A metric tensor for Riemann manifold Monte Carlo particularly suited for non-linear 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 non-linear 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.

翻译：针对非线性贝叶斯分层模型，本文提出一种特别适用的黎曼流形蒙特卡洛度量张量。该度量张量基于对称半正定对数密度梯度协方差（LGC）矩阵构建，本文同时对该矩阵进行深入探讨。LGC通过测量随机变量及其参数二者的联合信息含量与依赖结构，推广了Fisher信息矩阵。因此，基于Fisher/LGC的正定度量张量不仅可像当前实践那样通过观测似然构建，还能通过任意复杂的非线性先验/潜变量结构构建，前提是可为构建该结构的每个条件分布推导出LGC。所提方法具有高度自动化特性，且能充分利用模型关联的稀疏性。当与近期提出的数值广义随机化哈密顿蒙特卡洛过程的黎曼流形变体结合实现时，该方法展现出显著竞争力，尤其适用于贝叶斯分层模型中更具挑战性的目标分布。