Causal Learning for Continuous Variables with an Improved Bayesian Network Constructed by Symmetric Kernel Function Acceleration

Bayesian network-based causal structure learning provides an effective framework for uncovering causal relationships among continuous variables. However, many existing methods for continuous data still rely on strong parametric distribution assumptions, which may introduce information loss and reduce Bayesian network modeling accuracy. Kernel density estimation (KDE), a non-parametric statistical method that is more flexible in density estimation form, offers a versatile framework for conducting conditional independence (CI) tests. This approach enables the estimation of mutual information and conditional mutual information, thereby facilitating the identification of underlying structural relationships. Nevertheless, the high computational cost of KDE-based CI testing restricts its practical application in continuous-variable causal learning. To address this issue, this study introduces a radial symmetric kernel-based acceleration scheme within a Fast Fourier Transform (FFT) framework to improve the efficiency of density estimation. On this basis, an enhanced Bayesian network structure learning method is developed for continuous variables, enabling more efficient estimation of mutual information and conditional mutual information while improving the computational efficiency and empirical stability of variable dependency discovery. With proper bandwidth and grid resolution, the proposed MMHC-FFTKDE framework achieves a reduction in computational runtime and improves efficiency compared to MMHC-KDE in the ablation setting, while maintaining competitive F1-scores and SHD for causal structure discovery.