collection, thereby minimizing the need for IV validity testing.
The basic process is provided in Algorithm Due to limited space, the specific details of algorithm execution are provided in Appendix EL In practice, we check for the valid IV test, i.e., Eq. (I,), with the Pearson Correlation Test using the Fisher transformation [Anderson. 2003), denoted by PCT.
Algorithm 2 Find
Input: A dataset of measured genetic variants G = (Gi,..., Gg)T, two phenotypes X and Y, significance level a, and parameter W, maximum number of IVs consider.
1. Initialize valid IV sets V and G = G.
Find Vnew with |G] 2 according minimize the sum of correlation between pseudo-residual P G)))) and G,, where G, E G and G E G.
3. Update Vnew by incrementally adding genetic variant GR until it is impossible to add variables without passing PCT or the length of the set Vnew reaches our predetermined threshold W.
4. Repeat Step 1 and Step 2 to find another V If V is empty, add V to V. If not, and the combined set (V U yeew> passes the PCT, merge them; otherwise, discard Voew- Stop the iteration when V contains
sets of IVs or when there is fewer genetic variants left in
G.
Output: V. set that collects the valid IV sets.
4.2 Step II: Inferring Causal Direction and Estimating the Causal Effects
In this section, we infer the causal direction and estimate the causal effects given the identified valid IV sets. The basic process is provided in Algorithm B. Due to limited space, the specific details of algorithm execution are provided in Appendix E
Algorithm 3 InferCausllliircciionffeesss
Input:
Two phenotypes X and Y. Valid IV sets V.
1. Infer the causal direction corresponding to the valid IV set V based on Proposition
2. Calculate the final causal effects 3x-x and 3y by TSLS(.).
Output:
3x-YY and 4x
4.3
Correctness of PReBiM Algorithm
In this section, we show that, in the large sample limit, for the causal relationships of interest X ++ Y, the causal effects 3x-Y and B obtained through the proposed method are unbiased. Theorem (Correctness). Assume that the data {X.Y) and G strictly follow Eq.(Il) and Assump- tions W ~ B hold. Given infinite samples, the PReBiM algorithm outputs the true causal effects Bx and B correctly.
4.4 Complexity of PReBiM Algorithm
We give the computational complexity analysis follows. First, the complexity of find- ing a valid IV set Vnew of length 2 O (2()).. where g is the number of IVs. Then, (12--W-2220)) (2g-W-21(W1))
the complexity of expanding Vnew to the length W is O
where W is the maximum length of the IV set. For finding additional valid IV sets, the number of loops, i, must satisfy 1 < W. The complexity of finding the second valid IV set of length W is given by Ei (2(0-bW) 19-(2--1)-2)(0-
1))++
2
A cluster is considered a valid IV set, and the term Musion-like' refers to the specific process of identifying and merging these clusters.
7

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Therefore,
complexity of the PReBiM Ek=o 2 2 + (2g- (WW-(( + (2 (t--1) +1)
the total
computational
algorithm
is
2
5
Discussion
The preceding sections demonstrate that a valid IV set can be identified when any two genetic variants in G are independent. In this section, we explore whether the main results of Propositions D and B remain appli- cable when some genetic variants in G are dependent. As shown in Figure (b), we evaluate the condition of Proposition 2 and find that corr (Or((PR
G2
G2
G1
(a)
G1
(b)
G1) + O and co.((P........) G2) # 0. This indicates that G2] is an invalid IV set. Conversely, in Fig ure 4 (a), when assessing the condition of Proposition
Figure {G,, G2] is a valid IV set in (a), whereas invalid in (b).
4: Simple examples, where
G we observe that corr(PR (X.)))))) G1) = 0 and (P G2) = O, suggesting that G2) is a valid set. Consequently, is interesting to note that even with such dependence between genetic variants, our main results may still be effective in identifying valid IV sets. In Ap- pendix H+ E we further demonstrate the practicality of our method for this example. As for the complex scenarios where there is confounding among genetic variants or between the genetic variants and some phenotype, or where there exist genetic variants that are associated with another phenotype while this phenotype causes X and Y, leave them into our future work.
6 Experimental Results
In this section, we conduct various simulation studies to evaluate the performance the proposed PReBiM method. The methods compare against are: 1) NAIVE, the least-squares regression
method; 2) MR-Egger [Bowden et all 2015:: : 3) sisVIVE algorithm [Kang et all, 2016) B. 4) IV-
TETRAD algorithm(Ssiva and Shimizu, 2017 B. 5) TSHT algorithm [Gmoo_eta -uoo_eLal_ all 201] D. and 6)
PReBiM, our method. The second method, MR-Egger, is specifically for MR studies with invalid IVs. The last four methods capable of handling the one-sample MR model in the presence of unmeasured confounders. Our source code can be found in the Supplementary Materials.
Experimental setup. To compare the performance of these methods in realistic setting, analogous to Slob and Burgess [2020], the genetic variants are modeled as Single Nuclcotide Polymorphisms (SNPs), with varying minor allele frequency maf; j. and take values 0, 1, 2. The minor allele frequencies are drawn from a uniform distribution. Specifically, the data generation process for the
bi-directional model is follows:
U = GT +1,
(11)
X Y + G + UNxUU +
(12)
Y = XBx+Y + GTyy + UTY, + +
(13)
Gū Binomial maf ,), maf, ~ ((0.,,0.5)
(14)
where the error terms E1, 62, Ea each follow an independent normal distribution with mean 0 and unit variance. The causal effects ,y and xx are generated from uniform distribution between
[-1, -0.5] U [0.5, 1]. Note that by setting B = 0 in (I)) the data generation process for the one directional model can be obtained.
We here conducted experiments with the following tasks.
T1: Sensitivity to Sample Size. We evaluated the impact of different sample sizes: 2k, 5k, and 10k, where & equals 1, 000.
'For sisVIVE algorithm, algooiin, we used the implementations the sis VIVE package, which can downloaded at https:// cran pro ject. wwtoojecttrg///eeeeeeeeoo
'For IV-TERRAD method, we used the implementations in the R package, which be downloaded
https://wwx
http://www..oeppgesss homepages....................... ucl.ac. ac. u&/ wwggttdd/cooee/eeeeeeoooooe discovery/
For the TSHT algorithm, we used the implementations in the R RobustIV package, which can be down- loaded at https::/ cran. r-raaannrrroeopoeeeeeeee perttetttt.....

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T2: Sensitivity to the Number of Valid IVs. We explored three distinct IV combination scenar- ios: S(2, 6). S(3, 3, 8), and S(4, 10). The first two elements in the parentheses represent the number of valid instrumental variables for causal relationships X - Y and Y + X, respec- tively. The last element in the parentheses represents the total number of genetic variants.
For a comprehensive analysis, we examined the data from the Bi-directional MR model and the One-directional MR model, detailed in Section 6 and 6.2, respectively. For each experimental setting, we repeated the simulation 500 times and averaged the results.
Metrics: We adopted the following evaluation metries to evaluate the results of all methods.
(i) Cerrect-Selecting Rate (CSR): the number of correctly identified valid IVs divided by the total number of genetic variants, for the direction X - Y or Y - X.
(ii) Mean Squared Error (MSE): The squared difference between the estimated causal effect and the true causal effect.
To further validate the effectiveness and practicality of our algorithm, we have expanded our ex- perimental settings to include additional scenarios such as small sample sizes, different numbers of valid IVs, large-scale datasets, and cases where all IVs are valid. Within these settings, we com- pare our algorithm with the five aforementioned methods to assess algorithm performance. Detailed experimental results are provided Appendix
6.1 Evaluation on Bi-Directional MR Model
In this section, we evaluate the performance of six methods on bi-directional MR data, i.e., 3x-Y * O and 3y # O. Dable I summarizes the results of metrics CSR and MSE in bi- directional MR model with various sample sizes. is noted that VIVE, IV-TERRAD, and TSHT are not specifically designed for the bi-directional model, so we run them in both forward and re- verse directions to obtain the corresponding results. As expected, our proposed PReBiM method gives the best results in all three scenarios, with all sample sizes, indicating that it can identify valid IVs and derive consistent causal effects from purely observational data, even without knowledge of the causal direction between X and Y. Furthermore, the performance of the other three comparison methods of selecting IVs was poor, indicating that they are not suitable for bi-directional MR models. The MSE of the NAIVE method does not tend toward O, indicating that if unobserved confounding not considered, the estimated causal effect certainly biased. Similarly, the MSE of MR-Egger is poor if estimated with all instrumental variables.
Table 1: Performance comparison of NAIVE, MR-Egger, SiSVIVE, IV-TERRAD, TSHT, and PRe- BiM in estimating bi-directional MR models across various sample sizes and three scenarios.
8(2, 6)
313,3.8)
S14. 4, 10)
Size
Algorithm
CSR
MSE,
CSR
MSE,
CSR
MSE,
CSR
MSE.
CSR
MSE
CSR
MSEJ
NAIVE
0.354
0.349
0.351
0.335
0.304
MR-Egger
0,800
0.734
0.571
0.569
0.457
0.494
2k
SisVIVE.
0.30
0.398
0.25
0.447
0.34
0.379
0.32
0.379
0.37
0.346
0.38
0.330
IV-TETRAD
0.60
0.859
0.30
0.867
0.60
0.773
0.28
0.782
0.67
0.688
0.23
0.622
TSHT
0.07
0.606
0.06
0.660
0.07
0.549
0.07
0.613
0.12
0.653
0.11
0.689
PReBiM
0.070
0.083
0.87
0.58
0.039
0.057
NAIVE
0.350
0.339
0.347
0.119
0.357
0.306
MR. Ugger
0.737
0.520
5k
SisVIVE
0.32
0.421
0.30
0.407
0.33
0.378
0.35
0.384
0.39
0.439
0.38
0.365
IV-TETRAD
0.62
0.804
0.844
0.836
0.29
0.659
0.67
0.686
0.25
0.640
TSHT
0.04
0.544
0.06
0.565
0.08
0.528
0.06
0.570
0.10
0.549
0.07
0.541
PReBiM
0.93
0.020
0.91
0.027
0.90
0.019
0.92
0.020
0.90
0.009
0.91
0.010
NAIVE
0.366
0,348
0.319
0.342
0.349
0.300
MR- Egger
0.800
0.763
0.524
0.620
0-487
0.424
10k
SisVIVE
0.31
0.457
0.29
0.503
0.37
0.383
0.31
0.394
0.41
0.399
0.41
0.328
IETTRD
0.64
0.811
0.31
0,804
0.64
0.748
0.29
0.696
0.71
0.635
0.24
0.577
TSHT
0.449
0.475
0.525
0.05
0.07
PReBiM
0.95
0.044
0.93
0.018
0.93
0.026
0.93
0.039
0.93
0.027
0.94
0.011
Note: The symbol
means methods have output. "1" means higher value is better, and vice versa.
6.2 Evaluation on One-Directional MR Model
In this section, we evaluate the performance of six methods on one-directional MR data, e.g.. 3x-YY * 0 and 3y = 0. For fairness, we assume that the causal direction X >> Y known for all methods. Figure summarizes the results of metrics CSR (solid lines) and MSE (dashed

