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.

中文翻译：

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黎曼流形蒙特卡罗方法的对数密度梯度协方差和自动度量张量

提出了一种特别适合非线性贝叶斯分层模型的黎曼流形蒙特卡罗度量张量。度量张量是由对称正半定对数密度梯度协方差（LGC）矩阵构建的，这里也提出并进一步探讨了该矩阵。LGC 通过测量随机变量和所述变量的参数的联合信息内容和依赖结构来概括 Fisher 信息矩阵。因此，基于正定 Fisher/LGC 的度量张量不仅可以根据当前实践中的观测似然来构造，还可以根据任意复杂的非线性先验/潜变量结构来构造，前提是可以为用于构造的每个条件分布导出 LGC。所说的结构。所提出的方法是高度自动化的，并且允许利用与所讨论的模型相关的任何稀疏性。当与最近提出的数值广义随机哈密顿量蒙特卡罗过程的黎曼流形变体结合实施时，所提出的方法具有很强的竞争力，特别是对于与贝叶斯分层模型相关的更具挑战性的目标分布。