Simultaneously performing variable selection and inference in high-dimensional models is an open challenge in statistics and machine learning. The increasing availability of vast amounts of variables requires the adoption of specific statistical procedures to accurately select the most important predictors in a high-dimensional space, while being able to control some form of selection error. In this work we adapt the Mirror Statistic approach to False Discovery Rate (FDR) control into a Bayesian modelling framework. The Mirror Statistic, developed in the classic frequentist statistical framework, is a flexible method to control FDR, which only requires mild model assumptions, but requires two sets of independent regression coefficient estimates, usually obtained after splitting the original dataset. Here we propose to rely on a Bayesian formulation of the model and use the posterior distributions of the coefficients of interest to build the Mirror Statistic and effectively control the FDR without the need to split the data. Moreover, the method is very flexible since it can be used with continuous and discrete outcomes and more complex predictors, such as with mixed models. We keep the approach scalable to high-dimensions by relying on Automatic Differentiation Variational Inference and fully continuous prior choices.

We introduce CatNet, an algorithm that effectively controls False Discovery Rate (FDR) and selects significant features in LSTM with the Gaussian Mirror (GM) method. To evaluate the feature importance of LSTM in time series, we introduce a vector of the derivative of the SHapley Additive exPlanations (SHAP) to measure feature importance. We also propose a new kernel-based dependence measure to avoid multicollinearity in the GM algorithm, to make a robust feature selection with controlled FDR. We use simulated data to evaluate CatNet's performance in both linear models and LSTM models with different link functions. The algorithm effectively controls the FDR while maintaining a high statistical power in all cases. We also evaluate the algorithm's performance in different low-dimensional and high-dimensional cases, demonstrating its robustness in various input dimensions. To evaluate CatNet's performance in real world applications, we construct a multi-factor investment portfolio to forecast the prices of S\&P 500 index components. The results demonstrate that our model achieves superior predictive accuracy compared to traditional LSTM models without feature selection and FDR control. Additionally, CatNet effectively captures common market-driving features, which helps informed decision-making in financial markets by enhancing the interpretability of predictions. Our study integrates of the Gaussian Mirror algorithm with LSTM models for the first time, and introduces SHAP values as a new feature importance metric for FDR control methods, marking a significant advancement in feature selection and error control for neural networks.