---

lines) for all methods in one-directional MR model with different scenarios. As expected, the metries CSR and MSE of our method approach the expected value as the sample size increases. The overall results are comparable to the IV-TETRAD algorithm and superior to the sisVIVE algorithm and TSHT algorithm. This is because our method, similar to the IV-TETRAD method, requires fewer valid IVs in the system (at least 2), whereas the other two methods require the number of valid IVs that are related to the overall genetic variants, namely the majority rule assumption (over 50% of the candidate IVs are valid) and plurality rule assumption (the number of valid IV' is larger than any number of invalid IVs). It has been noted that under the IV-TETRAD method for scenario S 0, 6), the metries CSR and MSE outperform our method. This is because the IV-TETRAD method requires a predetermined number of IV sets to be specified in advance, and in this case, we have predetermined the number of valid IVs for it. However, our method does not have this prior information.
0.5
1.0
0.2
0.8
0.25
033
0.20
0.0
0.8
INTETRAD
TSHT
0.1
0.7
0.02
34
10c
20
Sample
Stzes
Sample Sizes
Sample
(a) S(2 0, 6)
(b) S(3 0, 8)
(c) S(4, 0, 10)
Figure 5: Performance comparison of SisVIVE, IV-TETRAD, TSHT, and PReBiM in estimating one-directional MR models across various sample sizes and three scenarios.
6.3 Evaluation on Real-World Data
We first apply our method to analyze the bi-directional causal relationships between obesity and vitamin D status using the GWAS data from Vimaleswaran et al [20[]] Note that the original data only consists of summary data, and thus we generate individual-level data based on the data generation mechanism described in Eq.(I.. The data we used comprises 16 related SNPs as can- didate IVs, including seven SNPs associated with obesity (BMI): rs9939609 (FTO). rs2867125 (TMEM18), rs4074134 (BDNF), rs7647305 (ETV5), rs71 (FAIM2), rs3101336 (NEGR1), and rs 1 09388397 (GNPDA2); four SNPs related to 25-hyyrroxyvitamin D (25(OH)D): rs12785878 (DHCR7), rs 1577411657 (CYP2RI), rs2282679 (GC), and 386013897 (CYP24AI); and five SNPs as- sociated with BMI and 25(OH)D: r84498364, rs r8138166 1660, rs4549702, rs12760109, and 1877913838. We observed that FTO and FAIM2 serve valid IVs for the X + Y. while DHCR7 and CYP24A1 are valid IVs for the Y - X, with estimated causal effects of -1.15 and -0.05, respectively. These findings are consistent with those findings in |Vimaleswaran ala 20T3).
Next, we apply our method to analyze the causal effect of institutions on economic development using the Colonial Origins dataset from [Acemoglu et all, 2001]. The data includes eight candidate IVs: absolute latitude (lat_ abst), the density of European settlements in 1900 (euro 1900), the level of democracy in 1900 (democl), constitutional constraints in 1900 (consl), the timing of industri- alization (indtime), democracy in 2000 (democ00a), constitutional constraints in 2000 (cons00a), and fog-transformed European settler mortality rates (logem4). Our method identifies euro1900 and logem4 as valid IVs, with an estimated causal effect of 0.861, which is consistent with the findings reported in Acemoglu et al. [200T].
7 Conclusion
We investigated the identifiability conditions for the bi-directional MR model using purely observa- tional data, First, we proposed a testable constraint for identifying valid IV sets, and we demon- strated that, under appropriate conditions, these valid IV sets can be partially or even fully identified by checking the proposed constraints. Experimental results on both datasets further verified the ef- fectiveness of our algorithm. Currently, we assume that genetic variants are randomized. Allowing for general dependent genetic variants is a direction for future work. Additionally, while we focus on linear bi-directional MR model, in future research, we will generalize to address nonlinear or even more complex data.
10

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Acknowledgements
Feng Xie would like to acknowledge the support by the National Natural Science Foundation of China (62306019). Yan Zeng would like to acknowledge the support of the Beijing Municipal Ed- ucation Commission Science and Technology Program General Project (KM20241001 1016). Feng Xie and Yan Zeng acknowledge the support by the funding from the Disciplinary funding of Bei- jing Technology and Business University. the Beijing Municipal Education Commission for the Emerging Interdisciplinary Platform for Digital Business at Beijing Technology and Business Uni- versity, and the Beijing Key Laboratory of Applied Statisties and Digital Regulation. Zhen Yao acknowledges the support of the Graduate Research Ability Enhancement Program Project Funding Beijing Technology and Business University in 2024. We appreciate the constructive comments from all anonymous reviewers, which greatly helped to improve this paper.
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A More Details on MR
The basic goal of epidemiologic studies is to assess the effect of changes in exposure on outcomes. The causal effect of exposure on outcome is often different from the observed correlation due to con- founding factors. Correlations between exposure and outcome cannot be used as reliable evidence for inferring causality. In contrast, Mendelian Randomization (MR) studies use genetic variants to infer the causal effect of exposure on outcome. The idea behind MR studies is to find genetic variant (or multiple genetic variants) that is associated with exposure but not with other confounders and that does not directly affect the outcome. Such a genetic variant in principle fulfills the basic as- sumptions of instrumental variables (IVs). It is then used to assess the causal effect of the exposure on the outcome [Lhomma and Contr 200E Since the assumptions of IVs cannot be fully tested and may be violated during IVs' selection, number of methods have emerged to identify valid IVs.
MR studies can be conducted using a single sample, i.e., one-sample Mendelian Randomization, in which individuals are tested for genetic variation, exposure, and outcome in the same population. On the contrary, in two-sample Mendelian Randomization, the association between genetic vari- ation and exposure is estimated in one dataset while that between genetic variation and outcome is estimated in another dataset. Compared with the Two-sample MR, one-sample MR provides more direct evidence by eliminating potential biases that can arise from individual-level data, which turns out to be more meaningful yet challenging.
Additionally, most existing MR methods assume a one-directional causal relationship between expo- sure and outcome, whereas bi-directional relationships are ubiquitous in real-life scenarios. For in- stance, there exist bi-directional relationships between obesity and vitamin D status [Vimal [Vimalesward eswaran atll 2013) body mass index (BMI) and type 2 diabetes (T2D), diastolic blood pressure and stroke [uu and Pan 2022]. insomnia and five major psychiatric disorders [Gao et all 2011), smoking and BMI [Carreras- lorres all 2018), etc. It is thus desirable take further research on bi-directional MR. Bi-directional Mendelian Randomization assesses not only the causal effect of the exposure on the outcome but also the effect of the outcome on the exposure. To achieve this, valid IVs for both the exposure and the outcome are needed. However, this is tough because genetic variants associated with the exposure are often also associated with the outcome, and vice versa. Our goal here is to identify and differentiate valid IVs for each direction within one-sample MR framework and infer bi-directional causal effects.
Regarding our bi-directional MR causal model, we give further relative descriptions in the following. (i) Similar to Li Yo [2022], our model in Eq. (II) describes bi-directional causality in equilib- rium, and in econometries, it is known as the simultaneous equation model. According to HRUsman [1983], to avoid the recursive formalization, (II) equivalently is transformed into Eq. with By * 1. (ii) For such bi-directional (cyclic) structure, Mooij et al. [20]]]] take an Or- dinary Differential Equation (ODEs) perspective and show under which conditions the equilibrium state of system of the first-order ODEs can be described by deterministic structural causal model (SCM). This is a very interesting work. Please note that our setting is different from those in Moorj aa [201]] our model is motivated by bi-directional MR studies, other than dynamical system from the control field. Further, due to the specificity of MR studies, we can only obtain the snapshot data or equilibrium data, other than dynamical ones for the treatment, the outcome well as genetic variants.
14

