Bayesian Bidirectional Mendelian Randomization Under Correlated and Uncorrelated Pleiotropy Using GWAS Summary Statistics

ABSTRACT

Mendelian randomization (MR) uses genetic variants as instruments to study causal relationships between traits. Standard unidirectional MR approaches are vulnerable to pleiotropy and typically assume that pleiotropic effects are independent of instrument strength, assumptions that may fail when causation operates in both directions. We propose BayBiMR, a Bayesian bidirectional MR framework that jointly estimates causal effects in both directions while modeling both correlated and uncorrelated pleiotropy through a hierarchical spike‐and‐slab prior on pleiotropic effects and an Inverse‐Wishart prior on the joint covariance of direct and pleiotropic SNP effects. The likelihood is based on a small‐feedback reduced‐form approximation to the underlying simultaneous‐equation model, which preserves conjugacy and enables efficient closed‐form posterior computation. Posterior inference is carried out via a blocked Gibbs sampler, and a data‐perturbation extension, BayBiMR(DP), is provided to improve empirical frequentist calibration in finite samples. Simulation studies show that BayBiMR and BayBiMR(DP) provide better type I error control than existing methods in the settings considered, particularly under correlated and mixed pleiotropy. Applications to large‐scale GWAS summary statistics on white‐matter (WM) microstructures, BMI, sleep duration, and neuroticism illustrate the method's practical utility as an exploratory and sensitivity tool.

1. Introduction

Mendelian randomization (MR) is a popular approach for causal inference that utilizes genetic variants as instrumental variables (IVs) to mimic randomization in clinical trials [1]. By design, MR helps distinguish correlation from causation while identifying biases arising from confounding and reverse causation [2]. In the MR framework, genetic variants serve as IVs with validity depending on three assumptions: (i) (relevance) the instruments are associated with the exposure; (ii) (exclusion restriction) the instruments affect the outcome only through the exposure; and (iii) (independence) the instruments are not correlated with the unmeasured confounders. Violating these assumptions can lead to problematic causal estimates [3]. For example, uncorrelated pleiotropy occurs when a genetic variant affects the outcome through pathways independent of the exposure, whereas correlated pleiotropy arises when pleiotropic effects are related to the variant's effect on the exposure [4, 5].

A variety of methods have been developed to handle pleiotropy in unidirectional MR. IVW [6], MR‐Egger [3], and Weighted Median [7] provide some robustness under different assumptions about the proportion or pattern of invalid instruments, but all assume a unidirectional causal structure and are not designed for settings where causation operates in both directions.

While conventional MR assumes a unidirectional effect, many biological systems involve feedback loops, motivating bidirectional MR to test causality in both directions [2, 8, 9, 10]. However, pleiotropy can bias inference even in unidirectional MR, and these challenges are amplified in the bidirectional setting, where invalid IVs may distort estimates in both directions.

Bidirectional MR faces several challenges: instruments that are valid in one direction will fail in the other; pleiotropy can bias estimates in both directions; and shared loci associated with both traits may generate reciprocal bias. Existing approaches address these challenges only partially. Steiger filtering [8] infers the likely causal direction from SNP effect sizes but is sensitive to measurement error and does not estimate effect magnitudes. CD‐cML [10] jointly estimates bidirectional effects by penalizing invalid instruments, but does not model the covariance between direct and pleiotropic effects and therefore cannot accommodate correlated pleiotropy [9, 11], which remains particularly problematic as observed associations may reflect shared underlying etiologies rather than genuine bidirectional causality. Addressing this limitation requires new statistical methods to ensure robust inference.

To address this issue, we propose BayBiMR in this research, which is a hierarchical Bayesian framework for bidirectional MR. BayBiMR jointly estimates causal effects in both directions while explicitly modeling both uncorrelated and correlated pleiotropy, providing a unified approach that improves robustness in complex genetic architectures. By leveraging GWAS summary statistics, BayBiMR accommodates scenarios with overlapping or shared invalid IVs, offering coherent uncertainty quantification and interpretable summaries of pleiotropic effects. The key methodological contributions of BayBiMR are: (i) a four‐dimensional latent effect representation that jointly parameterizes direct and pleiotropic SNP effects for both traits within a single hierarchical model; (ii) spike‐and‐slab priors on pleiotropic effects that adaptively shrink invalid instruments toward zero while preserving sensitivity to genuine causal signals; (iii) an Inverse‐Wishart prior on the joint covariance of direct and pleiotropic effects that explicitly captures correlated pleiotropy without invoking the InSIDE assumption; and (iv) a data‐perturbation extension, BayBiMR(DP), that improves the empirical frequentist calibration of uncertainty estimates in finite samples. Through simulation studies, we demonstrate that BayBiMR achieves better control of type I error than existing methods, particularly under correlated pleiotropy. We further illustrate the utility of BayBiMR by applying it to large‐scale genomic datasets, highlighting its ability to uncover potential bidirectional causal relationships.

2. Methods

This section introduces a Bayesian hierarchical framework for bidirectional MR analysis, aimed at estimating the causal effects between two continuous traits, $X$ and $Y$, while allowing for both correlated and uncorrelated pleiotropy. Posterior inference is carried out via a blocked Gibbs sampler.

2.1. Model Specification

For each SNP $j=1,\dots ,p$, the available data are the GWAS summary statistics $({\widehat{\beta }}_{X_j},{\widehat{s}}_{X_j})$ and $({\widehat{\beta }}_{Y_j},{\widehat{s}}_{Y_j})$ for traits $X$ and $Y$, respectively. We also define the corresponding precisions as $u_{X_j}={\widehat{s}}_{X_j}^{-2}$ and $u_{Y_j}={\widehat{s}}_{Y_j}^{-2}$. In this framework, ${\beta }_{X_j}$ and ${\beta }_{Y_j}$ represent the true but unobserved SNP effects on traits $X$ and $Y$, whereas ${\widehat{\beta }}_{X_j}$ and ${\widehat{\beta }}_{Y_j}$ are their estimated counterparts derived from GWAS, which are subject to sampling error.

To explicitly define the data‐generating process, we assume the individual‐level phenotypes are determined by the following structural equation model (SEM):

$$\begin{array}{cc}\hfill X& ={\theta }_{YX}Y+G\mathit{\alpha }+G\mathit{\delta }+{\mathit{ϵ}}_X\hfill \\ \hfill Y& ={\theta }_{XY}X+G\mathit{\eta }+G\mathit{\gamma }+{\mathit{ϵ}}_Y\hfill \end{array}$$ (1)

where $G$ is the $N\times p$ genotype matrix, $\mathit{\alpha }$ and $\mathit{\eta }$ are vectors of direct genetic effects on $X$ and $Y$ (with elements ${\alpha }_j$ and ${\eta }_j$), $\mathit{\delta }$ and $\mathit{\gamma }$ denote vectors of uncorrelated pleiotropic effects (with elements ${\delta }_j$ and ${\gamma }_j$), and ${\mathit{ϵ}}_X,{\mathit{ϵ}}_Y$ are individual‐level disturbance terms. In general, unmeasured confounders acting on both traits may induce $Cov({\mathit{ϵ}}_X,{\mathit{ϵ}}_Y)\ne 0$; this individual‐level dependence is absorbed at the SNP level through the joint covariance ${\mathbf{\Sigma }}_4$ of the latent effects (see below), while the two‐sample design of our likelihood ensures that sampling errors in the GWAS estimates remain independent across cohorts (see the comment following Equation (4)). To jointly capture these effects, we introduce a four‐dimensional latent vector for each SNP, ${\mathbf{L}}_j={({\alpha }_j,{\gamma }_j,{\eta }_j,{\delta }_j)}^{\top }$, where ${\theta }_{XY}$ and ${\theta }_{YX}$ act on the direct effects ${\alpha }_j$ and ${\eta }_j$ to encode the mediated causal pathways between the traits.

We emphasize that, as written in Equation (1), the per‐SNP coefficients ${\alpha }_j$ and ${\delta }_j$ enter the $X$‐equation symmetrically, and similarly ${\eta }_j$ and ${\gamma }_j$ enter the $Y$‐equation symmetrically. Without further structure they would be unidentified and could be merged into a single coefficient ${\alpha }_j+{\delta }_j$ on the $X$ side (and ${\eta }_j+{\gamma }_j$ on the $Y$ side). What separates them in our framework is the prior structure rather than the form of Equation (1): ${\alpha }_j$ and ${\eta }_j$ are modeled with a dense multivariate Gaussian slab through ${\mathbf{L}}_j$ (reflecting the prior belief that valid instruments typically carry non‐negligible direct effects on their target trait), whereas ${\delta }_j$ and ${\gamma }_j$ are modeled with a spike‐and‐slab prior through the indicators $w_j$ and $z_j$ (reflecting the prior belief that pleiotropic effects on the nontarget trait are sparse). Posterior inference therefore partitions each variant into one of two regimes: typical‐magnitude contributions to the $X$‐ or $Y$‐coefficient are absorbed by ${\alpha }_j$ or ${\eta }_j$, while sparse, additional pleiotropic contributions are absorbed by ${\delta }_j$ or ${\gamma }_j$. Identifiability of the pair $(\alpha ,\delta )$ (and symmetrically $(\eta ,\gamma )$) thus follows from the contrast between the dense slab and the sparse spike‐and‐slab, not from Equation (1) alone.

The reduced‐form coefficients ${\beta }_{X_j}$ and ${\beta }_{Y_j}$ cannot be obtained by simple substitution into Equation (1), because the two equations form a simultaneous system with feedback. Writing (1) in matrix form,

$$\left[\begin{array}{l}X\\ Y\end{array}\right]=\left[\begin{array}{ll}0& {\theta }_{YX}\\ {\theta }_{XY}& 0\end{array}\right]\left[\begin{array}{l}X\\ Y\end{array}\right]+\left[\begin{array}{l}G(\mathit{\alpha }+\mathit{\delta })+{\mathit{ϵ}}_X\\ G(\mathit{\eta }+\mathit{\gamma })+{\mathit{ϵ}}_Y\end{array}\right],$$

and solving the linear system under $|{\theta }_{XY}{\theta }_{YX}|<1$ (so that $I-\mathrm{\Theta }$ is invertible with determinant $\mathrm{\Delta }=1-{\theta }_{XY}{\theta }_{YX}$) gives the exact per‐SNP reduced‐form coefficients

$$\begin{array}{cc}\hfill {\beta }_{X_j}^{\text{exact}}& =\frac{{\alpha }_j+{\delta }_j+{\theta }_{YX}({\eta }_j+{\gamma }_j)}{1-{\theta }_{XY}{\theta }_{YX}},\hfill \\ \hfill {\beta }_{Y_j}^{\text{exact}}& =\frac{{\eta }_j+{\gamma }_j+{\theta }_{XY}({\alpha }_j+{\delta }_j)}{1-{\theta }_{XY}{\theta }_{YX}}.\hfill \end{array}$$ (2)

Equation (2) makes the symmetric roles of $({\alpha }_j,{\delta }_j)$ and $({\eta }_j,{\gamma }_j)$ explicit, and shows that, through the feedback loop, ${\gamma }_j$ contributes to ${\beta }_{X_j}$ (and ${\delta }_j$ to ${\beta }_{Y_j}$) via the cross‐trait effect.

