论文标题
对数密度梯度协方差和Riemann歧管蒙特卡洛方法的自动度量张量
Log-density gradient covariance and automatic metric tensors for Riemann manifold Monte Carlo methods
论文作者
论文摘要
提出了一种特别适合非线性贝叶斯分层模型的Riemann歧管蒙特卡洛的度量张量。度量张量是由对称正极半足限对数密度协方差（LGC）矩阵构建的，该矩阵也被提出并在此处进一步探索。 LGC通过测量随机变量的关节信息含量和依赖性结构和所述变量的参数来概括Fisher信息矩阵。因此，如果目前的实践也可以构建基于阳性的Fisher/LGC基于的度量标准张量，那么也可以根据目前的实践来构建，而且还可以由任意复杂的非线性先验/潜在变量结构来构建，前提是对于每个用于构造上述结构的条件分布的LGC，也可以得出LGC。所提出的方法是高度自动的，可以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随机汉密尔顿蒙特卡洛过程的Riemann歧管变体结合使用时，提出的方法具有很高的竞争力，特别是对于与贝叶斯层次模型相关的更具挑战性的目标分布。
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.
