In many research fields, researchers aim to identify significant associations between a set of explanatory variables and a response while controlling the false discovery rate (FDR). To this aim, we develop a fully Bayesian generalization of the classical model-X knockoff filter. Knockoff filter introduces controlled noise in the model in the form of cleverly constructed copies of the predictors as auxiliary variables. In our approach we consider the joint model of the covariates and the response and incorporate the conditional independence structure of the covariates into the prior distribution of the auxiliary knockoff variables. We further incorporate the estimation of a graphical model among the covariates, which in turn aids knockoffs generation and improves the estimation of the covariate effects on the response. We use a modified spike-and-slab prior on the regression coefficients, which avoids the increase of the model dimension as typical in the classical knockoff filter. Our model performs variable selection using an upper bound on the posterior probability of non-inclusion. We show how our model construction leads to valid model-X knockoffs and demonstrate that the proposed characterization is sufficient for controlling the BFDR at an arbitrary level, in finite samples. We also show that the model selection is robust to the estimation of the precision matrix. We use simulated data to demonstrate that our proposal increases the stability of the selection with respect to classical knockoff methods, as it relies on the entire posterior distribution of the knockoff variables instead of a single sample. With respect to Bayesian variable selection methods, we show that our selection procedure achieves comparable or better performances, while maintaining control over the FDR. Finally, we show the usefulness of the proposed model with an application to real data.

翻译：在许多研究领域中，研究者旨在识别解释变量与响应变量之间的显著关联，同时控制错误发现率（FDR）。为此，我们开发了经典模型-X knockoff滤波器的完全贝叶斯推广。Knockoff滤波器通过巧妙构造预测变量的副本作为辅助变量，在模型中引入受控噪声。在我们的方法中，我们考虑协变量与响应变量的联合模型，并将协变量的条件独立结构纳入辅助knockoff变量的先验分布。我们进一步整合了协变量间图模型的估计，这有助于knockoffs的生成并改进协变量对响应变量影响的估计。我们在回归系数上使用改进的spike-and-slab先验，避免了经典knockoff滤波器中常见的模型维度增加问题。我们的模型利用非包含后验概率的上界进行变量选择。我们展示了模型构建如何产生有效的模型-X knockoffs，并证明所提出的表征足以在有限样本中控制BFDR至任意水平。我们还证明了模型选择对精度矩阵估计具有鲁棒性。通过模拟数据，我们证明相较于经典knockoff方法，我们的方法提高了选择的稳定性，因为它依赖于knockoff变量的整个后验分布而非单一样本。与贝叶斯变量选择方法相比，我们的选择程序实现了相当或更好的性能，同时保持对FDR的控制。最后，我们通过实际数据应用展示了所提出模型的有效性。