In the small‐feedback regime $|{\theta }_{XY}{\theta }_{YX}|\ll 1$, which covers the bulk of biological MR settings (at least one direction is typically modest or null), $1/(1-{\theta }_{XY}{\theta }_{YX})\approx 1$ and the second‐order cross‐pleiotropy terms ${\theta }_{YX}{\gamma }_j$ and ${\theta }_{XY}{\delta }_j$ are dominated by the first‐order terms. Equation (2) then reduces to the working decomposition we use in the likelihood:

$$\begin{array}{cc}\hfill {\beta }_{X_j}& =\underset{\text{Direct}}{\underbrace{{\alpha }_j}}+\underset{\text{Causal}}{\underbrace{{\theta }_{YX}{\eta }_j}}+\underset{\text{Uncorrelated pleiotropy}}{\underbrace{{\delta }_j}},\hfill \\ \hfill {\beta }_{Y_j}& =\underset{\text{Direct}}{\underbrace{{\eta }_j}}+\underset{\text{Causal}}{\underbrace{{\theta }_{XY}{\alpha }_j}}+\underset{\text{Uncorrelated pleiotropy}}{\underbrace{{\gamma }_j}}.\hfill \end{array}$$ (3)

In this approximation, the dropped cross‐pleiotropy terms (i.e., ${\theta }_{YX}{\gamma }_j$ and ${\theta }_{XY}{\delta }_j$) are absorbed into the spike‐and‐slab priors on ${\delta }_j$ and ${\gamma }_j$, so that the operative meaning of these coefficients in the working model is “pleiotropic effect on the nontarget trait, possibly inflated by a second‐order feedback contribution.” Equation (3) is therefore the small‐feedback approximation to the exact reduced form (2), and is the form used throughout the rest of Section 2. We adopt this approximation because (i) it preserves linearity in the latent effects and yields closed‐form Gibbs updates for $({\theta }_{XY},{\theta }_{YX})$ and ${\mathbf{L}}_j$, and (ii) the dominant scientific regime of interest in two‐sample MR has $|{\theta }_{XY}{\theta }_{YX}|$ well below unity. The implications of this approximation are discussed further in Section 5.

Conditional on the latent effects, the observed associations ${({\widehat{\beta }}_{X_j},{\widehat{\beta }}_{Y_j})}^{\top }$ are modeled as independent bivariate normal draws across the SNPs:

$$\left[\begin{array}{l}{\widehat{\beta }}_{X_j}\\ {\widehat{\beta }}_{Y_j}\end{array}\right]|{\mathbf{L}}_j,{\theta }_{XY},{\theta }_{YX}\sim {𝒩}_2\left(\left[\begin{array}{l}{\alpha }_j+{\theta }_{YX}{\eta }_j+{\delta }_j\\ {\eta }_j+{\theta }_{XY}{\alpha }_j+{\gamma }_j\end{array}\right],\left[\begin{array}{ll}{\widehat{s}}_{X_j}^2& 0\\ 0& {\widehat{s}}_{Y_j}^2\end{array}\right]\right).$$ (4)

The diagonal sampling covariance in Equation (4) reflects the two‐sample design adopted throughout this work: ${\widehat{\beta }}_{X_j}$ and ${\widehat{\beta }}_{Y_j}$ are obtained from two nonoverlapping GWAS cohorts (see Section 3 and the real‐data harmonization in Section 4), so their sampling errors are independent across cohorts regardless of whether ${\mathit{ϵ}}_X$ and ${\mathit{ϵ}}_Y$ are correlated at the individual level. Any individual‐level confounding between ${\mathit{ϵ}}_X$ and ${\mathit{ϵ}}_Y$ is propagated into the latent‐effect covariance ${\mathbf{\Sigma }}_4$ (Section 2.2) rather than into the sampling‐error covariance. In a single‐sample design with overlapping cohorts, Equation (4) would need to be extended to include an off‐diagonal sampling covariance reflecting the shared individuals; this extension is outside the scope of the present paper.

This formulation is symmetric with respect to $X$ and $Y$ and explicitly separates distinct sources of genetic effect. To accommodate the empirical observation that pleiotropy is often sparse, we apply spike‐and‐slab priors to ${\gamma }_j$ and ${\delta }_j$ through binary indicators $z_j$ and $w_j$:

$$\begin{array}{ll}\hfill & {\gamma }_j=0\text{if}{z}_j=0,{\delta }_j=0\text{if}{w}_j=0,\\ \hfill & z_j\sim \text{Bernoulli}({\pi }_g),w_j\sim \text{Bernoulli}({\pi }_d).\end{array}$$ (5)

To further address correlated pleiotropy, we allow

$$\text{Cov}({\alpha }_j,{\gamma }_j)\ne 0\text{and}\text{Cov}({\eta }_j,{\delta }_j)\ne 0,$$

which relaxes the InSIDE assumption [12], similar to the approach taken by [13] in modeling correlated pleiotropy.

This hierarchical approach separates direct, mediated, and pleiotropic components of SNP‐trait associations while accounting for sampling uncertainty. By combining a bivariate likelihood with a sparsity‐inducing prior on pleiotropy, it learns which variants act pleiotropically versus causally, yielding clear posterior estimates of bidirectional effects and properly handling invalid instruments.

2.2. Priors and Likelihood

To regularize inference on the bidirectional causal parameters, we assign independent zero‐mean Gaussian priors, which shrink estimates toward the null while allowing data‐driven departures:

$${\theta }_{XY},{\theta }_{YX}\sim 𝒩(0,{\tau }_{\theta }^2).$$

The inclusion probabilities in Equation (5) for uncorrelated horizontal pleiotropy are modeled with conjugate Beta priors, reflecting uncertainty about the proportion of variants with nonzero pleiotropic effects:

$${\pi }_g,{\pi }_d\sim Beta(a_{\pi },b_{\pi }).$$

Each SNP's latent genetic‐effect vector ${\mathbf{L}}_j$ is modeled with a multivariate Gaussian slab prior, which is hierarchical:

$${\mathbf{L}}_j\sim {𝒩}_4(0,{\mathrm{\Sigma }}_4),{\mathrm{\Sigma }}_4\sim ℐ𝒲({\nu }_0,S_0).$$

In our implementation, we set ${\tau }_{\theta }=0.3$ in the Gaussian priors for the causal parameters ${\theta }_{XY}$ and ${\theta }_{YX}$, providing moderate shrinkage toward zero while allowing the data to identify non‐negligible causal effects. For the pleiotropic inclusion probabilities, we used Beta priors with $(a_{\pi },b_{\pi })=(1,19)$, which induce a sparse prior structure with prior mean 0.05 while permitting the data to increase inclusion probabilities when supported by the evidence. For the covariance of the latent SNP effects, we specified an Inverse‐Wishart prior ${\mathrm{\Sigma }}_4\sim ℐ𝒲({\nu }_0,S_0)$ with ${\nu }_0=6$ and $S_0=0.05I_4$, ensuring positive definiteness while providing a diffuse baseline scale for the four‐dimensional latent effect vector.

The $4\times 4$ covariance matrix ${\mathrm{\Sigma }}_4$ is crucial, as its off‐diagonal elements capture dependencies between the different genetic effects. Specifically, the covariances $\text{Cov}({\alpha }_j,{\gamma }_j)={\mathrm{\Sigma }}_{12}$ and $\text{Cov}({\eta }_j,{\delta }_j)={\mathrm{\Sigma }}_{34}$ directly model the correlated pleiotropy. If these terms are nonzero, it indicates a systematic relationship between a SNP's instrument strength and its pleiotropic effect. More generally, ${\mathbf{\Sigma }}_4$ can be written as

$${\mathbf{\Sigma }}_4=\left[\begin{array}{llll}{\mathrm{\Sigma }}_{11}& {\mathrm{\Sigma }}_{12}& {\mathrm{\Sigma }}_{13}& {\mathrm{\Sigma }}_{14}\\ {\mathrm{\Sigma }}_{12}& {\mathrm{\Sigma }}_{22}& {\mathrm{\Sigma }}_{23}& {\mathrm{\Sigma }}_{24}\\ {\mathrm{\Sigma }}_{13}& {\mathrm{\Sigma }}_{23}& {\mathrm{\Sigma }}_{33}& {\mathrm{\Sigma }}_{34}\\ {\mathrm{\Sigma }}_{14}& {\mathrm{\Sigma }}_{24}& {\mathrm{\Sigma }}_{34}& {\mathrm{\Sigma }}_{44}\end{array}\right],$$

where the diagonal entries represent the marginal variances of the direct and pleiotropic effects (${\alpha }_j,{\gamma }_j,{\eta }_j,{\delta }_j$), while the off‐diagonal entries capture all pairwise covariances. Thus, in addition to ${\mathrm{\Sigma }}_{12}$ and ${\mathrm{\Sigma }}_{34}$, the model can also estimate cross‐trait correlations such as ${\mathrm{\Sigma }}_{13}=\text{Cov}({\alpha }_j,{\eta }_j)$, which reflect the correlation between instrument strengths for $X$ and $Y$, and ${\mathrm{\Sigma }}_{14},{\mathrm{\Sigma }}_{23},{\mathrm{\Sigma }}_{24}$, which reflect more complex cross‐dependencies. In particular, when individual‐level confounders inject dependence between ${\mathit{ϵ}}_X$ and ${\mathit{ϵ}}_Y$ in Equation (1), the corresponding SNP‐level signature is absorbed into the off‐diagonals of ${\mathbf{\Sigma }}_4$ rather than into the sampling‐error covariance. ${\mathbf{\Sigma }}_4$ is unknown and can be estimated from the data, providing a flexible way to model correlated pleiotropy and instrument strength heterogeneity across SNPs.

Conditional on the latent vectors and causal parameters, the full likelihood for the observed data factorizes across all $p$ SNPs:

$$\begin{array}{ll}\hfill & ℒ\left({\{{\widehat{\beta }}_{X_j},{\widehat{\beta }}_{Y_j}\}}_{j=1}^p|\mathrm{\Theta },\mathrm{\Psi }\right)=\prod _{j=1}^p𝒩\left({\widehat{\beta }}_{X_j}|{\alpha }_j+{\theta }_{YX}{\eta }_j+{\delta }_j,{\widehat{s}}_{X_j}^2\right)\\ \hfill & 𝒩\left({\widehat{\beta }}_{Y_j}|{\eta }_j+{\theta }_{XY}{\alpha }_j+{\gamma }_j,{\widehat{s}}_{Y_j}^2\right),\end{array}$$ (6)

where $\mathrm{\Theta }=\{{\theta }_{XY},{\theta }_{YX}\}$ are the causal parameters and $\mathrm{\Psi }={\{{\mathbf{L}}_j\}}_{j=1}^p$ collect the latent effects.

2.3. Posterior Computation

We estimated the joint posterior distribution of all unknown quantities using a blocked Gibbs sampling scheme. The parameter set comprised the bidirectional causal effects $\mathrm{\Theta }=\{{\theta }_{XY},{\theta }_{YX}\}$, the pleiotropy parameters $\mathrm{\Phi }=\{{\pi }_g,{\pi }_d,{\mathbf{\Sigma }}_4\}$, the binary inclusion indicators $Z={\{z_j,w_j\}}_{j=1}^p$, and the latent SNP‐level effect vectors $\mathrm{\Psi }={\{{\mathbf{L}}_j\}}_{j=1}^p$. The joint posterior density factorized as

$$\begin{array}{ll}\hfill & p(\mathrm{\Theta },\mathrm{\Phi },Z,\mathrm{\Psi }|·)\propto ℒ(\{{\widehat{\beta }}_{X_j},{\widehat{\beta }}_{Y_j}\}|\mathrm{\Theta },\mathrm{\Psi })p(\mathrm{\Psi }|Z,{\mathbf{\Sigma }}_4)\\ \hfill & p(Z|{\pi }_g,{\pi }_d)p(\mathrm{\Theta })p(\mathrm{\Phi }),\end{array}$$ (7)

combining the likelihood of the observed GWAS summary statistics with the priors on latent effects, indicators, and hyperparameters.