---

B
Notation List
Table 2: List of major notations with descriptions.
NOTATION
DESCRIPTION
U
Unmeasured Confounders
G
Genetic variants
G C
Candidate genetic variants
X
Exposure or treatment
Y
Outcome
Sample size
7X. T
Effects of G
on and of G on respectively
A
reduction factor calculated with 1 -8
9x.6Y
Error terms of X, and of Y, respectively
8x-Y,,8y-
True effects of X on Y, and of Y on X, respectively
8x-Y,, 8x
Estimated causal effects from candidate IVs for the directions X - Y and Y - X,
respectively
GX-Y G -
set consisting of all valid IVs relative to -> Y, and Y + X, respectively A set consisting at least one invalid IV in G relative to X - Y or Y - X, respectively
G
-Y-Y"
G7 -
TSLS(-)
The procedure estimating causal effects using two-stage least square method Estimated causal effect derived from G Pseudo-residual of {XX, YJ relative to G\G,
We PR(XYIGG
PCT
Pearson Correlation Test
C More Details on Observations 1 and 2
C.1
Observation
Since G (G1, is a valid IV set, employing G1 or G3 in TSLS(.) entails unbiased causal effects, i.e., W(.)) = TSLS(X,Y, Bx-, with i {1,3). Thus, we have,
Y -WiWG Y XwyGs) Y --V-xXxx -X3x-x G27y:2 G4tya + Gsyy, EY-
X G22y2 + Giyya + Goyrs +e,,
Because both G1 and G3 are valid IVs, they are independent of ey; and any two genetic variants are assumed be independent of each other, we get,
corr(Y - XW(G)), G1) = 0,
corr(Y - XWiGI> Gz) = 0.
C.2 Observation 2
Without loss of generality, here give details about how corr(Y-XwiG,, XwiG., G2) & is derived. The other two inequalities could be obtained in the same way. Following the procedure of TSLS(-), we first get the estimate of X with (G4, G5). denoted by X, and obtain W(G,,Gs) using X and Y.
X = =G484 + Gr3;
-Gal7x,1 + + + 7x,3
C.rrx + Gs7x,s + Gsty.By
where since G4 and G3 are neither correlated with the latent confounder U nor the other IVs as shown in Figure Z. the second equality holds. Then, we could get the estimate of W(GG,GG),ie.
cov(G4, GA)( + my + 7x,43x-1)
((4.GG)) = cov(G,, G.) ,r, cou(Ss, Go) + xx) cou(Gs, G5) + corCss,. 7x,
cov(G1, Ga)(TxA 7,,08=xx,++ G3) NY.5
15

---

Then we get,
COLT(Y - Gx)
(1r22 +x -x Gs)
- |
Since,
8x-Y WiG.,Gs)
3x (x-+((1x14 + ( Bx 7y - Peouu Gs)
(yx,4 ) (1x.o (xx0 x Gs))
+ 1ow.
(7x,4 +
+ h| )ooo(GG)
(ixA + 1 ) 2000 x -x).0/XXXxxXX/GoGGor/
(Bx-y By 1
+ -y,58y
(xx,4 -x (7x.55 -
-
(Bx-Y -x xx + Cs5 x,stYss
Cn X,4 + Cas X.5
and
1 --BYxXW(GnGG))
(Tx.A +
G4) (1x.5 + mr xx Gs, Gs)
(TX, + 7r, +X Cow G4) (1ix,s 1y,80y G5)
+ (r-- (xx,1 1r, + 2co0 + bow.
(7x,55 (x [p) Gs)
(ΛxL+ PC4 G4 ( Gb)
(xxA(7Xx, + 7,,4 3r, (Gs, G4) + (xx + Tr, Jcou (ooDGG Gs)] G4,, Ga) + (1x,s + +y, 2 Poou Gs)
(yx,1
A(CuTxx,IxA + x,57x,))
Curl3 x, + Cul X,5
Where we define Fx,; = 9x.s + By TY, YYs + 8x,88x-Y, C₁₁ cov Gy) and é xx we can finally obtain corr(Y Xwic..G.) G2), that is,
corr(Y XwIGx. G2)
1x C41l X,, + Caal X,5
()(CTTXX + x.5
Mx2(1 - xxx a
x Cin + CasTx..7xxs)
C Css X,5
12
7x,)722) CuIxx,(7xxX71 -7,,712)) + Crsl' (7x,)x,, 7x,5772)
40.
Caa x + C55l X,5
D More Details on the Example Provided in Section 5
Figure gives an illustration of our idea, demonstrating that even when IVs are correlated, we can still identify them as valid IV' using our proposed method. In subgraph (a), there are two valid IVs: G and G2. However, in subgraph (b), these two IV' become invalid as G2 correlated with Y. rendering them invalid.
16

---

C2
C2
G1
G1
(a)
(b)
Figure 6: A simple example, where the set [G1. G23 is valid IV set in subgraph (a), whereas is not valid in subgraph (b).
Assume the generating process of X and Y in subgraph (a) follows:
X Y + G17x,1 + G2yx22 + Ex.
Y = Xix-Y +
(15)
G2 G1,,t
where xx and By correlated, and 82 represents the independent term.
For G1 E G, we check the condition of Proposition 3:
((G.) = TSTS(XXYY, (G:)) cov(X,Gi) = Bx + cov(ey, G1)
Cov(Y,Gi))
cov(X,G1) 9x
PR (G.)) = ((G)) X = ( G2) =0
Similarly, for G2 E G, we check the condition of Proposition 3:
((G,) Y, (G2)) Cov G2)
COv(Ey, G2)
cov(X,G2) 8x_- + cov(X, G2) -8
PR(XY (Ga)) = Y - W(G)) X =
cort(PR ((Ga)): Gt) 00
Assume the generating process of X and Y in subgraph (b) follows Equation 16
X Y3y-x + G17x.1 + G2Tx,2 +
Y X8x + G17y. +EY.
(16)
G2 = Gr
where x and ey are correlated, and e2 represents the independent emoo term.
Then, Equation 16 can be reorganized to avoid recursive formulations as follows:
X = A (Gi(1x1 +
+ 527x.2 + + ty
A [ txx
(17)
Y = A + + + - + +ex)
A + + xx +
where [x,1 = 7x,1 + 71,7Xx2 + rr.1rr- and Py. noted that Xx11 & and [y. # O.
Tra + Bx-x((xx, + 7x,271) Is
In subgraph (b), for G2 G. check the condition of Proposition 3:
17

---

cov(Y.
cov(G1, G2)
W(G.) -
Cov(XX
8x--y + 7y,1 cov (0,(X, G2) Cov(GI,
71771,
= -x-Y +
A
cov(G1, G1 (,, cov(
=xx+r 71,7112
A
7 X11 + 7x2M
(G2)) -
71,121
=G +
721 + Xx,2 X
A
=GmM + 2 + 2 + tex + By
7,2l',,1 x,1 + 7x2M
7
(xx(PR(XX G1) = .Y,1 n,2l'x, x,1 + 1x22
.... x,1 + 7 2
71,21x211 x,1 + 7
2
2,2 7
(18)
where M
cov( cov(G1.G1) # O.
Based on these two observations, we can conclude that the two IVs in subgraph (a) can be deemed valid. However, in subgraph (b), they can be deemed invalid.
18

---

E
More Details on Vess Algorithm (in Section 2.1)
noteworthy that, mitigate the unreliability of statistical tests in finite samples, in practice. we choose to take the smallest correlation coefficient to select valid IV set, as in Line 4 or 10 of Algorithm
Algorithm 4 VandVVViVSSests
Input: A dataset of measured genetic variants G (G1,... Gg)J, two phenotypes X and Y, significance level and parameter W, maximum number of IVs to consider.
1: Initialize valid IV sets V and G G.
2: repeat
3: For all subsets G E G with | 2, compute the correlation between pseudo-residual PR(x,y GGAG)) and Gj, where G, E G.
4: Vnew arg min corr(PR(x.y | (,)a)) G.). GEG
5: repeat
6:
G'+G
7:
For each variable Gg E G', remove all variables from G' corresponding to those that set [V, Gx fail to the PCT.
8:
if G' * 9 then
9:
Compute the correlation between pscudo-residual PR (V) and Gk-
10:
G' f arg min corr(PR,,,y (V) Gx).
G+EG'
11:
Vnew + Vnew U {G+).
12:
else
13:
Break the loop of line 5;
14:
end if
until the length of the set Vaew has reached our predefined threshold W.
16:
if Set V = 0 then
Add Vnew into V. else
18:
Set (neW fail to PCT then Add set Vnew into V.
20:
else
22:
Delete set Vnew from G. end if
24:
end if
until contains two sets or G S Output: V, set that collects the valid IV sets.
19

---

More Details on InferCCausaliircciinEetttsss Algorithm (in Section 4.2)
Algorithm 5 InferCausslDirrciioffffeeett
Input: Two phenotypes X and Y. Valid IV sets V.
G 2: for each Gj E y do
1: Initialize
0 and G = 0. and the causal effects 8x_. and f - 0.
3: if' [corr /oorrr X) then
4: Add Gj into G.-
5: else
6: Add Gj into G -
7: end if
8: end for
9: BX-Y t TS,
10: BY-x TSIS(Y, G -)
Output: 3x-YY and x
G Proofs
G.1 Proof Proposition M1
Proof. From procedure "two-stage least squares", we obtained the fitted values X from the regression the first stage: X = TG. According to OLS regression,
X = (G [ - x] - JG " X
=XX
Then run the OLS regression for Y on X in second stage: Y = -88 where:
β (XTX))
-(PX))X))"
-(XTPX))(XXPP)
= -XXPXX (XBx-Y + (GB
+Y ( +
- (XTPX)
(XTPX)
Since G is valid IV set, for Gj +Y both my 0 (A2) and cou(Gj,ev) 0 (A3).
Hence, we obtained
(XTPX) (XTPX) = (XT
=0,
which implies that 9 = BX-YY.
G.2 Proof Remark
Proof. Similar to G.I. from procedure "two-stage least squares", we obtained the fitted values X from the regression in the first stage: X (G) +G According OLS regression,
x = [G (G) ( x]
= (GG
= PX.
20

---

Then run the OLS regression for Y on X in second stage: Y = =8X. where:
B (XTX))
= -(XPX))
-XXT) (X3x-r + NTx
= -x--r+ + XTPx XTP(G +er)
Poias
=x ( (XTX
+ (XTXx)
Since G CX is valid IV set, for each Gj G either Yys + 0 (A2), cov(G + 0 (A3), or both Ty.J o and cov(G),ey) # (A2 and A3). Hence, we obtained
( (ax)
(XTPX)- =
- #0, 0,
which implies that Boias # 0. That to say the estimated causal effect A 3x-+Y + Boias biased.
G.3 Proof of Proposition 2
Proof. We prove it by contradiction, i.e., if G is valid IV for of the causal relationships, then for each Gj E G. cor((PR(xx G,) = 0. Without loss of generality, we consider the case for X Y. The proof with the inverse causal relationship
be derived accordingly.
i) assume that G is the valid IV set relative to the causal relationship X + Y. i.e., G S G Let Gj be genetic variant in G. We can see that G) is also valid IV set. Because the size of participants n + oo, and according to the consistent estimator of TSLS, we have
W(G)G,) G,GG)) 8x-YY
Based on Eq. we have
PRXYIIGG, Y Xu((oa,) GTyY +
Because C valid IV relative to X +, for GR E G, k: + j, either TY. 0 (if G. E G) or corr(Ga, Gj) = 0 (if G2 & G.. is noted that G& can not be correlated with Gj, otherwise, the genetic variant G, will dissatisfy A2 A3. Hence, we obtain Tor(GTTy + EY, Gy) 0, which implies that (x.
G2) = O.
ii) assume that G is the valid IV set relative to - Y variant in G. We can see that G, is also valid IV set relative to Y + X. It noted that we here still need to verify (5). According to the TSLS estimator and EI,( we have
i.., G C GØ Let G, be genetic
W(G)G,) TSLS(X, T G),)
XTTXX XTPY
[XTPX] XTP(I/Br - GTyx ex)
(1- (TP(GTTx+Ex)
= (1/8r=x)
1 XTPX] XTP +Ex) +EX) BY
where P - (G)G)) [GGG x, and ex are n x vectors, G is n X matrix and 7x is a g x 1 vector.
21

