Log‐density gradient covariance and automatic metric tensors for Riemann manifold Monte Carlo methods
提出一种适用于非线性贝叶斯分层模型的黎曼流形蒙特卡洛度量张量，基于对数密度梯度协方差矩阵构建，可自动利用模型稀疏性，提升采样效率。
Abstract A metric tensor for Riemann manifold Monte Carlo particularly suited for nonlinear Bayesian hierarchical models is proposed. The metric tensor is built from symmetric positive semidefinite log‐density gradient covariance (LGC) matrices, which are also proposed and further explored here. The LGCs generalize the Fisher information matrix by measuring the joint information content and dependence structure of both a random variable and the parameters of said variable. Consequently, positive definite Fisher/LGC‐based metric tensors may be constructed not only from the observation likelihoods as is current practice, but also from arbitrarily complicated nonlinear prior/latent variable structures, provided the LGC may be derived for each conditional distribution used to construct said structures. The proposed methodology is highly automatic and allows for exploitation of any sparsity associated with the model in question. When implemented in conjunction with a Riemann manifold variant of the recently proposed numerical generalized randomized Hamiltonian Monte Carlo processes, the proposed methodology is highly competitive, in particular for the more challenging target distributions associated with Bayesian hierarchical models.