2.3.1. Causal Effects

Updates for the causal parameters ${\theta }_{XY}$ and ${\theta }_{YX}$ followed directly from the normal form of Bayesian linear regression, yielding closed‐form posterior distributions:

$$\begin{array}{cc}\hfill {\theta }_{XY}|-& \sim 𝒩\left(\frac{{\sum }_{j=1}^p{u}_{Y_j}{\alpha }_j\left({\widehat{\beta }}_{Y_j}-{\eta }_j-{\gamma }_j\right)}{{\sum }_{j=1}^p{u}_{Y_j}{\alpha }_j^2+{\tau }_{\theta }^{-2}},\right.\left.{\left(\sum _{j=1}^p{u}_{Y_j}{\alpha }_j^2+{\tau }_{\theta }^{-2}\right)}^{-1}\right),\hfill \\ \hfill {\theta }_{YX}|-& \sim 𝒩\left(\frac{{\sum }_{j=1}^p{u}_{X_j}{\eta }_j\left({\widehat{\beta }}_{X_j}-{\alpha }_j-{\delta }_j\right)}{{\sum }_{j=1}^p{u}_{X_j}{\eta }_j^2+{\tau }_{\theta }^{-2}},\right.\left.{\left(\sum _{j=1}^p{u}_{X_j}{\eta }_j^2+{\tau }_{\theta }^{-2}\right)}^{-1}\right).\hfill \end{array}$$ (8)

2.3.2. Latent Vectors

For each SNP $j$, the full conditional for ${\mathbf{L}}_j$ is a multivariate normal distribution, ${\mathbf{L}}_j|-\sim 𝒩({\mathit{\mu }}_j,{\mathbf{Prec}}_j^{-1})$. The posterior precision matrix is ${\mathbf{Prec}}_j={\mathbf{J}}_4+{\mathbf{Q}}_j$, where ${\mathbf{J}}_4={\mathbf{\Sigma }}_4^{-1}$ is the prior precision and ${\mathbf{Q}}_j$ is the precision contribution from the likelihood. The posterior mean is ${\mathit{\mu }}_j={\mathbf{Prec}}_j^{-1}{\mathbf{h}}_j$. The terms ${\mathbf{Q}}_j$ and ${\mathbf{h}}_j$ can be expressed compactly as

$${\mathbf{Q}}_j=u_{X_j}{\mathbf{a}}_X{\mathbf{a}}_X^{\top }+u_{Y_j}{\mathbf{a}}_Y{\mathbf{a}}_Y^{\top },{\mathbf{h}}_j=u_{X_j}{\widehat{\beta }}_{X_j}{\mathbf{a}}_X+u_{Y_j}{\widehat{\beta }}_{Y_j}{\mathbf{a}}_Y,$$

where the vectors ${\mathbf{a}}_X$ and ${\mathbf{a}}_Y$ encode the linear structure of the model means:

$${\mathbf{a}}_X={\left[\begin{array}{llll}1& 0& {\theta }_{YX}& 1\end{array}\right]}^{\top },{\mathbf{a}}_Y={\left[\begin{array}{llll}{\theta }_{XY}& 1& 1& 0\end{array}\right]}^{\top }.$$

If $z_j=0$ or $w_j=0$, the corresponding elements are removed from ${\mathbf{L}}_j$, and the update proceeds using the relevant sub‐matrices of ${\mathbf{Prec}}_j$ and ${\mathbf{h}}_j$.

2.3.3. Hyperparameters

The remaining updates follow standard conjugate forms. The inclusion indicators $z_j,w_j$ are updated from Bernoulli distributions where the probabilities are proportional to the marginal likelihood of the data under each state. The slab covariance ${\mathbf{\Sigma }}_4$ and sparsity probabilities ${\pi }_g,{\pi }_d$ are updated from their Inverse‐Wishart and Beta conjugate posteriors, respectively. Because ${\gamma }_j$ and ${\delta }_j$ are set to zero under the spike component, the latent vector ${\mathbf{L}}_j$ has structural zeros at inactive coordinates. In our implementation, the Inverse‐Wishart update accumulates outer products ${\mathbf{L}}_j{\mathbf{L}}_j^{\top }$ across SNPs with at least one active pleiotropic component ($z_j+w_j>0$), which has the effect of contributing information only to the active blocks of ${\mathbf{\Sigma }}_4$ at each iteration. The full closed‐form conditional, with explicit selector‐matrix notation, is given in Supplemental Notes S1. The update can be written as

$${\mathbf{\Sigma }}_4|-\sim ℐ𝒲\left({\nu }_0+n_{\text{act}},{\mathbf{S}}_0+\sum _{j:z_j+w_j>0}{\mathbf{L}}_j{\mathbf{L}}_j^{\top }\right),$$

where ${\mathbf{L}}_j={({\alpha }_j,{\gamma }_j,{\eta }_j,{\delta }_j)}^{\top }$ is evaluated with ${\gamma }_j=0$ when $z_j=0$ and ${\delta }_j=0$ when $w_j=0$ (so the scatter matrix has zero contributions on inactive coordinates by construction), and $n_{\text{act}}={\sum }_j\mathbf{1}\{z_j+w_j>0\}$ counts SNPs with at least one active pleiotropic component. Only SNPs with $z_j+w_j>0$ contribute to the update in our implementation; SNPs with $z_j=w_j=0$ are not used to refresh ${\mathbf{\Sigma }}_4$ at the current iteration. This is a deliberate choice that ties the covariance update to the population of variants currently exhibiting active pleiotropy, and is the behavior implemented in the publicly available BayBiMR R/C++ code. The Beta updates take the standard conjugate form,

$$\begin{array}{cc}\hfill {\pi }_g|-& \sim \text{Beta}\left(a_{\pi }+\sum _j{z}_j,b_{\pi }+p-\sum _j{z}_j\right),\hfill \\ \hfill {\pi }_d|-& \sim \text{Beta}\left(a_{\pi }+\sum _j{w}_j,b_{\pi }+p-\sum _j{w}_j\right).\hfill \end{array}$$

This blocked Gibbs sampler alternated between causal parameters, latent effects, and hyperparameters, allowing full posterior inference under the hierarchical model.

2.3.4. Sensitivity to Hyperparameters

Because information accumulates across many SNPs, posterior inference is primarily driven by the likelihood rather than the prior specification. For example, the conditional posterior precision of ${\theta }_{XY}$ is ${\sum }_{j=1}^p{u}_{Yj}{\alpha }_j^2+{\tau }_{\theta }^{-2},$ where ${\alpha }_j$ denotes the latent direct effect of SNP $j$ on trait $X$ (the first component of ${\mathbf{L}}_j$). Provided the prior on ${\mathbf{L}}_j$ assigns positive mass to nonzero ${\alpha }_j$ values, the expected value of ${\sum }_{j=1}^p{u}_{Yj}{\alpha }_j^2$ grows with $p$, so the likelihood precision dominates the fixed prior precision ${\tau }_{\theta }^{-2}$ as the number of variants increases. Consequently, inference for ${\theta }_{XY}$ and ${\theta }_{YX}$ becomes largely insensitive to moderate changes in ${\tau }_{\theta }$.

A similar argument applies to the remaining hyperparameters. The posterior distributions of ${\pi }_g$ and ${\pi }_d$ depend on counts of order $p$, so the Beta hyperparameters $(a_{\pi },b_{\pi })$ contribute only $O(1)$ pseudo‐counts relative to the $O(p)$ information from the data. Likewise, the covariance update

$${\mathbf{\Sigma }}_4|-\sim ℐ𝒲\left({\nu }_0+n_{\text{act}},{\mathbf{S}}_0+{\tilde{\mathbf{S}}}_L\right),{\tilde{\mathbf{S}}}_L=\sum _{j:z_j+w_j>0}{\mathbf{L}}_j{\mathbf{L}}_j^{\top },$$

is dominated by the empirical scatter matrix ${\tilde{\mathbf{S}}}_L$ when the number of active SNPs $n_{\text{act}}$ is large relative to ${\nu }_0$, reducing the influence of the prior scale ${\mathbf{S}}_0$. Overall, posterior estimates are therefore stable under moderate changes in the hyperparameter values when $p$ is reasonably large and the active‐SNP count $n_{\text{act}}$ is not too small.

2.4. Posterior Inference and Implementation

The joint posterior distribution is approximated using a blocked Gibbs sampler that iteratively updates each parameter block from its full conditional distribution (Algorithm 1). This procedure generates samples from which the marginal posterior distributions of all unknowns can be approximated. Inference on the bidirectional causal effects $({\theta }_{XY},{\theta }_{YX})$ is based on their marginal posteriors, typically summarized by posterior means and 95% credible intervals. Variant‐level inference is obtained from posterior inclusion probabilities (PIPs). These probabilities quantify the evidence that SNP $j$ has a direct pleiotropic effect. The posterior distribution of the covariance matrix ${\mathbf{\Sigma }}_4$ further characterizes correlated pleiotropy. All analyses were implemented in R, with core computational routines written in C++ using the Rcpp and RcppArmadillo packages to improve computational efficiency [14, 15].

ALGORITHM 1. Gibbs sampler for the BayBiMR bi‐directional model.

Additional details are provided in the Supporting Information. Supplemental Notes S1 presents the full model specification, notation, and derivations of all Gibbs updates, including selector matrices for active coordinates and closed‐form conditionals for $\mathrm{\Theta }$, ${\mathbf{L}}_j$, ${\mathbf{\Sigma }}_4$ and the inclusion indicators.

Remark 1

(Robustness to pleiotropy) By jointly updating latent genetic effects and their inclusion indicators, the Gibbs sampler separates pleiotropic from mediated causal components. This structure permits valid posterior inference for ${\theta }_{XY}$ and ${\theta }_{YX}$ even under violations of the InSIDE assumption. The posterior distribution of the slab covariance matrix ${\mathbf{\Sigma }}_4$ offers a direct summary of correlated pleiotropy: off‐diagonal elements that shrink toward zero indicate weak correlation, whereas substantial departures from zero signal stronger violations of instrument independence.

2.5. Posterior Propriety and Regularization

The hierarchical model defined in Sections 2.1, 2.2 combines proper priors with a likelihood based on the small‐feedback reduced‐form approximation. Under this specification, the joint posterior is well defined, and the full conditionals of the causal parameters are nondegenerate normal distributions whenever the latent instrument‐strength components are not all zero. We state this as a propriety and regularization result rather than as a likelihood identifiability theorem, because the working likelihood does not in itself separate the causal and pleiotropic contributions to ${\widehat{\beta }}_{X_j}$ and ${\widehat{\beta }}_{Y_j}$: changes in ${\theta }_{XY}{\alpha }_j$ can be partially absorbed by ${\gamma }_j$, and likewise for the reverse direction. The separation of causal from pleiotropic effects in BayBiMR is therefore model‐assisted, achieved through the contrast between the dense Gaussian slab on $({\alpha }_j,{\eta }_j)$ and the sparse spike‐and‐slab prior on $({\delta }_j,{\gamma }_j)$, rather than purely identified from the marginal summary statistics.