---

Hence, we have
PR(XY1G())) =Y Y XuyGIG)
=Y 1 XTPX| XTP(GTTx+eN) s7-x
=Y - X X XTPX XTP(GT7x tEx)
= = + Ex ex) + X [XTX]] XT +
---++X ( XTPX|
xTp (GT (xx
where I is a n x n identity matrix. Because @G is valid IV set relative to Y - X, for G2 E G. + j. either 7x 0 (if G G) or co G₁) = 0 (if G2 & G).
Hence,
corr y +xx),
Therefore, we obtain
corr((X/ XTPX]]
XTP +Ex),G,) =0
which implies that con((R((x.))))))) Gy) =0
G.4 Proof of Proposition E
Proof. We prove it by contradiction, i.e., if is an invalid IV set for one of the causal relationships,
then there exists at least one G3 G such that corr(PR,xxy | G2) * 0. Without loss of generality, we consider the case for X - Y. The proof with the inverse causal relationship Y +>> X can be derived accordingly. There are three cases to be considered. i) Suppose Gj is an invalid IV G and G GG is a valid IV set; ii) Suppose G, is an invalid IV in G and G)G, is an invalid IV set; iii) Suppose Gj is valid IV in G and G) is invalid IV set.
Suppose G, invalid and all IVs in G. are valid. Due to the validity of all IVs G\G,, according to the consistent estimator of TSLS, we have,
((G),)) = TSLS(X, G)G,, = 8x-
In order to consider the case of all violating assumptions for IVs, we extend Eq. Eq.(D), we have
PR(X.1 = Xw(Grc,) = GT + GTay UNY. + E3.
where make Ey = UNY,U + Ey to tell the influential difference between the latent confounder U and the external noise E2 on Y. Here Gj IL Ey but G may be correlated with U by Yu, Denote by G the other IV with i # j, and we have G₁ Il G.. Hence,
cort(PRxx.y1eG, G,) Ty.,oou(G,, GG) + TY, G,)
(Var(PR (o,, (wi + wco0)))oooG
(Var((PRx.y Var((G,)
Due to the faithfulness assumption, corr | (G, G3) #0.
ii) Suppose G, invalid and G) is an invalid IV set, where there are m geneties in GVG, (GI,... Gm J. Following the procedure of TSLS, we first get the estimate of X with G G,,, denoted
22

---

by X, and obtain W using X and Y.
m
m
x
G Σ GLal(Tx, + 7x (1x,0 + )xx
ンk= 1 7 A + +
Σm=1 (3x))
13 + XX +x A + 7X. Xx-
Σ
k== (3mm)
112
1 BAlLTYTA + xX v) + (TVU + Xx
\x=1(Bx)²
DA E Ax, + Ay,
2t=1(32)
where Bg = Ba/A = (7x,* + TY + TU.KTTX.U + Ty, Ax. + A
Ax,* = X + 3y and Ax,u 7x,U + Yy Ay = YY + 7x,88x- and
AY. = YY,U + Xx and Cak = Go, GR). Note that such notations A. enjoy some
great properties shown as below,
Ax,* - Av, 7x -x-x
Ax - Av, = (xx - Ax,1 Ax, -1 11-xx-1++=x/xxxx1 -x221)))))))1)1))
(19)
Hence,
cor(PR(x.1/11c))..
cou(G,.G)) Gj)
((Ya + +xx08xx)4 (xx. + m
+ + - (.,
GA
- BY -y (Va(PRRYIBc))VVVGG) (1
(1- + +x, -x000==
= =dA[(1 - BY Br- + manuus) + (8x -((a,,)))(xx + (xwrrss)),
Var(C,))
where 8
0. Employing the properties in Eqs. (I)) get
1 - By-xw(6iG,
Σm (Ax.* + Ax,UYU, CR By-x((Axx (x, )
(3x)
(1- 1-Bx-y0y=xx)2 Ex (Ax,r k + Axx.nUx))xx + 7x.UTU,))
Ck=1 (Bx
and
3x-Y WCC),,,)
3x-+ - + )x, - Ei + Ax, + Ay.u
Σm 2
(y 1) "m (Ax. + (x.UNUN)))TY NY. C+a
(Bx
23

---

Thus,
coT((PRxxy
81 DK=1 CAA(Ax, Axx 70.U
(nY,s + (Y.U)))))
(
Ek=1 CRE(AX, + Ax, M.UTUU (7x. +xx,uno)))
1i=1(3a)
2k=1 CRE((TT (x.0 + (YUTU (NY mmuoa)((xs + 7x,0))))
Ek=1 (B
8 EE_ CCCalllxAAYa--AAAA. www.::/wwww........///././///./. 40.
Because of the faithfulness assumption, 3x * 0. Due to Assumption D. for G₁ G)G, and the confounder U, EE k==
(
(U.)) # O. iii) Suppose Gj is valid IV in G and G is an invalid IV set. Similar to ii), we get ((GG)))
m (Ax,s + Ax. + )
W(GGG))
Since the valid IV Gj is uncorrelated with neither other invalid IVs nor the confounder, we then derive,
corT(PR G, G, + xx,58xxx) (1x, + (r,8r-x))
=0xxxx(xx-x -
where:
BX-Y - W(GGG))
Bx- EiT Cuxl(Ax, + Ax, - (Ax,a + Axwytu))(Axx + Av,U)U,s)
Em (3x)
(3x-y Em CAx + Ax, m....U
Ek PCa
Thus, get
(PT( (xxx
Eh CRe(Ax, + Ax, ) t.o..,)
Ets (Bx)2
T k= B.CHA + mYunua)xx
Em (Bx)
Since Gj a valid IV (Ty, Y0 0), and from Assumption EL we have Σm Br C )Y, # 0. Proposition B is proved.
+ 11
G.5 Proof Proposition 4
Proof. We prove this proposition by contradiction, i.e., i) if G is the valid IV set relative to the causal relationship + X, then GG E G satisfies
corr G, 1; and ii) if G is the valid IV set comr(G..Y)
conr(G
relative to the causal relationship X + Y, then 3 G3 G satisfies corr(C C < 1.
24

---

i) Suppose G is the valid IV set relative to the causal relationship Y - X. According to Assumption 3, we have 18y+xl=
Var(X)
Hence,
Var(Y)
corr corr(G,,Y Y)
Cov(G,,
v Var(G))Va((X))
corr Gj.X)
/Var(G,) Cov(G,, X)
A
Var(X)
Var(Y) TY ABx-x
Var(X) Var(Y)
2 1,
which proves case i).
ii) Suppose G is the valid IV relative to the causal relationship X - Y. According to Assump-
tion have |3
Var(Y) Hence.
V)r(X))
corr (j,
Cov(G,,
corr(Gj,, X)
Cov(G,; X)
ABx Var(X)
VVrr(Y)
Yx,
Var(X) Var(Y)
<1,
which proves case ii) and concludes the proposition.
G.6 Proof of [Teerrm
Proof. Firstly, given Assumptions MI and in light of Propositions 2 and B. we can identify all sets of valid instrumental variables (IVs). Next, given Assumption BL by Proposition g identify what the causal relationships corresponding to these IV sets. single causal relationship. such X -> Y, we conclude
further
important to note that
if all the IV sets correspond that the model is a one-directional MR model. Finally, with the corresponding IV set identified for each causal influence, can employ the classical IV estimator, namely the two-stage least squares (TSLS) estimator, to accurately estimate the causal effects.
Based on the above analysis, we can conclude that the bi-directional MR model is fully identifiable.
G.7
Proof of [Thoorem
Proof. The correctness of PReBiM originates from the following observations:
Firstly, Step (Algorithm M) by Propositions and BL valid IV sets have been exactly discovered in V (Lines 2 ~ 25 of Algorithm E).
Secondly, in Step 2 (Algorithm 50, by the condition of Proposition Bl the direction of causal influence can be determined (Line 2 ~ of Algorithm 5)
Lastly, by (I) of Proposition MI unbiased causal effects can be estimated exactly given the identified IV sets (Lines 10 of Algorithm ()).
Based on the above analysis, we can identify valid IV sets and obtain the causal effects correctly.
25

