Most existing multimodal palm recognition methods have shown notable advancements in closed-set identification performance; nevertheless, their performance declines dramatically when applied to out-of-domain (OOD) datasets. Ensuring the robustness and efficacy of algorithms against variations in the image domain remains a crucial unresolved issue in multimodal palm recognition. This paper proposes a novel Bayesian approach (BYNet) to achieve OOD robustness for palmprint and palm vein classification. BYNet comprises two symmetric, trainable feature extraction branches designed to learn distinctive palmprint texture and vein vascular distribution patterns. To further enhance model robustness against OOD interference in real-world scenarios, we incorporate the von Mises-Fisher (vMF) distribution to probabilistically model the direction of palmprints and the geometric properties of palm veins, enabling the model to learn a transition dictionary of vMF kernels between source-target domains for adaptive feature selection. Furthermore, as cross-entropy loss struggles to construct compact intra-class and separable inter-class feature spaces, we combine contrastive loss to jointly optimize the network to reduce the source-target domain shift. Validation on three public datasets demonstrates that the proposed BYNet exhibits superior recognition performance and OOD robustness compared to other state-of-the-art methods.