Theorem 1

(Posterior propriety and nondegenerate conditional updates) Suppose (i) ${\widehat{s}}_{X_j},{\widehat{s}}_{Y_j}>0$ for all $j$;(ii) the slab prior on ${\mathbf{L}}_j$ has positive variance and assigns positive prior probability to $\{{\alpha }_j\ne 0\}$ for at least one $j$ and to $\{{\eta }_j\ne 0\}$ for at least one $j$;(iii) the priors on $({\theta }_{XY},{\theta }_{YX})$ are proper Gaussians, and those on $({\pi }_g,{\pi }_d,{\mathbf{\Sigma }}_4)$ are proper with finite moments. Then the joint posterior distribution under the working likelihood is proper, and the full conditional distributions of ${\theta }_{XY}$ and ${\theta }_{YX}$ are nondegenerate normal distributions whenever the latent instrument‐strength components are not all zero.

Conditional on $\{{\mathbf{L}}_j\}$, the log‐likelihood for $({\theta }_{XY},{\theta }_{YX})$ decomposes as a sum over SNPs. Because ${\theta }_{XY}$ enters only the $Y$‐likelihood and ${\theta }_{YX}$ enters only the $X$‐likelihood, the two parameters are conditionally independent given $\{{\mathbf{L}}_j\}$, and the conditional Fisher information is block‐diagonal:

$$\begin{array}{ll}\hfill I({\theta }_{XY},{\theta }_{YX}|\mathbf{L})& =\left[\begin{array}{cc}\sum _j{u}_{Y_j}{\alpha }_j^2& 0\\ 0& \sum _j{u}_{X_j}{\eta }_j^2\end{array}\right],\\ \hfill & u_{X_j}={\widehat{s}}_{X_j}^{-2},u_{Y_j}={\widehat{s}}_{Y_j}^{-2}.\end{array}$$

Under assumption (ii), the slab prior assigns positive probability to $\{{\alpha }_j\ne 0\}$ for at least one $j$ and the slab variance is positive, so $𝔼[u_{Y_j}{\alpha }_j^2]>0$ for that $j$; hence, ${\sum }_j{u}_{Y_j}{\alpha }_j^2>0$ with positive prior probability. The same argument applies to ${\sum }_j{u}_{X_j}{\eta }_j^2$. Therefore $I$ is almost surely positive definite, and together with the proper Gaussian priors on $({\theta }_{XY},{\theta }_{YX})$, the conditional posteriors are nondegenerate Gaussians:

$$\begin{array}{ll}\hfill & {\theta }_{XY}|\{{\mathbf{L}}_j\},-\sim 𝒩\left(\frac{B_{XY}}{A_{XY}},A_{XY}^{-1}\right),\\ \hfill & {\theta }_{YX}|\{{\mathbf{L}}_j\},-\sim 𝒩\left(\frac{B_{YX}}{A_{YX}},A_{YX}^{-1}\right),\end{array}$$

where $A_{XY}={\sum }_j{u}_{Y_j}{\alpha }_j^2+{\tau }_{\theta }^{-2}>0$, $B_{XY}={\sum }_j{u}_{Y_j}{\alpha }_j({\widehat{\beta }}_{Y_j}-{\eta }_j-{\gamma }_j)$, and $A_{YX}$, $B_{YX}$ are defined symmetrically. Because the priors on $({\pi }_g,{\pi }_d,{\mathbf{\Sigma }}_4)$ are proper with finite moments and the slab prior on ${\mathbf{L}}_j$ is proper, the joint posterior integrates to a finite normalizing constant [16], completing the proof.

Remark 2

(Model‐assisted separation of causal and pleiotropic effects) Theorem 1 should not be interpreted as likelihood identifiability of the causal and pleiotropic components. Under the working likelihood, the causal contribution ${\theta }_{XY}{\alpha }_j$ and the pleiotropic contribution ${\gamma }_j$ enter ${\widehat{\beta }}_{Y_j}$ additively, so they can trade off on a per‐SNP basis. Separation is achieved across SNPs through the prior structure: ${\alpha }_j$ is treated as a dense, typically non‐negligible direct effect, whereas ${\gamma }_j$ is treated as sparse via the spike‐and‐slab prior. Posterior conclusions about ${\theta }_{XY}$ and ${\theta }_{YX}$ therefore depend on the sparse‐pleiotropy assumption being approximately satisfied. Sensitivity to this assumption is explored empirically in Section 3.

Remark 3

(Posterior consistency) As the number of variants $p$ increases and the direct effects $\{{\alpha }_j,{\eta }_j\}$ remain bounded away from degeneracy, the information terms ${\sum }_j{u}_{Y_j}{\alpha }_j^2$ and ${\sum }_j{u}_{X_j}{\eta }_j^2$ diverge to infinity. Under these conditions, the posterior variance of $({\theta }_{XY},{\theta }_{YX})$ vanishes, so the posterior mass concentrates around the population values that minimize the working‐model Kullback–Leibler divergence to the true data‐generating distribution [17]. Under correct model specification and approximately sparse pleiotropy, these limits coincide with the true causal parameters; in general, however, the limiting values reflect the regularization induced by the prior as well as any model misspecification.

2.6. Data Perturbation (DP) via Jittered‐Summary Resampling

When GWAS sample sizes are modest and weak or invalid instruments are common, the selection properties of BayBiMR may not fully hold, allowing some pleiotropic variants to escape detection and bias the estimated effects. To help guard against this, we applied a data‐perturbation scheme inspired by parametric resampling following [18, 19]. Rather than resampling from a fitted model in the strict sense of a parametric bootstrap, this scheme jitters the observed summary statistics using independent Gaussian noise calibrated to the reported standard errors, and re‐fits BayBiMR on each perturbed dataset. This procedure relies on the assumption that GWAS summary statistics are approximately normally distributed, with the reported standard errors treated as the true noise standard deviations. By repeatedly generating pseudo‐datasets and re‐estimating the causal effects with the full BayBiMR model, we can assess the stability of the results. This provides a practical, approximate frequentist view of the uncertainty arising from sampling variation in the summary statistics.

For each replicate $b=1,\dots ,B$, perturbed statistics are generated as

$${\widehat{\beta }}_{X_j}^{\ast (b)}={\widehat{\beta }}_{X_j}+{ϵ}_{X_j}^{\ast },{ϵ}_{X_j}^{\ast }\sim 𝒩(0,{\widehat{s}}_{X_j}^2),$$

$${\widehat{\beta }}_{Y_j}^{\ast (b)}={\widehat{\beta }}_{Y_j}+{ϵ}_{Y_j}^{\ast },{ϵ}_{Y_j}^{\ast }\sim 𝒩(0,{\widehat{s}}_{Y_j}^2),$$

where $({\widehat{\beta }}_{X_j},{\widehat{s}}_{X_j})$ and $({\widehat{\beta }}_{Y_j},{\widehat{s}}_{Y_j})$ are the observed marginal estimates and reported standard errors in GWAS summary statistics.

Each perturbed dataset ${\{({\widehat{\beta }}_{X_j}^{\ast (b)},{\widehat{\beta }}_{Y_j}^{\ast (b)})\}}_{j=1}^p$ is then analyzed with the same MCMC sampler as the original data, producing point estimates ${\widehat{\theta }}_{XY}^{(b)}$ and ${\widehat{\theta }}_{YX}^{(b)}$ for that replicate. Although the DP procedure requires running the Gibbs sampler $B$ times, the computation remains efficient due to the C++ implementation. The complexity of the Gibbs sampler is $O(pT)$, where $p$ denotes the number of SNPs and $T$ the number of MCMC iterations. In a typical setting with $p=50$ SNPs and $T=2000$ iterations, a single BayBiMR run takes approximately 0.04 s, while BayBiMR(DP) with $B=100$ perturbations requires about 3.5 s in total. All computations were performed on a MacBook Pro with an Apple M3 Max chip and 36 GB RAM. From the $B$ bootstrap replicates, the averages are given by

$${\bar{\theta }}_{XY}=\frac{1}{B}\sum _{b=1}^B{\widehat{\theta }}_{XY}^{(b)},{\bar{\theta }}_{YX}=\frac{1}{B}\sum _{b=1}^B{\widehat{\theta }}_{YX}^{(b)},$$

which are used as alternative point estimates. The empirical standard deviations across replicates serve as frequentist standard error estimates,

$$\widehat{\text{sd}}({\theta }_{XY})={\left\{\frac{1}{B-1}\sum _{b=1}^B{\left({\widehat{\theta }}_{XY}^{(b)}-{\bar{\theta }}_{XY}\right)}^2\right\}}^{1/2},$$

with an analogous expression for ${\theta }_{YX}$. Equal‐tailed percentile intervals are given by the $2.5\%$ and $97.5\%$ quantiles of the bootstrap distribution.

2.7. Asymptotic Properties of the DP Estimators

Let ${\{{\widehat{\theta }}^{(b)}\}}_{b=1}^B$ denote data‐perturbation replicates of an estimator $\widehat{\theta }$ for a causal effect parameter $\theta \in \{{\theta }_{XY},{\theta }_{YX}\}$, generated by the DP scheme described above. Conditional on the observed data, these replicates are independent draws from the perturbation distribution $P^{\star }$, which treats the observed GWAS summary statistics and their reported standard errors as the data‐generating mechanism.

Two limiting arguments support the use of $P^{\star }$ to approximate sampling variability. As $B\to \infty$, the empirical mean and variance of the replicates converge almost surely to their population counterparts under $P^{\star }$ by the strong law of large numbers, so that Monte Carlo error in the DP summaries vanishes with sufficient replicates. As the original GWAS sample size $n\to \infty$, under regularity conditions, and when the perturbation law adequately approximates the local sampling distribution of the summary statistics, standard resampling theory [20] suggests that the empirical distribution of the perturbed estimates provides a practical approximation to frequentist uncertainty, provided $\widehat{\theta }$ is a sufficiently regular function of the summary statistics. We emphasize that this is an approximation argument rather than a formal consistency result: the DP scheme does not resample from a fitted model in the strict parametric‐bootstrap sense, and formal calibration guarantees would require additional conditions on instrument validity, model correctness, and the alignment of the perturbation law with the true sampling distribution. The simulation studies in Section 3 indicate that the DP scheme improves empirical frequentist calibration of BayBiMR in the settings considered, but it should not be interpreted as ensuring calibration in general.

3. Simulation Studies

We study finite‐sample performance under a bidirectional SEM that allows both uncorrelated and correlated pleiotropy. Each replication generates individual‐level data, constructs two nonoverlapping GWAS samples, forms two‐sample summary statistics, and evaluates competing methods on type I error and power.

Although BayBiMR is a Bayesian framework, we primarily evaluate its performance using frequentist metrics (Type I error and Power) to facilitate a direct and fair comparison with established frequentist methods such as IVW, MR‐Egger, and CD‐cML. Bayesian coverage probabilities are reported in the Supporting Information (Tables S1–S4) to assess the calibration of the credible intervals.

In all simulations, the Gibbs sampler was run for 4000 iterations, with the first 1500 discarded as burn‐in. The remaining chain was thinned by a factor of 2 to reduce autocorrelation.