---

H More Experimental Results
H.1 Evaluating the Performance on Bi-directional MR Model with Dependent Valid IVs
Table Bl summarizes the CSR and MSE metrics' results across six methods within a bi-directional MR model with varying sample sizes. As expected, our proposed method exhibits superior per- formance, even in scenarios where the included valid instrumental variables (IVs) are correlated. This highlights its capability to accurately identify effective IVs and provide consistent causal effect estimates solely from observational data, regardless of whether the relevant IVs are correlated. Im- portantly, this effectiveness persists without prior knowledge of the causal direction between and Y. Conversely, the performance of the other three methods remains suboptimal in such contexts, indicating their limited applicability in cases with correlated valid IVs.
Table 3: Performance comparison of NAIVE, MR-Egger, ViSE,VE, IV-TETRAD, TSHT, and PRe- BiM in estimating bi-directional MR models across various sample sizes and three scenarios.
2().2.6)
3(8.3,8)
5(1,4, 10)
X
Y
X
X
y
Size
Algorithm
CSR
CSR
MSE]
MSEJ
CSR
CSR
MSEJ
MSEL
2k
NAIVE
0.409
0.341
0.350
0.372
0.409
0.314
MR Fager
1.198
1.320
1,033
1.322
1.087
1.391
siVVIVE
0.31
0.478
0.29
0.395
0.34
0:408
0.34
0.388
0.35
0,450
0.36
0.328
IV-RAD
0.40
1.224
0.42
0.900
0.52
0.830
0.29
0.903
0.44
1.033
0.32
0.779
TSHT
0.04
0.696
0.12
0.708
PReBiM
0.84
0.014
0.79
0.167
0.89
0.058
0.88
0.097
0.84
0.053
0.83
0.079
5k
NAIVE
0.399
0.326
0.371
0.362
0.391
0.333
MR-Egger
1.171
1.358
dsVIVE
0.30
0.484
0.31
0.426
0.35
0.452
0.34
0.381
0:38
0.375
0.37
0.327
IV-TETRAD
0.50
1,042
0.40
0.700
0.49
1.041
0.38
0.778
0.47
0.977
0.36
0.778
ISHT
0.701
0.05
0.531
0.09
0.689
0.06
0.660
0.11
0.653
0.09
PReBIM
0.90
0.113
0.84
0
0.91
0,050
0.90
0.061
0,88
0.048
0.89
0,061
T0K
NAIVE
0,408
0.323
D.339
MR. Rager
1,052
1.307
1.017
1,313
1.185
1.303
SiSVIVE
0.33
0,465
0.34
0.427
0,35
0,448
0.34
0.358
0.38
0.431
0.41
0.299
TETRAD
0.39
0.553
TSHT
0.04
0.641
0.03
0.543
0.04
0.734
0.04
0.449
0.09
0.654
0.07
0.567
PReDIM
0.90
0.058
0.57
0,089
0.94
0,018
0.93
0.026
0.93
0.022
0.92
0.056
Note: This scenario represents bi-directional model with correlated valid IVs. The symbol means that methods do not output information. "4" means a higher value is better, and vice versa.
H.2 Evaluating the Performance on One-directional MR Model with Dependent Valid IVs
In this section, we assess the efficacy of four methods within one-directional Mendelian Random- ization (MR) framework. For a fair comparison, we operate under the assumption that the causal pathway from X to Y established for all methods. Figure I displays the CSR and MSE metrics for these methods, considering diverse sample sizes and numbers of valid instrumental variables (IVs). As expected, our approach maintains strong performance, even amidst correlated valid IVs. The outcomes are comparable with those of the TSHT algorithm, with our method notably surpassing both the sis VIVE and IV-TETRAD algorithms under various conditions.
1.0
150
1.0
05
1.0
0.30
125
025
04
030
04
02
0.4
030
02
025
0.55
00
00
00
Sample Sibes
Sample Sibes
Sample Sibes
(a) S22 6)
(b) S(3.0,
(c) S(4,0, 10)
Figure 7: Performance comparison of Sis, IV-TETRAD, TSHT, and PReBiM in estimating one-directional MR models incorporating valid IVs across various sample sizes and three scenarios.
H.3 Evaluating the Performance Bi-directional MR Model with Small Sample Sizes
In this section, we will evaluate the performance of these six methods in a bi-directional Mendelian randomization framework with small samples. Table E displays the CSR and MSE metrics for
26

---

these methods, considering three scenarios with a sample size of 200. We find that though the performances of all four methods are not satisfactory, ours still outperforms other baselines. also reflects an open problem of our work with small sample sizes. We leave it as our future in-depth direction.
Table Performance comparison of NAIVE, MR-Egger, SISV,VE, IV-TETRAD, TSHT, and PRe- BiM in estimating bi-directional MR models across small sample size (200) and three scenarios.
2,))
$(3,3,8)
S(4,4, 4, 10)
X
y
X
X
+Y
V
X
X
V
y
X
Size Algorithm NAIVE
CSR
MSE
CSR)
MSE]
CSR
MSEJ
CSR
MSEL
CSR)
MSEL
CSR
MSEJ
0,360
0.330
0.343
0.331
0.320
0.336
0.941
0.657
0.579
0.594
0.433
0.535
200
saVIVE
0.21
0.389
0.34
0.383
0.29
0.331
0.28
0.336
0.34
0.318
0.30
0.329
IV.TETRAD
0.30
1.057
0.26
0.797
0.26
0.8111
0.20
0.886
0.15
0.723
0.12
0.787
TSHT
0.677
0.16
0.504
0.26
0.452
PReBIM
0.54
0.339
0.55
0.350
0.55
0.313
0.57
0,281
0.51
0.270
0.52
0.288
Note: The symbol "-" means that methods do not output this information. "f" means higher value is
better, and vice versa.
H.4 Evaluating the Performance Bi-directional MR Model with Different Numbers of Valid IVs
Table summarizes the CSR and MSE metrics' results across six methods within a bi-directional MR model with different numbers of valid IVs. Specifically, S(2, 3. 7) indicates that in the bi- directional model, there are two valid instrumental variables estimated for the direction X Y. and three valid instrumental variables estimated for the direction Y + X, with the remaining two being invalid instrumental variables. The same interpretation applies to S(2, 4, 8) and S(3, 5, 10). It can be seen that whether the number of valid IVs in these two directions is the same or not does not affect the performance of our algorithm theoretically. The experimental results also empirically verify it. Overall, In cases where the number of IVs in both the X + Y and Y + X directions are different, our PReBiM algorithm outperforms other baselines in terms of CSR and MSE.
Table 5: Performance comparison of NAIVE, MR-Egger, SISVIVE, IV-TERAAD TSHT, and PRe- BiM in estimating bi-directional MR models across various sample sizes and three scenarios.
3,2,3,7)
8(2,4,8)
8(3,5, 10)
X
X
Size Algorithm NAIVE
CSR
MSRL
CSR
MSR
CSR
MSE,
CSR
MSEL
CSR)
MSR
CSR
MSE
0.433
0.267
0.464
0.227
0.431
0.237
MR-Fage Egger
0.702
0.646
0.560
0.610
0.465
0.476
2k
sisVIVE
0.52
0,958
0.14
0.65
0.099
IV-TETRAD
0.37
1.490
0.55
0.368
0.24
1.886
0.71
0.188
0.35
1.474
0.56
0.150
TSHT
0.00
2.226
0.72
0.066
0.00
2.467
0.85
0.004
0.00
2-453
0.84
0.003
PReDiM
0.59
0.78
0.063
0.50
0
NAIVE
0.364
0.307
0.179
0.225
0.437
0.228
Egger
0.509
0.691
0.593
0.546
0.540
0.486
5k
SiVVIVE
0.20
0.611
0.49
0.274
0.09
1.085
0.72
0.113
0.13
0.917
0.66
0.089
IV-TERAAD
0.37
1,479
0.56
0.312
0.25
1,869
0.69
0,165
0.30
1.41
0.63
164
TSHT
2.296
0.80
2.476
0.00
0:001
PReBIM
0.86
0.067
0.93
0.008
0.83
0.035
0.93
0.023
0.86
0.021
0.91
0.003
NAIVE
0,434
0.273
0.463
0,234
0.434
D.226
Fager
0.550
10k
ViVE
0.17
0.807
0.50
0.230
0.10
1.083
0.71
0.088
0.15
0.932
0.77
0.079
IV TETRAD
0.38
1,495
0.59
0.322
0.26
1.753
0.69
0.244
0.34
1.429
0.62
0.204
TSHT
0.00
2,370
0.85
0.022
0.00
2.494
0.92
0.001
0.00
2.486
0.93
0.000
PReDiM
0.90
0.022
0.97
0.005
0.91
0.013
0.95
0.009
0.92
0.038
0.93
0.008
Note:
The symbol
means that methods do not output this information. "?" means higher value
better, and vice versa.
H.S Evaluating the Performance on Bi-directional MR Model with Large-Scale Datasets
In this section, we evaluate the performance of these methods in a bi-directional MR framework with large-scale datasets. Table E show results of the settings S(10, 30) and S(15, 15, 40). We observe that the overall performances of all methods decrease with large-scale datasets, whereas our method still performs more superior than other baselines.
27

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Table 6: Performance comparison of NAIVE, MR-Egger, VisV IV-TETRAD, TSHT, and PRe-
BiM in estimating bi-directional MR models with large-scale scale datasets.
S(10, 10,30)
S(15, 15,10) 10)
X
X
X
X
Size
Algorithm
CSR
MSEJ
CSR
MSEJ
CSR
MSEJ
CSR
MSEJ
NAIVE
0.298
0.292
0.288
0.299
MR-Egger
0.337
0.316
0.276
0.285
2k
sisVIVE
0.15
0.275
0.15
0.246
0.24
0.188
0.22
0.223
IV-TETRAD
0.22
0.802
0.22
0.647
0.11
0.666
0.07
0.700
TSHT
0.18
1.029
0.28
0.874
0.27
1.012
0.28
0.995
PReBiM
0.59
0.244
0.57
0.173
0.67
0.152
0.71
0.157
NAIVE
0.297
0.299
0.301
0.285
MR-Egger
2.307
0.323
0.293
0.278
5k
sisVIVE
0.15
0.247
0.14
0.281
0.23
0.236
0.23
0.200
IVTETRAD
0.49
0.635
0.48
0.495
0.38
0.572
0.36
0.546
TSHT
0.20
1.020
0.21
1.083
0.22
0.967
0.19
1.042
PReBiM
0.72
0.079
0.73
0.102
0.81
0.094
0.77
0.075
NAIVE
0.301
0.284
0.307
0.279
MR-Egger
0.338
0.325
0.280
0.272
10k
SisVIVE
0.16
0.271
0.17
0.264
0.24
0.216
0.25
0.196
IV-TETRAD
0.58
0.443
0.62
0.390
0.53
0.535
0.53
0.432
TSHT
0.19
0.957
0.20
0.863
0.22
0.886
1.024
PReBiM
0.80
0.062
0.80
0.062
0.85
0.061
0.84
0.075
Note: The symbol means that methods do not output this information. "f" means a higher value
better, and vice versa.
H.6 Evaluating the Performance on Bi-directional MR Model with All IVs Being Valid
In this section, we evaluate the performance of these methods in a bi-directional MR framework with all IVs being valid. That is, only valid IVs are included in the model, either to estimate the direction X - Y or the direction - The results Table D below show the settings S(2, 2, 4). To ensure the thoroughness of the experiment, we' ve added another setting, S(3, 3, 6). As expected, our method gives the best results across all sample sizes and directions. As the sample size increases, the metric CSR approaches 1, indicating that our method can identify valid IVs and also validating our theoretical guarantees.
28

