A Bayesian Hierarchical Framework for Non-Linear and Iterative Transfer Learning of Gaussian Process

Classical multi-output GP (MGP) models often suffer from limitations such as the assumption of linear inter-process correlations, sensitivity to data imbalance, and lack of support for online updates. This paper proposes a novel Bayesian hierarchical framework to address these challenges. Specifically, we introduce a three-layer architecture that models shared hyper-parameters as random variables and updates them asymmetrically through a hierarchical structure. The proposed framework treats source data as historical information to form informative priors and uses target data for posterior update, which facilitates a flexible and process-specific knowledge transfer. We theoretically establish that this treatment induces nonlinear inference for the target process and allows efficient, iterative updates without re-optimizing the entire model. Comprehensive evaluations on synthetic data and real-world case studies in battery diagnostics and nano-sensor design demonstrate that our method significantly outperforms existing single-output and hierarchical MGP baselines in both accuracy and computational efficiency. The results suggest that our approach offers a principled and scalable solution for transfer learning under data-scarce and heterogeneous conditions.