3.1. Data‐Generating Mechanism

We generated data under a bidirectional structural equation model (SEM) with explicit instrument and pleiotropic SNP sets. In total, $p=p_X+p_Y+p_U$ SNPs were divided into instruments for $X$ ($G_X$), instruments for $Y$ ($G_Y$), and pleiotropic variants ($G_U$), with typical configurations such as $(p_X,p_Y,p_U)\in \{(5,5,5),(5,5,10)\}$. Genotypes were simulated independently as $G_{ij}\sim \mathrm{Bin}(2,0.3)$ with a minor allele frequency (MAF) of 0.3. Direct genetic effects were assigned random magnitudes in $(0.2,0.3)$ and random signs drawn from independent Rademacher variables, $s_k,t_{\ell },u_m\sim \text{Rademacher}(\pm 1)$:

$$\begin{array}{cccc}\hfill {\alpha }_k& \sim \text{Unif}(0.2,0.3)·s_k,\hfill & \hfill k& =1,\dots ,p_X,\hfill \\ \hfill {\eta }_{\ell }& \sim \text{Unif}(0.2,0.3)·t_{\ell },\hfill & \hfill \ell & =1,\dots ,p_Y,\hfill \\ \hfill c_m,d_m& \sim \text{Unif}(0.2,0.3)·u_m,\hfill & \hfill m& =1,\dots ,p_U.\hfill \end{array}$$

Here ${\alpha }_k$ and ${\eta }_{\ell }$ denote the direct genetic effect magnitudes for $G_X$ and $G_Y$ instruments (consistent with the model notation in Section 2), and $c_m$ and $d_m$ denote the simulation‐level pleiotropic effect magnitudes for $G_U$ SNPs (distinct from the model parameters ${\gamma }_j$ and ${\delta }_j$). This specification yields two sets of valid instruments and a third set of SNPs with pleiotropic effects on both traits, mimicking realistic scenarios of pleiotropy in GWAS‐based MR studies.

Correlated pleiotropy was introduced by making the pleiotropic effects on the $G_U$ SNPs depend on a SNP‐specific weight ${\xi }_m\sim \text{Unif}(-0.3,0.3)$ for $m=1,\dots ,p_U$. Specifically, for SNPs in $G_U$ we set the per‐SNP pleiotropic effects so that ${\gamma }_m$ is correlated with the direct effect magnitude $c_m$ and ${\delta }_m$ is correlated with $d_m$ through ${\xi }_m$. This produces SNP‐level dependence between instrument‐strength components and pleiotropic components on $G_U$, which is precisely the structure that the off‐diagonal entries of ${\mathbf{\Sigma }}_4$ are designed to capture. Phenotypes were generated under the bidirectional structural equation model

$$\begin{array}{ll}\hfill X& ={\theta }_{YX}Y+G_X\mathit{\alpha }+G_U\mathit{\delta }+{\epsilon }_X,\\ \hfill Y& ={\theta }_{XY}X+G_Y\mathit{\eta }+G_U\mathit{\gamma }+{\epsilon }_Y,\end{array}$$

where $\mathit{\alpha }$ and $\mathit{\eta }$ represent the vectors of direct SNP effects on $X$ and $Y$ (consistent with the model notation in Section 2), and $\mathit{\delta }$ and $\mathit{\gamma }$ correspond to pleiotropic effects (uncorrelated when generated independently of ${\xi }_m$; correlated with instrument‐strength components when generated as described above). In the simulation, direct effects are nonzero only for their designated instrument groups ($G_X$ for $\mathit{\alpha }$, $G_Y$ for $\mathit{\eta }$), while pleiotropic effects are nonzero only for $G_U$. Error terms were independently distributed as ${\epsilon }_X,{\epsilon }_Y\sim 𝒩(0,1)$. In matrix form,

$$\left[\begin{array}{l}X\\ Y\end{array}\right]=\left[\begin{array}{ll}0& {\theta }_{YX}\\ {\theta }_{XY}& 0\end{array}\right]\left[\begin{array}{l}X\\ Y\end{array}\right]+\left[\begin{array}{l}{G}_X\mathit{\alpha }+G_U\mathit{\delta }+{\epsilon }_X\\ G_Y\mathit{\eta }+G_U\mathit{\gamma }+{\epsilon }_Y\end{array}\right].$$

Solving this system yields the reduced‐form expressions:

$$\begin{array}{cc}\hfill X& =\frac{(G_X\mathit{\alpha }+G_U\mathit{\delta }+{\epsilon }_X)+{\theta }_{YX}(G_Y\mathit{\eta }+G_U\mathit{\gamma }+{\epsilon }_Y)}{1-{\theta }_{XY}{\theta }_{YX}},\hfill \\ \hfill Y& =\frac{(G_Y\mathit{\eta }+G_U\mathit{\gamma }+{\epsilon }_Y)+{\theta }_{XY}(G_X\mathit{\alpha }+G_U\mathit{\delta }+{\epsilon }_X)}{1-{\theta }_{XY}{\theta }_{YX}}.\hfill \end{array}$$

The denominator $1-{\theta }_{XY}{\theta }_{YX}$ captures the feedback between $X$ and $Y$; when $|{\theta }_{XY}{\theta }_{YX}|<1$, the reduced‐form solutions are well defined. We note that these simulation reduced‐form expressions agree with the exact reduced form in Equation (2), and that the inference procedure based on the small‐feedback approximation (3) is evaluated against data generated from this exact reduced form. The simulation settings reported in Section 3 use $|{\theta }_{XY}{\theta }_{YX}|\le 0.04$, so the approximation factor $1/(1-{\theta }_{XY}{\theta }_{YX})$ is within 5% of unity throughout.

3.2. GWAS Summary Statistics Generation

For each replication, $2n$ individuals are sampled and split into two independent cohorts of size $n$. In the first sample, we regress $X$ on each SNP to obtain $({\widehat{\beta }}_{X_j},{\widehat{s}}_{X_j})$. In the second sample, we regress $Y$ on each SNP to obtain $({\widehat{\beta }}_{Y_j},{\widehat{s}}_{Y_j})$. These marginal estimates serve as input summary statistics for all methods in both directions.

3.3. Methods and Evaluation

We compared BayBiMR, BayBiMR(DP) (with $B=100$ data perturbations), IVW, MR‐Egger, Weighted Median, CD‐cML [10], and an oracle two‐stage estimator. The oracle estimator uses the true valid SNP sets ($G_X$ for $X\to Y$ and $G_Y$ for $Y\to X$). Because SNPs are generated independently in the simulations, linkage disequilibrium (LD) is absent and does not affect inference. The oracle, therefore, represents an approximate upper bound on achievable performance under perfect knowledge of valid instruments. For fair comparisons across competing methods, we used the DP version of CD‐cML to obtain uncertainty estimates. For the proposed method, BayBiMR(DP) applies DP to account for the additional post‐selection uncertainty introduced by the spike‐and‐slab prior.

For each method and causal direction, type I error and empirical power were defined as the proportions of replicates excluding zero from the 95% interval at $\alpha =0.05$ using Bayesian credible intervals for BayBiMR and Wald‐type confidence intervals for all others. Although BayBiMR is formulated within a Bayesian framework, evaluating its frequentist operating characteristics under repeated sampling is standard practice for assessing calibration and enabling direct comparison with existing MR methods.

3.4. Power and Type I Error Across Effect Settings

We first considered scenarios with no causal effect from $X$ to $Y$ (${\theta }_{XY}=0$) and varied the true effect of $Y$ on $X$ over $\{0,0.05,0.10,0.20\}$ with $n=10000$ per GWAS. We also set the number of pleiotropic variants to be $(p_X,p_Y,p_U)=(5,5,5)$. In this case, the $X\to Y$ direction represents a null setting for assessing type‐I error, while the $Y\to X$ direction allows us to study power as the effect increases. Results are presented in Table 1 and Figure 1. As expected, the oracle estimator provides the theoretical upper bound of performance. BayBiMR and BayBiMR(DP) closely track the oracle in type I error control for $X\to Y$, while showing marked power gains for $Y\to X$. In contrast, CD‐cML and Weighted Median achieve higher power at the cost of inflated error rates, and both IVW and MR‐Egger demonstrate liberal tendencies when pleiotropic effects are present.

TABLE 1.

Empirical type‐I error for $X\to Y$ (with ${\theta }_{XY}=0$) and power for $Y\to X$ across ${\theta }_{YX}\in \{0,0.05,0.10,0.20\}$, based on $n=10000$ per GWAS. Reported values are rejection probabilities at the $\alpha =0.05$ level.

Method	$X\to Y$				$Y\to X$
	$0$	$0.05$	$0.10$	$0.20$	$0$	$0.05$	$0.10$	$0.20$
BayBiMR	0.082	0.082	0.084	0.082	0.078	0.138	0.402	0.832
BayBiMR(DP)	0.044	0.058	0.054	0.056	0.064	0.110	0.332	0.768
IVW	0.092	0.134	0.182	0.312	0.092	0.132	0.178	0.308
MR‐Egger	0.168	0.106	0.114	0.150	0.192	0.216	0.252	0.298
Weighted Median	0.288	0.338	0.386	0.456	0.322	0.420	0.568	0.872
CD‐cML	0.120	0.104	0.102	0.098	0.112	0.174	0.370	0.714
Oracle	0.046	0.046	0.044	0.044	0.080	0.222	0.518	0.932

FIGURE 1.

Simulation results under independent SNPs ($n=10000$ per GWAS; $p_X=p_Y=p_U=5$) showing empirical type I error for the null $X\to Y$ direction (${\theta }_{XY}=0$) and empirical power for the $Y\to X$ direction as the true effect ${\theta }_{YX}$ increases over $\{0,0.05,0.10,0.20\}$.

We then fixed ${\theta }_{XY}=0.2$ to represent a nonzero effect from $X$ to $Y$, while again varying ${\theta }_{YX}$ with the same sample size. Here, the $X\to Y$ direction allows us to study power, and the $Y\to X$ direction once more reflects type‐I error (or power when nonzero). As shown in Table 2 and Figure 2, BayBiMR and BayBiMR(DP) deliver high power for $X\to Y$ while controlling errors in the reverse direction. Weighted Median and CD‐cML also perform strongly in terms of power, though with inflated error, whereas IVW and MR‐Egger remain less competitive under pleiotropy. Oracle again sets the upper bound in both directions.

TABLE 2.

Power for $X\to Y$ with ${\theta }_{XY}=0.2$ and type‐I error/power for $Y\to X$ across ${\theta }_{YX}$, based on $n=10000$ per GWAS. Reported values are rejection probabilities at the $\alpha =0.05$ level.

Method	$X\to Y$				$Y\to X$
	$0$	$0.05$	$0.10$	$0.20$	$0$	$0.05$	$0.10$	$0.20$
BayBiMR	0.842	0.842	0.846	0.852	0.086	0.166	0.416	0.842
BayBiMR(DP)	0.762	0.758	0.764	0.764	0.048	0.114	0.316	0.750
IVW	0.308	0.396	0.472	0.632	0.314	0.388	0.474	0.632
MR‐Egger	0.274	0.182	0.132	0.150	0.154	0.142	0.142	0.148
Weighted Median	0.876	0.896	0.904	0.934	0.482	0.560	0.696	0.920
CD‐cML	0.674	0.684	0.682	0.700	0.098	0.156	0.358	0.734
Oracle	0.932	0.932	0.934	0.934	0.080	0.240	0.534	0.940

FIGURE 2.

Simulation results under independent SNPs ($n=10000$ per GWAS; $p_X=p_Y=p_U=5$) showing empirical power for the nonzero $X\to Y$ direction (${\theta }_{XY}=0.2$ fixed) and empirical type‐I error (or power when nonzero) for the $Y\to X$ direction as the true effect ${\theta }_{YX}$ varies over $\{0,0.05,0.10,0.20\}$.

3.5. Power and Type I Error Under Large‐Sample Conditions

We increased the sample size to $n=50000$ per GWAS, keeping all other settings unchanged. The qualitative conclusions are largely consistent with the $n=10000$ results: BayBiMR(DP) continues to control type I error for $X\to Y$ close to the oracle level, while BayBiMR without perturbation shows somewhat elevated error at this sample size; both methods gain substantially more power for $Y\to X$ as the sample grows. Weighted Median and CD‐cML again achieve high power at the cost of inflated error, and IVW/MR‐Egger remain liberal under pleiotropy. Complete numerical results are provided in Tables S5, S6 of the Supporting Information; the figures below summarize the key patterns (Figures 3 and 4).