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Table Performance comparison of NAIVE, MR-Egger, SisVIVE, IV-TETRAD, TSHT, and PRe- BiM in estimating bi-directional MR models across various sample sizes and three scenarios.
2,2,
$33.3.6 3,6)
X
Y
Y
X
X
Y
Y
X
Size
Algorithm
CSR
MSET
CSR
MSEJ
CSR
MSEJ
CSR
MSEJ
NAIVE
0.413
0.349
0.351
0.353
MR-Fgger
1.993
2.067
0.852
0.981
2k
sisVIVE
0.70
0.532
0.72
0.450
0.75
0.367
0.74
0.414
IV-TETRAD
0.63
0.822
0.35
0.792
0.68
0.651
0.29
0.716
TSHT
0.02
0.412
0.02
0.388
0.02
0.372
0.02
0.373
PReBiM
0.98
0.003
0.98
0.003
0.98
0.010
0.98
0.008
NAIVE
0.386
0.364
0.369
0.342
MR-Egger
2.145
2,570
0.943
0.952
5k
sisVIVE
0.71
0.578
0.74
0,502
0.79
0.383
0.73
0.442
IV-TETRAD
0.66
0.811
0.33
0.899
0.68
0.714
0.29
0.596
TSITT
0.01
0.415
0.02
0.387
0.01
0.355
0.01
0.370
PReBiM
0.99
0.009
0.99
0.008
0.99
0.001
0.99
0.001
NAIVE
0.390
0.374
0.336
0.364
MR MR-Egger
2.015
2.902
1.051
0.994
10k
sisVIVE
0.74
0.538
0.72
0.497
0.78
0.384
0.78
0.437
IV-TERAA
0.65
0.812
0.35
0.877
0.74
0.630
0.25
0.792
TSHT
0.02
0.423
0.02
0.398
0.01
0.335
0.01
0.387
PReBiM
0.99
0.001
0.979
0.000
0.99
0.008
0.99
0.009
Note: The symbol "-" means that methods do not output this information.
"1" means higher value
better, and vice
29

