We present a novel offline-online method to mitigate the computational burden of Bayesian inference, particularly in the regime where the posterior densities are computationally demanding to evaluate while real-time inference results are needed. In the offline phase, the proposed method learns the joint law of the parameter random variables and the observable random variables in the tensor-train (TT) format. Then, in the online phase, the resulting order-preserving transport can be conditioned on newly observed data to characterize the posterior random variables in real-time. Compared with the state-of-the-art normalizing flows techniques, our proposed method relies on function approximation, for which we can provide a thorough performance analysis. The function approximation perspective allows us to significantly improve the capability of transport maps in challenging problems with high-dimensional observations and high-dimensional parameters. Capitalizing on this, we present novel heuristics to either reorder or reparametrize the variables to enhance the approximation power of TT. We then integrate the TT-based transport maps and the parameter reordering/reparametrization into a layered composite map to further improve the performance of the resulting inference. We demonstrate the efficiency of the proposed method on various statistical learning tasks involving ordinary differential equations (ODEs) and partial differential equations (PDEs).
翻译:在离线阶段,拟议方法学习了参数随机变量和可观测随机变量的联合法,采用高压拖车(TT)格式。然后,在在线阶段,由此产生的定序保护运输可以以新观察到的数据为条件,以便实时确定后继随机变量的特征。与当前正常流动技术的状态相比,我们拟议方法依赖于功能近似,对此我们可以提供透彻的业绩分析。功能近似观点使我们能够大大提高运输图的能力,以应对高分辨率观测和高分辨率参数带来的难题。我们在此基础上提出了新颖的超常理论,以便重新排序或重新校正变异变量,从而增强TT的近似性能。我们随后将基于TT的运输图和参数重新定序/重新定值的流程技术结合起来,我们的拟议方法则依赖功能近似法,为此我们可以提供透彻的业绩分析。功能近似观点使我们得以大大提高运输图在高分辨率观察和高维度参数参数方面的能力。