FIGURE 3.

Simulation results under independent SNPs with increased sample size ($n=50000$ per GWAS; $p_X=p_Y=p_U=5$) showing empirical type I error for the null $X\to Y$ direction (${\theta }_{XY}=0$ fixed) and empirical power for the $Y\to X$ direction as the true effect ${\theta }_{YX}$ varies over $\{0,0.05,0.10,0.20\}$. Full numerical results are in Table S5.

FIGURE 4.

Simulation results under independent SNPs with increased sample size ($n=50000$ per GWAS; $p_X=p_Y=p_U=5$) showing empirical power for the nonzero $X\to Y$ direction (${\theta }_{XY}=0.2$ fixed) and empirical type I error (or power when nonzero) for the $Y\to X$ direction as the true effect ${\theta }_{YX}$ varies over $\{0,0.05,0.10,0.20\}$. Full numerical results are in Table S6.

3.6. Impact of a Larger Proportion of Invalid IVs With Mixed Pleiotropy

We increased the number of pleiotropic variants to $(p_X,p_Y,p_U)=(5,5,10)$ with $n=10000$, generating a mixture of invalid SNPs: 20% uncorrelated $(\gamma ,\delta \ne 0,\xi =0)$, 20% correlated $(\xi \ne 0,\gamma =\delta =0)$, and 60% with both. The true effects were set to $({\theta }_{XY},{\theta }_{YX})=(0.2,0)$.

We first examined a setting with no effect from $X$ to $Y$ $({\theta }_{XY}=0)$ while varying the $Y\to X$ effect over $\{0,0.05,0.10,0.20\}$. In this case, the $X\to Y$ direction assesses type‐I error, whereas $Y\to X$ reflects power. BayBiMR and BayBiMR(DP) maintain type I error near the oracle level while gaining power for $Y\to X$. Weighted Median achieves high power but elevated type I error, while CD‐cML shows lower power yet remains liberal in error control. Both IVW and MR‐Egger behave liberally under pleiotropy.

We then fixed ${\theta }_{XY}=0.20$ to represent a nonzero effect from $X$ to $Y$ and again varied ${\theta }_{YX}$. Here $X\to Y$ measures power and $Y\to X$ type I error. BayBiMR and BayBiMR(DP) control type‐I error in both directions. Weighted Median remains powerful but liberal; IVW and MR‐Egger underperform, and CD‐cML shows modest power with a mildly inflated type I error. Complete numerical results and figures are provided in Tables S7, S8 and Figures S7, S8 of the Supporting Information.

3.7. Weak‐Instrument Setting

We next consider a weak‐instrument scenario with $n=10000$, $(p_X,p_Y,p_U)=(5,5,6)$, and $({\theta }_{XY},{\theta }_{YX})=(0.2,0)$. For both $G_X$ and $G_Y$, at least half of the instrument effects are drawn from a uniform distribution on $(0.01,0.08)$ (weak instruments), while the remainder are sampled from $(0.20,0.30)$ (strong instruments), each assigned a random sign.

We first examined the case ${\theta }_{XY}=0$ while varying the $Y\to X$ effect over $\{0,0.05,0.10,0.20\}$. BayBiMR and BayBiMR(DP) retain type I error control close to the oracle estimator while maintaining competitive power for $Y\to X$. Weighted Median attains higher power but inflates type I error, while CD‐cML shows moderately elevated error rates. IVW and MR‐Egger continue to demonstrate liberal behavior under pleiotropy.

We then fixed ${\theta }_{XY}=0.20$ to study a nonzero effect from $X$ to $Y$ and again varied ${\theta }_{YX}$. BayBiMR and BayBiMR(DP) achieve high power for $X\to Y$ while controlling error in the reverse direction. Weighted Median remains powerful but liberal, whereas IVW and MR‐Egger underperform relative to BayBiMR methods. CD‐cML attains similar power to BayBiMR(DP) while showing a slightly inflated type I error. Complete numerical results and figures are provided in Tables S9, S10 and Figures S9, S10 of the Supporting Information.

Across the simulation scenarios, BayBiMR methods performed comparably to the Oracle in detecting the true causal direction and were consistently more reliable than classical MR approaches under correlated or mixed pleiotropy. The bootstrap version BayBiMR(DP) offered the most stable type I error control, generally close to the oracle estimator, though with some loss of power. Existing estimators such as IVW, MR‐Egger, Weighted Median, and CD‐cML often show noticeably inflated false‐positive rates, particularly in the presence of pleiotropy or weak instruments.

3.8. Supplemental Simulations

To further assess robustness, we conducted additional simulations designed to reflect more realistic genetic architectures and heterogeneous pleiotropy patterns (Supporting Information S2).

In Simulation S2.1, we generated 500 GWAS replicates with $n=10000$ individuals per trait and 15 instruments allocated via a multinomial draw to direct ($p_X$), reverse ($p_Y$), and pleiotropic ($p_U$) groups, ensuring at least three variants per group. Mixed pleiotropy was introduced by sampling pleiotropic effect magnitudes from $\text{Unif}(0.2,0.3)$ with random signs and correlated pleiotropy weights $\xi \sim \text{Unif}(-0.3,0.3)$. The true causal effects were set to ${\theta }_{XY}=0.2$ and ${\theta }_{YX}=0$. Uncertainty was quantified using 95% credible intervals for BayBiMR and Wald 95% confidence intervals for competing methods. Empirical performance was evaluated using bias, RMSE, coverage probability, and rejection rates (Supplementary Section S2).

BayBiMR and BayBiMR(DP) exhibited low bias and coverage probabilities close to the nominal level in both causal directions, with performance approaching the oracle benchmark. In contrast, IVW, MR‐Egger, and Weighted Median showed substantial positive bias, poor coverage, and inflated type I error rates, while CD‐cML demonstrated moderate bias and intermediate coverage performance.

Simulation S2.2 increased both the number of instruments and the degree of pleiotropy (45 total SNPs). BayBiMR and BayBiMR(DP) again maintained low bias and high coverage in both directions, with rejection rates for $X\to Y$ remaining high (0.96/0.924) and type I error for $Y\to X$ remaining close to the oracle estimator (0.086/0.032). Under these more challenging conditions, IVW and MR‐Egger produced highly variable and biased estimates with severely inflated false‐positive rates; Weighted Median tended to overestimate effects with low coverage; and CD‐cML underestimated effect sizes. The relative ranking of methods remained consistent, although the advantages of the Bayesian approaches became more pronounced as pleiotropy increased.

Overall, the supplemental simulations demonstrate that BayBiMR and its more conservative variant, BayBiMR(DP), maintain reliable type I error control and interval coverage while preserving strong power to detect nonzero causal effects. These results underscore the advantages of the proposed Bayesian methods in settings with large instrument sets and pervasive pleiotropy.

4. Real Data Applications

We applied our bidirectional MR frameworks to white‐matter (WM) microstructures, using the same set of instruments and estimators to evaluate causal links between the outcomes and complex traits in both directions. The corresponding GWAS data sources and sample sizes for all traits are summarized in Table 3.

TABLE 3.

Summary of GWAS data sources used for genetic traits in real data applications.

Trait	Study	Sample size ($n$)
White‐matter microstructures	UK Biobank [21]	25,415
Plasma APP	UK Biobank [22]	47,745
Sleep duration	UK Biobank [23]	446,118
BMI	UK Biobank [24]	532,396
Neuroticism	UK Biobank [25]	170,911
Depressive symptoms	UK Biobank [25]	161,460
Subjective well‐being	UK Biobank [25]	298,420

The six genetic traits examined were plasma amyloid precursor protein (APP), sleep duration (SLP), neuroticism (NEU), depressive symptoms (DS), subjective well‐being (SWB), and body mass index (BMI), each assessed for bidirectional relationships with WM microstructures. GWAS summary statistics for WM microstructures were obtained from 25,415 UK Biobank participants free of stroke, dementia, and other major central nervous system disorders [21]. Plasma APP was analyzed using summary statistics from study GCST90468344 [22]. SLP data were derived from a GWAS of 446,118 adults of European ancestry in the UK Biobank [23], and BMI from study GCST90029007 [24]. Summary statistics for SWB ($n=298420$), DS ($n=161460$), and NEU ($n=170911$) were obtained from a large GWAS meta‐analysis in the UK Biobank [25].

4.1. Various Phenotypes With WM Microstructures

In our analyses, SNPs were harmonized to the relevant GWAS, retained if they reached genome‐wide significance in either trait ($p\le 5\times 10^{-8}$), and then pruned for independence using LD clumping in PLINK with the 1000 Genomes EUR reference panel [26] ($r^2\le 0.001$, 10 Mb window). Across all six phenotypes, associations at the $p<0.05$ threshold (we use 95% CrIs for BayBiMR and Wald 95% CIs for all other methods) are shown in Figure 5. Our analyses reveal clear differences in the patterns of significant microstructure‐trait pairs across methods and directions of inference.

FIGURE 5.

Significant associations between WM microstructures and various MR methods in bidirectional analyses of APP, BMI, SLP, NEU, DS and SWB. The direction $X\to Y$ denotes effects from WM microstructures to the traits, and $Y\to X$ denotes effects from the traits to WM microstructures.

4.1.1. Plasma APP and WM Microstructures

Across all methods, the strongest pattern for plasma APP emerged for associations from WM microstructures to circulating APP levels. The most consistent finding was the mean diffusivity of the middle cerebellar peduncle (MCP‐MD), detected repeatedly by both BayBiMR and BayBiMR(DP). In the opposite direction, a broader but less consistent set of associations was identified, with BayBiMR most reliably highlighting MCP‐MD and SCP‐FA. These findings point to potential bidirectional effects of APP on specific cerebellar pathways.

4.1.2. BMI and WM Microstructures

Analyses of BMI revealed the most extensive pattern of associations. Both BayBiMR and BayBiMR(DP) detected a large set of microstructure‐trait links in each direction, spanning nearly the entire cerebellar peduncle panel (ICP‐FA/MD, MCP‐FA/MD, SCP‐FA/MD). In contrast, conventional estimators recovered only a limited subset of these associations, most often restricted to MCP‐MD. Taken together, these results suggest that BMI is broadly related to WM microstructure and that the proposed estimators capture the most comprehensive view of these effects.

4.1.3. NEU and WM Microstructures

Findings for neuroticism were considerably more selective. When modeling effects from WM microstructure to NEU, only BayBiMR detected a small cluster of associations (ICP‐MD and MCP‐MD), whereas BayBiMR(DP) yielded no signals in this direction. In the opposite direction, signals were sparse and varied by method, with MR‐Egger identifying associations at ICP‐FA and ICP‐MD. Overall, these results suggest that links between neuroticism and WM microstructure are more limited and heterogeneous than those observed for APP or BMI.

4.1.4. SLP and WM Microstructures

BayBiMR yielded the most extensive evidence for associations from WM microstructures to SLP, particularly in the inferior, middle, and superior cerebellar peduncles (ICP‐FA/MD, MCP‐FA/MD, and SCP‐MD). In the opposite direction, the Weighted Median estimator identified more associations (ICP‐FA/MD, MCP‐MD, and SCP‐MD) than BayBiMR, which detected only ICP‐MD. These results point to a robust but directionally asymmetric relationship between SLP and cerebellar microstructure.