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Reviews:
[{"review_id": "njctj0iumt", "reviewer": "Reviewer_Qa6w", "summary": "This paper studies the identifiability problem of the bi-directional Mendelian randomization (MR) model, where $X$ and $Y$ are a pair of phenotypes of interest and causes of each other, and $\\textbf{G}$ is the set of measured genetic variants, which may include invalid instrumental variables (IVs). Under some assumptions, the paper has identified and proved correct the sufficient and necessary conditions for identifying valid IV sets from $\\textbf{G}$  based on observational data, without requiring prior knowledge about which candidate IVs in $\\textbf{G}$ are valid or invalid. Supported by the theoretical result, an algorithm is proposed for finding the valid IV sets from the set of measured genetic variants  $\\textbf{G}$ using observation data and estimating the bi-directional causal effects using the found valid IVs. Experiments are conducted with synthetic data to show the effectiveness of the proposed algorithm.", "questions": "1. Line 106: Does the assumption regarding the multiplication of the two effects have any practical meaning/implication?\n2. Could you explain what \"cluster fusion\" means exactly in the paper and why the proposed algorithm is said to be \"cluster fusion-like\"?\n3. Section 6.2 - how the one-directional data used in this section generated?\n4. The work is based on the assumed structure in Figure 2 (plus some invalid IVs as illustrated in the other figures) , but in practice there would be more complicated situations than those, e.g. the vertical pleiotropy effect in biology where the IVs (genetic variants) are associated with another phenotype (or biological pathway) and this in turn causes the two phenotypes of interest ($X$ and $Y$).", "limitations": "Some limitations of the paper have been discussed briefly, but as mentioned above, the consequence and limitations due to the assumptions should be discussed a bit more.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper addresses a challenging and practical problem.\n2. The work is comprehensive, with both theoretical results and corresponding algorithm presented.\n3. The paper is very well written in general.", "weaknesses": "1. The experimental evaluation is done with synthetic data only. Although the presented experiments with synthetic data are comprehensive and the identification conditions have been theoretically proved, as the theoretical result relies on several assumptions, it would be necessary to conduct some case studies with real world data to evaluate how the method works in practice (where domain knowledge or literature can be used to justify the correctness of the found IV sets)\n\n2. It would be very helpful if the assumptions and their feasibility (and consequences/limitations) in practice can be illustrated and justified with real world examples.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your time dedicated to reviewing our paper and your thoughtful and encouraging comments. Below, please see our responses. We hope they can resolve your concerns. Note that we also summarize the main concerns of all reviewers.  Please refer to the general response if interested.\n\n>**W1. it's necessary to conduct some case studies with real world data.**\n\nWe additionally evaluated our method on two real-world datasets. One is derived from a study on the bi-directional causal relationships between obesity and Vitamin D Status [1], while the other one from an empirical study on the impact of colonial history on the economic development of various regions [2].\n- This first bi-directional dataset is produced based on the GWAS summary data from [1] and a publicly available website. With obesity (X) and Vitamin D Status (Y), we selected 16 related SNPs as candidate IVs, they are fat mass and obesity-associated-rs9939609(FTO), Fas apoptotic inhibitory molecule 2-rs7138803(FAIM2), 7-dehydrocholesterol reductase-rs12785878(DHCR7), cytochrome P450 family 24 subfamily A member 1-rs6013897(CYP24A1), etc. We entail FTO and FAIM2 to be the valid IVs related to $X\\to Y$ while DHCR7 and CYP24A1 are valid IVs related to $Y\\to X$, with causal effects -1.15 and -0.05, respectively. These results are in accordance with findings [1]. \n- The second one-directional dataset, the Colonial Origins dataset, consists of Institutions(X), and Economic Development(Y), with other 8 variables as candidate IVs [2]. They are Latitude (lat_abst), European settlements in 1900 (euro1900), Log European settler mortality(logem4), etc. We find that our method selects euro1900 and logem4 as valid IVs, with the estimated causal effect 0.861, which are both consistent with results in [2]. Will add the data details and results.\n\n>**W2. the assumptions and their feasibility (and consequences/limitations) in practice.**\n \n- **Assumption 1** can be easily justified in practice, since it allows the number of valid IVs to equal 2 for the bi-directional model which is much milder than existing methods. Please see examples in real-world experiments.\n- Honestly, it is hard to test **Assumptions 2 or 3** directly in real life, since usually we cannot obtain the ground truths of causal effects between any two variables, including latent confounders. It should be noted that Assumption 2 is satisfied mostly in reality as the set of conditions that meet this assumption occupies a very small portion of the entire space, making it very demanding to violate. If Assumption 3 is violated, we may fail to determine the causal direction for the identified IV set. Will add them.\n\n>**Q1: Line 106: Does the assumption regarding the multiplication of the two effects have any practical meaning/implication?**\n\nWe would like to clarify that if $\\beta_{X \\to Y} \\beta_{Y \\to X} = 1$, the causal effects $\\beta_{X \\to Y}$ and $\\beta_{Y \\to X}$ are not identifiable, even given the valid IV. In fact, this condition serves as the fundamental identification criterion for Eq.(1). For more detailed information, please refer to pages 402-407 of [Hausman, 1983]. We will include this discussion in the revision.\n\n>**Q2: Could you explain what \"cluster fusion\" means exactly in the paper and why the proposed algorithm is said to be \"cluster fusion-like\"?**\n\nA cluster is considered a valid IV set. The term \"fusion-like\" suggests a specific process for identifying and merging these clusters. We will add the explanation.\n\n>**Q3: Section 6.2 - how the one-directional data used in this section generated?**\n\nThe one-directional data in Section 6.2 is simply generated by setting $\\beta_{Y \\to X} = 0$ in Eq.(11), shown below. Will emphasize it in the revision.\n\n$$U=\\mathbf{G}^\\intercal\\gamma_U+\\varepsilon_1,X=\\mathbf{G}^\\intercal\\gamma_X+U\\gamma_{X,U}+\\varepsilon_2,$$\n$$Y=X\\beta_{X\\to Y}+\\mathbf{G}^\\intercal\\gamma_Y+U\\gamma_{Y,U}+\\varepsilon_3,$$\n$$G_{ij}\\sim Binomial(2,maf_j),maf_j\\sim\\mathcal{U}(0.1,0.5). \\tag{11}$$\n\n>**Q4: ...complicated situations, e.g. the vertical pleiotropy effect in biology where the IVs (genetic variants) are associated with another phenotype (or biological pathway) and this in turn causes the two phenotypes of interest (X and Y).**\n\nThanks for the insightful idea. When it comes to the complicated structure with the vertical pleiotropy effect from IV (denote another phenotype as T), we could find that such an IV still satisfies Assumption A2 [Exclusion Restriction] once given T. So we could upgrade Definition 1 of Pseudo-Residual conditional T, where $\\omega_{\\mathbb{G}}$ is obtained by Two-Stage Least Squares (TSLS) estimator but also needs to be conditional on T. We will add it with an example in Section 5 Discussion and regard it as our future work. Thanks again.\n\n**References**\n\n[1] Vimaleswaran K S, Berry D J, Lu C, et al. Causal relationship between obesity and vitamin D status: bi-directional Mendelian randomization analysis of multiple cohorts[J]. PLoS medicine, 2013, 10(2): e1001383.\n\n[2] Acemoglu D, Johnson S, Robinson J A. The colonial origins of comparative development: An empirical investigation[J]. American economic review, 2001, 91(5): 1369-1401.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tEsFNU0xNi", "reviewer": "Reviewer_XbcF", "summary": "The paper addresses the problem of estimating causal effects in bi-directional Mendelian randomization (MR) models with some invalid instrumental variables (IVs) and unmeasured confounding. It proposes a framework for identifying valid IV sets under the assumption that the IV set consists of genetic variants that are independent of each other and that at least two of them are valid IVs.  The authors introduce a cluster fusion-like algorithm based on this framework and demonstrate its effectiveness through theoretical proofs and experimental results.", "questions": "1) How does your method perform when the assumptions (e.g., independence of genetic variants) are violated in practice? Are there any robust techniques or adjustments to handle such cases?\n\n2) Regarding the construction of the IV set, do you find that a larger IV set generally leads to more robust causal estimates, or does it introduce more complexity and potential for bias with invalid instruments? \n\n3) Is the process of constructing the IV set dependent on the order in which instruments are considered? Specifically, does sequentially adding IVs versus a simultaneous assessment of all potential IVs impact the validity and effectiveness of the identified set?\n\n4) Can the proposed algorithm handle large-scale datasets efficiently? What are the computational complexities and potential bottlenecks?", "limitations": "See weaknesses.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The authors establish both necessary and sufficient conditions for identifying bi-directional Mendelian randomization (MR) models, which builds upon previous work focusing on uni-directional MR. \n\nThe proposed cluster fusion-like algorithm is well-founded. The experimental results on synthetic datasets show the algorithm's efficacy in estimating causal effects. These results support the theoretical claims and suggest that the method performs well in practice.", "weaknesses": "While the paper discusses various assumptions (such as the independence of genetic variants and existence of two valid IVs), it would benefit from a more in-depth exploration of the limitations and potential pitfalls of these assumptions in real-world data. Addressing how violations of these assumptions impact the results could strengthen the paper.\n\nThe experiments are performed on synthetic datasets. While this is a good starting point, additional validation on real-world datasets would provide more robust evidence of the method’s practical utility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful comments and suggestions. We have addressed the comments related to the empirical experiment and three assumptions in real-world data. Please see our responses below.\n\n>**W1.** ...it would benefit from a more in-depth exploration of the limitations...how violations of these assumptions impact the results could strengthen the paper.\n\n- **Assumption 1** can be easily justified in practice, since it allows the number of valid IVs to equal 2 for the bi-directional model which is much milder than existing methods. Please see examples in real-world experiments.\n- Honestly, it is hard to test **Assumptions 2 or 3** directly in real life, since usually we cannot obtain the ground truths of causal effects between any two variables, including latent confounders. It should be noted that Assumption 2 is satisfied mostly in reality as the set of conditions that meet this assumption occupies a very small portion of the entire space, making it very demanding to violate. If Assumption 3 is violated, we may fail to determine the causal direction for the identified IV set. \n\n>**W2.** real-world datasets?\n\nWe additionally evaluated our method on two real-world datasets. One is derived from a study on the bi-directional causal relationships between obesity and Vitamin D Status [1], while the other one from an empirical study on the impact of colonial history on the economic development of various regions [2].\n\n- The first bi-directional dataset is based on GWAS summary data from [1] and a publicly available website. For obesity (X) and Vitamin D status (Y), we selected 16 related SNPs as candidate IVs, including FTO, FAIM2, DHCR7, and CYP24A1. FTO and FAIM2 are valid IVs for \\(X \\to Y\\), while DHCR7 and CYP24A1 are valid IVs for \\(Y \\to X\\), with causal effects of -1.15 and -0.05, respectively. These results align with findings in [1].\n\n- The second one-directional dataset, the Colonial Origins dataset, consists of Institutions(X), and Economic Development(Y), with other 8 variables as candidate IVs [2]. They are Latitude, European settlements in 1900 (euro1900), Log European settler mortality(logem4), etc. We find that our method selects euro1900 and logem4 as valid IVs, with the estimated causal effect 0.861, which are both consistent with results in [2].  Will add the data details and results.\n\nWe will add the data description details and results in the revisions.\n\n[1] Vimaleswaran K S., et al. Causal relationship between obesity and vitamin D status: bi-directional Mendelian randomization analysis of multiple cohorts. PLoS Med, 2013.\n\n[2] Acemoglu D, et al. The colonial origins of comparative development: An empirical investigation. Am Econ Rev, 2001.\n\n>**Q1.** How does your method perform when the assumptions (e.g., independence of genetic variants) are violated in practice?\n\n**A1:** First, we conducted experiments with dependent genetic variants on both bi-directional and one-directional MR data, as illustrated in Appendix H.1-H.2. Table 3 shows that our method performs superiorly across various sample sizes and scenarios, even with correlated genetic variants.\n\nSecond, we performed empirical experiments with confounding among genetic variants. As shown in Table 2 of the supplementary PDF, our method remains effective across different sample sizes.\n\nThese results highlight its capability to accurately identify effective IVs and provide consistent causal effect estimates from observational data, regardless of assumption violations. This implies no need for adjustment.\n\n>**Q2.** a larger IV set or does it introduce more complexity and potential for bias with invalid instruments?\n\n**A2:** Thank you for your question. \n- A larger IV set can sometimes offer a broader range of IVs to better capture variations in the treatment variables, potentially enhancing the robustness of the estimates. However, the size of the IV set alone does not guarantee its validity for causal inference. Even with a large IV set, it might not effectively address unobserved confounders, which can introduce significant biases into the estimation results [3]. When constructing IV sets, it is crucial to ensure that the selected IVs are strongly correlated with the treatment variable while remaining conditionally independent of the outcome variable. If these conditions are not satisfied, robust causal estimates may remain elusive, regardless of the IV set size.\n\n- Moreover, a large IV set might introduce more complexity for our method (see the complexity of our method in the next answer). Therefore, introducing additional parameters, such as W in Algorithm 1, to control the set size can help manage complexity and reduce potential bias.\n\n[3] Zawadzki R S., et al. Frameworks for estimating causal effects in observational settings: comparing confounder adjustment and instrumental variables. BMC Med Res Methodol, 2023.\n\n>**Q3.** constructing the IV set dependent on the order in which instruments are considered? \n\n**A3:** We would like to clarify that our algorithm does not depend on the order of the candidate IVs. As demonstrated in Lines 3-4 and 9-11 of Algorithm 4 in Appendix E, we evaluate simultaneously all subsets of IVs and compute their corresponding correlations, ultimately selecting the subset with the minimum correlation. It ensures the robustness of the algorithm. \n\n>**Q4.** large-scale datasets efficiently?  computational complexities? \n\n**A4:** In summary, the computational complexity of our PReBiM algorithm is:\n\n $\\sum_{k=0}^{t} (2\\binom{g-kW}{2} + \\frac{(2g-(2k+1)W-2)(W-1)}{2}) + 2W(t-1)$, \n\nwhere g is the number of IVs, W is the maximum length of the IV set, and t is the number of loops. \n\nFor validation, we performed synthetic experiments on S(10,10,30) and S(15,15,40), with 30 and 40 IVs, respectively. Results are shown in Table 1 of the supplementary PDF. We observe that the overall performance of all methods decreases with larger-scale IVs, but our method still outperforms the baselines.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZIcchwz7Tr", "reviewer": "Reviewer_St5u", "summary": "The authors take up a _very useful_ topic, of trying to identify instruments in models where bidirectional adjacencies exist, at least for the Mendelian randomization application.", "questions": "No particular questions.", "limitations": "I did not see a discussion of societal impact.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The topic of the paper is on point--this is something we need to know more about, as bidirectional edges obviously exist in real data.\n\nThis was an excellent paper, thanks. The discussion made sense to me from start to finish, and the experimental results were compelling. Thanks.", "weaknesses": "From _my_ perspective, there were no glaring weaknesses to this paper. Perhaps other reviewers have issues to mention.\n\nThe only possible weakness I saw was the strong reliance on the assumption of linearity, though in the discussion this was mentioned as an assumption that could possibly be relaxed in future work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your inspiring commendation. We would like to mention that:\n\n(i) Identifying instrumental variables in bidirectional MR, both theoretically and practically, within the one-sample MR framework is a desirable but challenging research topic. We employed the **linearity assumption to entail the theoretical identifiability** of the bi-directional MR model, while the linearity model could enjoy some remarkable characteristics. \n\n(ii) **Linear models have also been widely explored and used** in many practical situations [Pearl, 2009, Spirtes et al., 2000, Imbens and Rubin, 2015], often providing meaningful results [Kang et al., 2016, Windmeijer et al., 2021, Silva and Shimizu, 2017, Li and Ye, 2022]. Hence, we focus primarily on linear models, other than nonlinear ones.\n\nFurthermore, how to develop a framework to **handle nonlinear causal relationships** is a significant future direction.\n\n\n\n\n**References**\n\nJudea Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, New York, 2nd edition, 2009.\n\nPeter Spirtes, Clark Glymour, and Richard Scheines. Causation, Prediction, and Search. MIT press, 2000.\n\nGuido W Imbens and Donald B Rubin. Causal inference for statistics, social, and biomedical sciences: An introduction. Cambridge University Press, 2015.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "P5Zkpk3LQh", "reviewer": "Reviewer_UQEn", "summary": "The paper addresses the challenge of estimating causal effects in bi-directional Mendelian randomization (MR) studies using observational data, where invalid instruments and unmeasured confounding are common. It investigates theoretical conditions for identifying valid instrumental variable (IV) sets and proposes a cluster fusion-like algorithm to discover these IV sets and estimate causal effects accurately. Experimental results demonstrate the effectiveness of the method in handling bi-directional causal relationships, providing insights crucial for improving causal inference in complex systems.", "questions": "I have few Questions/Suggestions for the Authors:\n\n* In line 106, the authors mention, \"Following Hausman [1983], we assume that $\\beta_{X ->Y} \\beta_{X->X} \\neq 1$.\" It would be useful to discuss in the main paper why this assumption is necessary and what happens when it is violated.\n\n* Similarly, regarding Assumption 3, the authors mention it as a very natural condition that one expects to hold for the unique identifiability of valid IVs. It would be useful to explain briefly in the main paper why this assumption is necessary for the identifiability of IVs.\n\n* The authors in Section 5 claim that with dependence between genetic variants, main results may still be effective in identifying valid IV sets. Does this claim still hold when there is confounding among the genetic variants or between the genetic variants and some phenotype? Or does the dependence just mean direct causal effect here?. \n\n* At the moment, the proposed solution is restricted to linear causal relationships. Can one apply the proposed method using some linearization technique for scenarios where causal relationships are not necessarily linear?", "limitations": "The authors clearly state all the assumptions. The paper could benefit from adding some more discussion on the necessity of these assumptions in the main paper. I don't think the paper has any potential negative societal impacts.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The main contribution of the paper is presenting sufficient and necessary conditions for the identifiability of the bi-directional model, enabling both valid IV sets for each direction. They also propose a practical and effective cluster fusion-like algorithm for unbiased estimation based on the theorems and prove the correctness of the algorithm. The paper also validates the theoretical findings using extensive experiments on synthetic data along with comparisons to baseline methods. Overall, the paper is well written and easy to follow.", "weaknesses": "The paper has no major weaknesses. However, the setting is restricted with causal relations limited to being linear and assuming genetic variants are randomized, which limits the practical applicability of the proposed approach. Additionally, the experimental results provided are mainly synthetic in nature.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We truly appreciate your insightful and encouraging comments. Please see below for our responses.\n\n>**W1. The setting is restricted with causal relations limited to being linear.** \n\nWe’d like to mention that, \n- Identifying instrumental variables in bidirectional MR, both theoretically and practically, within the one-sample MR framework is a desirable but challenging research topic. We employed the **linearity assumption to entail the theoretical identifiability** of the bi-directional MR model, while the linearity model could enjoy some remarkable characteristics. \n- **Linear models have also been widely explored and used** in many practical situations [Pearl, 2009, Spirtes et al., 2000, Imbens and Rubin, 2015], often providing meaningful results [Kang et al., 2016, Windmeijer et al., 2021, Silva and Shimizu, 2017, Li and Ye, 2022]. Hence, we focus primarily on linear models, other than nonlinear ones. We leave nonlinear models to be our future works.\n\n>**W2. the experimental results are mainly synthetic in nature.**\n\nWe performed experiments on two real-world datasets. One is derived from a study on causal relationships btw. obesity and Vitamin D Status [Vimaleswaran et al., 2013], while the other one from an empirical study on the impact of colonial history on the economic development of various regions [Acemoglu et al., 2001].\n- The first bi-directional dataset is produced based on the GWAS summary data from [Vimaleswaran et al., 2013] and a publicly available website. With obesity (X) and Vitamin D Status (Y), we selected 16 related SNPs as candidate IVs. They are fat mass and obesity-associated-rs9939609(FTO), Fas apoptotic inhibitory molecule 2-rs7138803(FAIM2), 7-dehydrocholesterol reductase-rs12785878(DHCR7), cytochrome P450 family 24 subfamily A member 1-rs6013897(CYP24A1), etc. We entail FTO and FAIM2 to be the valid IVs related to $X\\to Y$ while DHCR7 and CYP24A1 are related to $Y\\to X$. These results are in accordance with findings [Vimaleswaran et al., 2013]. \n- The second one-directional dataset, the Colonial Origins dataset, consists of Institutions(X), and Economic Development(Y), with other 8 variables as candidate IVs [Acemoglu et al., 2001]. They are Latitude (lat_abst), European settlements in 1900 (euro1900), Log European settler mortality(logem4), etc. We find that our method selects euro1900 and logem4 as valid IVs, with the estimated causal effect 0.861, which are both consistent with results in [Acemoglu et al., 2001]. Will add the data details and results.\n\n>**Q1. In line 106, why $\\beta_{X\\to Y}\\beta_{Y\\to X}\\neq1$ is necessary and what happens when violated.**\n\nWe’d like to clarify if $\\beta_{X \\to Y}\\beta_{Y\\to X} = 1$, causal effects $\\beta_{X\\to Y}$ and $\\beta_{Y\\to X}$ are not identifiable, even given the valid IV. This condition serves as the fundamental identification criterion for Eq.(1). For more details, please see pages 402-407 of [Hausman, 1983]. Will explain it. \n\n>**Q2. Why Assumption 3 is necessary for the identifiability of IVs?**\n\nNote that according to Proposition 3, we obtain the IV set for one of the causal relationships, either $X\\to Y$ or $Y\\to X$. To achieve full identifiability of valid IVs in a bi-directional MR model, we need to further determine which causal direction the IV set is related to. Thus, to render the causal effects identifiable, we introduce Assumption 3 [Xue and Pan, 2020]. It employs the correlations between IVs and phenotypes to find the related direction for the IV set. We have included the necessity of assumptions in the revision.\n\n>**Q3-W3. genetic variants are randomized...dependence btw. genetic variants. Does this claim still hold when there is confounding among the genetic variants or between the genetic variants and some phenotype? Or does the dependence just mean direct causal effect here?**\n\nThanks for your valuable comments. We conjecture that the claim still holds when there is confounding among the genetic variants or between the genetic variants and some phenotype. Following the example's proof in Appendix D, one can prove some simple examples of these cases (Due to space limitations, we do not provide detailed proofs here but will offer them in the revision).\n- In addition to the example proofs, we also performed empirical experiments to validate the first case. In Table 2 of the supplemented PDF, we see that when there’s confounding between genetic variants, our method remains effective with different sample sizes.\n- It's worth noting particularly that when there's confounding between a genetic variant pointing to X and X (not Y), our conjecture still holds; when the confounding is between the genetic variant pointing to X and Y, we would need this confounding factor to be observable to control the conditional independence between the genetic variant and Y. Will leave such general research into our future work.\n\n>**Q4. ...using some linearization technique for scenarios where causal relationships are not necessarily linear?** \n\nThis is a very good point. When causal relationships are not necessarily linear, one can practically apply some linearization technique, mapping nonlinear causal relationships to possibly linear ones. In such a case with possibly linearity data, our conclusions might be still feasible. And we leave it as our future work, i.e., how to deal with not necessarily linear data as well as complex nonlinear data. Will add this discussion.\n\n**References**\n\nJudea Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, New York, 2nd edition, 2009.\n\nPeter Spirtes, Clark Glymour, and Richard Scheines. Causation, Prediction, and Search. MIT press, 2000.\n\nGuido W Imbens and Donald B Rubin. Causal inference for statistics, social, and biomedical sciences: An introduction. Cambridge University Press, 2015.\n\nDaron Acemoglu, Simon Johnson, and James A. Robinson. The colonial origins of comparative development: An empirical investigation. American economic review, 2001.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all reviewers for their constructive suggestions and **overall positive comments**, especially for the acknowledgment of our writing quality, comprehensive theoretical analysis, and empirical experimental performance.

