Manifold Markov Chain Monte Carlo Methods for Bayesian Inference in Diffusion Models
针对离散观测的非线性扩散模型，提出一种结合统计物理和计算化学思想的流形约束哈密顿蒙特卡洛算法，实现潜在扩散路径和参数的后验推断，算法自动化程度高，适用于多种观测场景。
Abstract Bayesian inference for nonlinear diffusions, observed at discrete times, is a challenging task that has prompted the development of a number of algorithms, mainly within the computational statistics community. We propose a new direction, and accompanying methodology—borrowing ideas from statistical physics and computational chemistry—for inferring the posterior distribution of latent diffusion paths and model parameters, given observations of the process. Joint configurations of the underlying process noise and of parameters, mapping onto diffusion paths consistent with observations, form an implicitly defined manifold. Then, by making use of a constrained Hamiltonian Monte Carlo algorithm on the embedded manifold, we are able to perform computationally efficient inference for a class of discretely observed diffusion models. Critically, in contrast with other approaches proposed in the literature, our methodology is highly automated, requiring minimal user intervention and applying alike in a range of settings, including: elliptic or hypo-elliptic systems; observations with or without noise; linear or non-linear observation operators. Exploiting Markovianity, we propose a variant of the method with complexity that scales linearly in the resolution of path discretisation and the number of observation times. Python code reproducing the results is available at http://doi.org/10.5281/zenodo.5796148.