4.1.5. DS and WM Microstructures

Results for DS were sparse. In the direction from WM microstructures to DS, only the Weighted Median, CD‐cML, and MR‐Egger identified a few associations, mainly involving ICP‐MD and MCP/SCP‐MD. In the opposite direction, signals were limited to MR‐Egger at ICP‐FA and MCP‐MD. Both BayBiMR and BayBiMR(DP) produced no associations or were highly selective, indicating weak and inconsistent evidence for links between DS and WM microstructure.

4.1.6. SWB and WM Microstructures

In analyses treating SWB as the outcome, associations were limited to IVW and MR‐Egger, which identified effects at ICP‐FA, ICP‐MD, and SCP‐FA, while BayBiMR detected none. In the opposite direction, BayBiMR(DP), the Weighted Median, and IVW each captured a small number of signals (including ICP‐MD, SCP‐MD, and ICP‐FA), again with BayBiMR showing no significant signals. The overall pattern suggests scattered, method‐dependent links between SWB and WM microstructures.

BayBiMR detected more trait‐microstructure associations than BayBiMR(DP) across all six phenotypes, while BayBiMR(DP) produced a smaller but more conservative set of findings. This contrast was most evident for mean diffusivity in the middle cerebellar peduncle (MCP‐MD) and related cerebellar tracts. Whether the additional associations detected by BayBiMR but not BayBiMR(DP) represent true causal effects or reflect residual pleiotropy or false positives cannot be determined from these data alone, and replication in independent cohorts would be needed to distinguish these possibilities. The overall tally of significant associations appears in Table 4, with per‐trait, per‐direction breakdowns in Tables 5 and 6. For the affective traits (NEU, DS, and SWB), all methods yielded fewer associations, with BayBiMR(DP) retaining only the most robust pairs.

TABLE 4.

Number of significant trait‐microstructure associations detected by each MR method, aggregated across all six traits and all WM microstructures. The notation $X\to Y$ indicates effects from WM microstructures to the traits, and $Y\to X$ indicates effects from the traits to WM microstructures.

Method	$X\to Y$	$Y\to X$	Total
BayBiMR	14	10	24
BayBiMR(DP)	11	9	20
IVW	3	4	7
MR‐Egger	8	12	20
Weighted Median	9	15	24
CD‐cML	4	0	4

TABLE 5.

Number of significant trait‐microstructure associations by trait and method for the direction $X\to Y$ (effects from WM microstructures to the traits).

Trait	BayBiMR(DP)	BayBiMR	CD‐cML	MR‐Egger	IVW	Weighted Median
APP	2	1	0	0	1	2
BMI	6	6	1	2	1	4
SLP	3	5	2	0	0	2
NEU	0	2	0	0	0	0
SWB	0	0	0	3	1	0
DS	0	0	1	3	0	1

TABLE 6.

Number of significant trait‐microstructure associations by trait and method for the direction $Y\to X$ (effects from the traits to WM microstructures).

Trait	BayBiMR(DP)	BayBiMR	CD‐cML	MR‐Egger	IVW	Weighted Median
APP	2	4	0	4	1	6
BMI	5	5	0	2	2	4
SLP	0	1	0	1	0	4
NEU	0	0	0	2	0	0
SWB	2	0	0	1	1	1
DS	0	0	0	2	0	0

Repeated signals in the middle and inferior cerebellar peduncles suggest that metabolic and neuropsychiatric traits may converge on pathways supporting cortico‐cerebellar communication, which is central to motor coordination, cognitive integration, and emotional regulation [27, 28, 29]. These patterns are consistent with prior neuroimaging work linking WM integrity in these microstructures to both metabolic status [30] and affective symptomatology [31]. Figure 5 and Tables 4, 5, 6 show that our proposed Bayesian methods recover a broader range of putative causal effects than standard MR estimators, while the more conservative variant yields a narrower but higher‐confidence set of associations.

4.2. Causal Directions of SLP, BMI, and NEU

We examined potential bidirectional causal relationships among SLP, BMI, and NEU. These three traits were selected because they represent distinct phenotypic domains (behavioral, metabolic, and psychiatric) that span the range of association patterns observed across all six analyzed traits: BMI shows widespread and strong associations, SLP demonstrates moderately extensive but directionally asymmetric effects, and NEU represents sparse and heterogeneous signals. Detailed presentations for all six traits would be largely redundant; complete summary results for these three traits are provided in Figure 5 and Tables 4, 5, 6. For each pairwise comparison (SLP with BMI, BMI with NEU, and SLP with NEU), we applied BayBiMR and BayBiMR(DP) alongside CD‐cML, IVW, MR‐Egger, and the Weighted Median. Each analysis was repeated across multiple SNP inclusion thresholds to evaluate how instrument selection influenced the estimated effects and the inferred direction of association. Sleep, metabolic status, and affective traits are biologically interconnected through shared neuroendocrine, inflammatory, and neural circuit pathways [32, 33, 34], and these analyses therefore provide a framework for identifying potential causal links between behavioral, metabolic, and psychiatric phenotypes [35].

For the SLP–BMI analysis, SNPs were selected using inclusion thresholds of $5\times 10^{-6}$, $5\times 10^{-5}$, and $5\times 10^{-4}$. For the SLP–NEU and BMI–NEU analyses, thresholds of $5\times 10^{-7}$, $5\times 10^{-6}$, $5\times 10^{-5}$, and $5\times 10^{-4}$ were applied. For each trait pair and each threshold, causal effects were estimated in both directions, and the results are displayed as forest‐style panels stratified by method. BayBiMR reports 95% credible intervals (CrIs), whereas BayBiMR(DP), CD‐cML, IVW, MR‐Egger, and Weighted Median report Wald‐type 95% confidence intervals (CIs). Figures 6, 8 and 7 present the bidirectional effect estimates across all methods and thresholds. Complete numerical results for all significant estimates are provided in Tables S11–S13 of the Supporting Information.

FIGURE 6.

Bidirectional MR between BMI and SLP. Points represent causal effect estimates with 95% CrI for BayBiMR and 95% CI for other methods. Different shapes correspond to SNP‐selection $p$‐value thresholds.

FIGURE 8.

Bidirectional MR between sleep duration and neuroticism. Causal estimates with 95% CrI for BayBiMR and 95% CI for other estimators. Shapes indicate SNP‐selection $p$‐value thresholds.

FIGURE 7.

Bidirectional MR between BMI and neuroticism. Points represent causal effect estimates with 95% CrI for BayBiMR and 95% CI for other methods. Different shapes correspond to SNP‐selection $p$‐value thresholds.

4.2.1. Directional Effects Between SLP and BMI

In the SLP–BMI analyses (Figure 6 and Table S12 in the Supporting Information), BayBiMR and BayBiMR(DP) yielded relatively stable and precise effect estimates across all SNP inclusion thresholds. In both directions of analysis, their 95% credible intervals remained relatively narrow and, in most cases, excluded zero, suggesting consistent directional signals within the considered threshold range. By contrast, the classical estimators IVW and MR‐Egger showed marked sensitivity to pleiotropy: point estimates fluctuated widely across thresholds, and confidence intervals often included zero or reversed sign, suggesting instability of the estimated effects. CD‐cML and the Weighted Median occasionally agree with the proposed Bayesian estimates but show greater variability with SNP threshold and causal direction, reflecting moderate susceptibility to pleiotropy and sampling variation. These results indicate that, in this application, BayBiMR and BayBiMR(DP) produced narrower and more threshold‐stable intervals than several competing estimators; we treat the directional signals as suggestive rather than confirmatory.

From a biological perspective, the SLP–BMI findings are consistent with a reciprocal, negative relationship between SLP and BMI. The inverse effects estimated by BayBiMR and BayBiMR(DP) suggest that higher BMI may be associated with shorter sleep duration, and conversely, that altered sleep patterns could contribute to weight gain, though causal interpretation requires caution. These estimates align directionally with previous observational and experimental studies linking excess adiposity to sleep disturbances and weight gain through metabolic, hormonal, and inflammatory pathways [36, 37, 38].

4.2.2. Directional Effects Between BMI and NEU

In the BMI–NEU analyses shown in Figure 7 and Table S11 of the Supporting Information, BayBiMR and BayBiMR(DP) again displayed greater stability across SNP inclusion thresholds than those conventional MR estimators. For the BMI $\to$ NEU direction, our proposed methods produced moderate positive effects at the more stringent thresholds but shifted to a strong negative effect at the most relaxed threshold. This sign change across thresholds is not consistent with stable directional inference, and we therefore do not interpret it as evidence of a single direction of effect. Their 95% credible intervals were comparatively narrow within each threshold, but the across‐threshold sign reversal indicates sensitivity to instrument selection. By contrast, IVW sometimes produced very large standard errors, and MR‐Egger gave an unusually large estimate with a wide interval at the $5\times 10^{-6}$ threshold, reflecting large sensitivity to potential horizontal pleiotropy. The Weighted Median and CD‐cML methods sometimes align with the Bayesian estimates, but sometimes show greater variability with the SNP threshold and direction tested. For the NEU $\to$ BMI direction, BayBiMR yielded a mix of positive and negative effects depending on the threshold but remained more precise than existing methods, with very small standard errors at the most relaxed thresholds. BayBiMR(DP) produced a similar pattern but with somewhat wider intervals. Weighted Median and CD‐cML showed the same sign changes but with greater variation across thresholds. Overall, the BMI–NEU comparison is more accurately described as exploratory: the proposed methods give narrower within‐threshold intervals than several competing estimators, but the across‐threshold sign reversals preclude any firm causal conclusion in either direction from these data alone.

Our findings suggest a potentially bidirectional link between BMI and NEU, though the sign reversals across SNP thresholds indicate sensitivity to instrument selection and should be interpreted cautiously. At more stringent SNP thresholds, the positive effects of BMI on NEU may reflect shared genetic influences or horizontal pleiotropy. The shift to a negative effect at the most inclusive threshold may indicate that adding weaker instruments introduces different biological pathways or additional confounding. Likewise, the mix of positive and negative effects in the NEU $\to$ BMI direction suggests that neuroticism could influence body weight via several mechanisms, though residual pleiotropy cannot be ruled out. The tighter credible intervals of the proposed Bayesian methods relative to existing estimators reflect their regularization of pleiotropic effects, but do not on their own establish that the estimates are correct.

4.2.3. Directional Effects Between SLP and NEU

In the SLP–NEU analyses shown in Figure 8 and Table S13 of the Supporting Information, BayBiMR and BayBiMR(DP) again produced relatively stable and precise effect estimates compared with other estimators. For the SLP $\to$ NEU direction, both proposed methods yielded consistently positive effects across thresholds, with narrow 95% credible intervals that excluded zero, whereas Weighted Median showed weaker but concordant effects with wider intervals. Within the considered threshold range, this pattern is suggestive of a positive directional signal, though we do not interpret it as confirmatory causal evidence. In the reverse NEU $\to$ SLP direction, BayBiMR displayed a mixture of positive and negative effects depending on the inclusion threshold, the credible intervals remained comparatively tight; BayBiMR(DP) tracked a similar pattern but with larger uncertainty. MR‐Egger produced an implausibly large effect estimate at the most stringent threshold, reflecting high sensitivity to pleiotropy. Weighted Median and CD‐cML showed intermediate performance. Taken together, these panels are consistent with a complex bidirectional relationship between SLP and NEU. The Bayesian estimators produce narrower within‐threshold intervals than several competing estimators in the SLP$\to$NEU direction, but the sign reversals in the reverse direction preclude firm causal conclusions and the findings should be treated as exploratory.