We have taken carefully the reviewers' feedback into account and responded to each question with detailed explanations and additional experimental results. Please see below the summarized main concerns.

# Experiments
Following the suggestions from reviewers, we provide additional results on two **real-world datasets** to further validate the effectiveness of our method and enhance our paper. One is derived from a study on the bi-directional causal relationships between obesity and Vitamin D Status [1], while the other one from an empirical study on the impact of colonial history on the economic development of various regions [2]. Experimental results on both datasets revealed that our method could find valid IVs as well as obtain causal effects, which are consistent with those findings from existing literature.

Moreover, we performed additional **synthetic experiments** to validate the efficacy of our method, with **results shown in the supplemented PDF**.

# Assumptions 
Following the suggestions from reviewers, we provide in-depth discussion and exploration of the necessity of assumptions, to strengthen the paper.

- Compared with existing methods that constrain the number of valid IVs to be larger than 2, **Assumption 1** of our method allows the number of valid IVs to equal 2 for the bi-directional model, which is much milder. If it is violated, our method as well as other mentioned methods would fail to identify a valid IV set theoretically.

- Note that **Assumption 2** is satisfied mostly in reality as the set of conditions that meet this assumption occupies a very small portion of the entire space, making it very demanding to violate.

- To derive the full identifiability of valid IVs in our bi-directional MR model, i.e., determining which causal direction the IV set is related to, we further introduce **Assumption 3**. If it is violated, we may fail to determine the causal direction for the identified IV set. 

- **Linearity assumption**. Identifying instrumental variables in bidirectional MR, both theoretically and practically, within the one-sample MR framework is a desirable but challenging research topic. We employed the linearity assumption to entail the theoretical identifiability of the bi-directional MR model, while the linearity model could enjoy some remarkable characteristics. 

We sincerely thank the reviewers and the AC for their time and thoughtful feedback on our paper. We hope that our responses have effectively addressed all the questions and concerns.

**References**

[1]Vimaleswaran K S, Berry D J, Lu C, et al. Causal relationship between obesity and vitamin D status: bi-directional Mendelian randomization analysis of multiple cohorts[J]. PLoS medicine, 2013, 10(2): e1001383.

[2] Acemoglu D, Johnson S, Robinson J A. The colonial origins of comparative development: An empirical investigation[J]. American economic review, 2001, 91(5): 1369-1401.

Meta-review: The paper addresses the challenging problem of causal effect estimation from observational data in Mendelian randomization (MR). It studies the identification problem in bi-directional MR and presents complete theoretical results with both necessary and sufficient conditions. Additionally, the paper introduces an algorithm and reports experimental results on both synthetic and real-world datasets.

All reviewers agree that the work tackles a challenging and practical problem. The theoretical results are sound and complete, the algorithm is well-founded, and the evaluations are comprehensive. However, there was some concern about the initial lack of experiments on real-world datasets. In response, the authors added two new experiments involving real-world data in the discussion phase. Reviewers were also interested in understanding the impact of the strong assumptions on the feasibility of the algorithm. The authors provided explanations for these assumptions, which were satisfactory to the reviewers.
Decision: Accept (oral)