ABSTRACT We present geodesic Lagrangian Monte Carlo, an extension of Hamiltonian Monte Carlo for sampling from posterior distributions defined on general Riemannian manifolds. We apply this new algorithm to Bayesian inference on symmetric or Hermitian positive definite (PD) matrices. To do so, we exploit the Riemannian structure induced by Cartan's canonical metric. The geodesics that correspond to this metric are available in closed-form and – within the context of Lagrangian Monte Carlo – provide a principled way to travel around the space of PD matrices. Our method improves Bayesian inference on such matrices by allowing for a broad range of priors, so we are not limited to conjugate priors only. In the context of spectral density estimation, we use the (non-conjugate) complex reference prior as an example modelling option made available by the algorithm. Results based on simulated and real-world multivariate time series are presented in this context, and future directions are outlined.

中文翻译：

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正定矩阵空间上的测地拉格朗日蒙特卡罗：应用于贝叶斯谱密度估计

摘要 我们提出了测地线拉格朗日蒙特卡罗，这是哈密顿蒙特卡罗的扩展，用于从一般黎曼流形上定义的后验分布进行采样。我们将这种新算法应用于对称或埃尔米特正定 (PD) 矩阵的贝叶斯推理。为此，我们利用嘉当规范度量导出的黎曼结构。与该度量相对应的测地线以封闭形式提供，并且在拉格朗日蒙特卡罗的背景下提供了一种在 PD 矩阵空间中移动的原则性方法。我们的方法通过允许广泛的先验来改进对此类矩阵的贝叶斯推理，因此我们不仅限于共轭先验。在谱密度估计的背景下，我们使用（非共轭）复参考先验作为算法提供的示例建模选项。在此背景下，提出了基于模拟和现实世界多元时间序列的结果，并概述了未来的方向。