From a biological perspective, the positive SLP $\to$ NEU effects are consistent with mechanisms by which altered sleep patterns influence mood regulation and may predispose individuals to neurotic traits through hormonal or inflammatory pathways [32, 33]. Conversely, the mixed and sometimes negative NEU $\to$ SLP effects suggest that personality characteristics such as neuroticism may influence sleep in opposing ways, perhaps promoting insomnia or hypersomnia depending on context [39]. The sign reversals across thresholds in the reverse direction indicate sensitivity to instrument selection, and causal conclusions in either direction should be interpreted with caution pending replication in independent samples.

5. Discussion and Conclusions

In this study, we introduce BayBiMR (and its extension BayBiMR(DP)), a Bayesian framework for bidirectional MR that estimates causal effects in both directions while explicitly accounting for latent correlated pleiotropy through correlated direct genetic effects. By placing directional effects and pleiotropic pathways within a single hierarchical model, BayBiMR provides coherent uncertainty quantification and interpretable posterior summaries of causal parameters, pleiotropic structure, and instrument validity. This joint formulation allows information to be shared across directions, stabilizing estimation under complex genetic architectures and yielding probabilistic statements about the presence and direction of causality.

In extensive simulations designed to mimic realistic genetic architectures, BayBiMR and BayBiMR(DP) consistently showed better control of type I error than conventional estimators, particularly under mixed and correlated pleiotropy, where standard approaches such as IVW and MR‐Egger are prone to bias and inflated false‐positive rates.

Applications to large‐scale summary data on traits such as plasma APP, BMI, SLP, NEU, DS, and SWB illustrate how the framework operates in practice. Across these examples, BayBiMR and BayBiMR(DP) identified associations that were attenuated or nonsignificant under existing methods; whether these represent genuine causal effects or artifacts of model assumptions cannot be established from observational GWAS data alone.

We note differences in the behavior of CD‐cML between simulation studies and real‐data analyses. In simulations, inflated type I error occurred under controlled settings where pleiotropic effects violated modeling assumptions. In real data, however, true causal effects are unknown, and effect sizes may be small, heterogeneous, or influenced by residual confounding. Under such conditions, CD‐cML may detect fewer associations, leading to reduced apparent power relative to BayBiMR. In contrast, BayBiMR incorporates pleiotropic correlations through the covariance structure ${\mathbf{\Sigma }}_4$, which may allow it to retain information from variants exhibiting complex pleiotropic patterns. These differences highlight how methodological assumptions interact with realistic genetic architectures. These findings underscore the potential of our proposed approaches to provide a more reliable and nuanced picture of directionality and pleiotropy in complex trait genetics.

We acknowledge several limitations of the present implementation. First, the current framework assumes independent genetic instruments and does not explicitly model LD. To satisfy this assumption, we applied LD clumping, which may reduce statistical efficiency by excluding informative but correlated variants, particularly in regions with extensive LD. Additionally, the model is restricted to linear causal effects between two traits. Second, the working likelihood adopted in Equations ((3), (4)) corresponds to the small‐feedback approximation of the exact reduced form (2): the second‐order cross‐pleiotropy terms ${\theta }_{YX}{\gamma }_j$ and ${\theta }_{XY}{\delta }_j$ and the feedback denominator $1/(1-{\theta }_{XY}{\theta }_{YX})$ are absorbed into the spike‐and‐slab pleiotropic effects rather than modeled explicitly. This approximation preserves linearity in the latent effects, which enables the closed‐form Gibbs updates in Section 2.3, and is well justified whenever $|{\theta }_{XY}{\theta }_{YX}|$ is small relative to unity, which is the regime of primary scientific interest in two‐sample MR. In settings with strong bidirectional feedback (i.e., $|{\theta }_{XY}{\theta }_{YX}|$ close to one), the exact reduced form (2) would need to be inserted into the likelihood, at the cost of replacing the closed‐form normal updates for $({\theta }_{XY},{\theta }_{YX})$ with a Metropolis step. Third, the framework assumes a two‐sample design with nonoverlapping cohorts Equation (4); extending it to single‐sample or partially overlapping designs would require modeling an off‐diagonal sampling‐error covariance. Future work could address these limitations by incorporating explicit LD structures, accommodating the exact reduced form in the likelihood, and extending the framework to multivariate and nonlinear MR settings.

Together, our developments establish a coherent and flexible Bayesian framework for bidirectional MR. By coupling the estimation of directional effects with the modeling of pleiotropic covariance, BayBiMR aims to improve the robustness of causal inference in genetic epidemiology, though its practical performance will depend on the degree to which its modeling assumptions are met in a given application.

Funding

The author has nothing to report.

Conflicts of Interest

The author declares no conflicts of interest.

Supporting information

Data S1: Table S1: Performance in the $X\to Y$ direction (true effect ${\theta }_{XY}=0.2$) under Simulation S2.1 (15 SNPs, mixed pleiotropy). Table S2: Performance in the $Y\to X$ direction (true effect ${\theta }_{YX}=0$) under Simulation S2.1 (15 SNPs, mixed pleiotropy). Table S3: Simulation S2.2: X→Y direction (true effect ${\theta }_{XY}=0.2$) under 45 SNPs with mixed pleiotropy. Table S4: Simulation S2.2: Y→X direction (true effect ${\theta }_{YX}=0$) under 45 SNPs with mixed pleiotropy. Table S5: Empirical type‐I error for X→Y (with ${\theta }_{XY}=0$) and power for $Y\to X$ across ${\theta }_{YX}\in \{0,0.05,0.10,0.20\}$, based on n = 50 000 per GWAS ($p_X=p_Y=p_U=5$). Reported values are rejection probabilities at the $\alpha =0.05$ level. Table S6: Power for $X\to Y$ with ${\theta }_{XY}=0.2$ and type‐I error/power for $Y\to X$ across ${\theta }_{YX}\in \{0,0.05,0.10,0.20\}$, based on n = 50 000 per GWAS ($p_X=p_Y=p_U=5$). Reported values are rejection probabilities at the $\alpha =0.05$ level. Table S7: Empirical type‐I error for $X\to Y$ (with ${\theta }_{XY}=0$) and empirical power for $Y\to X$ across ${\theta }_{YX}\in \{0,0.05,0.10,0.20\}$ under mixed pleiotropy (20% uncorrelated, 20% correlated, 60% both; $p_X=p_Y=5$, $p_U=10$, n = 10 000 per GWAS). Values are rejection probabilities at the $\alpha =0.05$ level. Table S8: Empirical power for $X\to Y$ (${\theta }_{XY}=0.2$ fixed) and empirical type‐I error/power for $Y\to X$ across ${\theta }_{YX}\in \{0,0.05,0.10,0.20\}$ under mixed pleiotropy (20% uncorrelated, 20% correlated, 60% both; $p_X=p_Y=5$, $p_U=10$, n = 10 000 per GWAS). Values are rejection probabilities at the $\alpha =0.05$ level. Table S9: Empirical type‐I error for $X\to Y$ (with ${\theta }_{XY}=0$) and empirical power for $Y\to X$ under a weak‐instrument scenario (n = 10 000 per GWAS; $(p_X,p_Y,p_U)=(5,5,6)$). Reported values are rejection probabilities at the $\alpha =0.05$ level across ${\theta }_{YX}\in \{0,0.05,0.10,0.20\}$. Table S10: Empirical power for $X\to Y$ (with ${\theta }_{XY}=0.2$) and empirical type‐I error/power for $Y\to X$ under a weak‐instrument scenario (n = 10 000 per GWAS; $(p_X,p_Y,p_U)=(5,5,6)$). Reported values are rejection probabilities at the $\alpha =0.05$ level across ${\theta }_{YX}\in \{0,0.05,0.10,0.20\}$. Table S11: Significant effect estimates, standard errors, and 95% confidence or credible intervals for the $X\to Y$ (BMI $\to$ NEU) and $Y\to X$ (NEU $\to$ BMI) associations across methods and SNP inclusion thresholds. Table S12: Significant effect estimates, standard errors, and 95% confidence or credible intervals for the $X\to Y$ (SLP $\to$ BMI) and $Y\to X$ (BMI $\to$ SLP) associations across methods and SNP inclusion thresholds. Table S13: Significant effect estimates, standard errors, and 95% confidence or credible intervals for the $X\to Y$ (SLP $\to$ NEU) and $Y\to X$ (NEU $\to$ SLP) associations across methods and SNP inclusion thresholds. Figure S1: Estimated causal effects in Simulation S2.1 for $X\to Y$ (left) and $Y\to X$ (right). Violin and boxplots show the distribution of effect estimates across 500 replications for each method. The red dashed line indicates the true effect. Figure S2: Empirical 95% interval coverage in Simulation S2.1 for $X\to Y$ (left) and $Y\to X$ (right). Bars represent the proportion of intervals covering the true effect across 500 replications for each method. Figure S3: Empirical rejection rates at $\alpha =0.05$ in Simulation S2.1 for $X\to Y$ (left) and $Y\to X$ (right). Bars represent the proportion of replications rejecting the null for each method. Figure S4: Estimated causal effects in Simulation S2.2 for $X\to Y$ (left) and $Y\to X$ (right). Violin and boxplots show the distribution of effect estimates across 500 replications for each method. The red dashed line indicates the true effect. Figure S5: Empirical 95% interval coverage in Simulation S2.2 for $X\to Y$ (left) and $Y\to X$ (right). Bars represent the proportion of intervals covering the true effect across 500 replications for each method. Figure S6: Empirical rejection rates at $\alpha =0.05$ in Simulation S2.2 for $X\to Y$ (left) and $Y\to X$ (right). Bars represent the proportion of replications rejecting the null for each method. Figure S7: Simulation results under mixed pleiotropy (20% uncorrelated, 20% correlated, 60% both; $p_X=p_Y=5$, $p_U=10$, n = 10 000 per GWAS) showing empirical type‐I error for the null $X\to Y$ direction (${\theta }_{XY}=0$ fixed) and empirical power for the $Y\to X$ direction as ${\theta }_{YX}$ varies over $\{0,0.05,0.10,0.20\}$. Figure S8: Simulation results under mixed pleiotropy (20% uncorrelated, 20% correlated, 60% both; $p_X=p_Y=5$, $p_U=10$, n = 10 000 per GWAS) showing empirical power for the nonzero $X\to Y$ direction (${\theta }_{XY}=0.2$ fixed) and empirical type‐I error (or power when nonzero) for the $Y\to X$ direction as ${\theta }_{YX}$ varies over $\{0,0.05,0.10,0.20\}$. Figure S9: Empirical type‐I error for $X\to Y$ and empirical power for $Y\to X$ under a weak‐instrument scenario ($n=10000$ per GWAS; $(p_X,p_Y,p_U)=(5,5,6)$). At least half of instrument effects were drawn from uniform$(0.01,0.08)$ (weak) and the remainder from uniform$(0.20,0.30)$ (strong) with random sign. The null setting fixes ${\theta }_{XY}=0$ and varies ${\theta }_{YX}\in \{0,0.05,0.10,0.20\}$. Figure S10: Empirical power for $X\to Y$ and empirical type‐I error/power for $Y\to X$ under a weak‐instrument scenario (n = 10 000 per GWAS; $(p_X,p_Y,p_U)=(5,5,6)$). Here ${\theta }_{XY}=0.20$ is fixed and ${\theta }_{YX}$ varies over $\{0,0.05,0.10,0.20\}$.

Data Availability Statement

The data that support the findings of this study are openly available in GitHub at https://github.com/chen‐siyi7/BayBiMR.