A Riemannian geometric framework for Markov chain Monte Carlo (MCMC) is developed where using the Fisher-Rao metric on the manifold of probability density functions (pdfs) informed proposal densities for Metropolis-Hastings (MH) algorithms are constructed. We exploit the square-root representation of pdfs under which the Fisher-Rao metric boils down to the standard $L^2$ metric on the positive orthant of the unit hypersphere. The square-root representation allows us to easily compute the geodesic distance between densities, resulting in a straightforward implementation of the proposed geometric MCMC methodology. Unlike the random walk MH that blindly proposes a candidate state using no information about the target, the geometric MH algorithms effectively move an uninformed base density (e.g., a random walk proposal density) towards different global/local approximations of the target density. We compare the proposed geometric MH algorithm with other MCMC algorithms for various Markov chain orderings, namely the covariance, efficiency, Peskun, and spectral gap orderings. The superior performance of the geometric algorithms over other MH algorithms like the random walk Metropolis, independent MH and variants of Metropolis adjusted Langevin algorithms is demonstrated in the context of various multimodal, nonlinear and high dimensional examples. In particular, we use extensive simulation and real data applications to compare these algorithms for analyzing mixture models, logistic regression models and ultra-high dimensional Bayesian variable selection models. A publicly available R package accompanies the article.

翻译：本文发展了一种基于黎曼几何框架的马尔可夫链蒙特卡洛（MCMC）方法，利用概率密度函数（pdf）流形上的Fisher-Rao度量构造信息化的Metropolis-Hastings（MH）算法建议分布。我们利用概率密度函数的平方根表示，该表示将Fisher-Rao度量简化为单位超球面正卦限上的标准$L^2$度量。平方根表示使我们能够轻松计算密度之间的测地距离，从而直接实现所提出的几何MCMC方法。与随机游走MH算法盲目使用无目标信息的建议状态不同，几何MH算法有效地将无信息的基础密度（例如随机游走建议密度）向目标密度的不同全局/局部近似移动。我们通过多种马尔可夫链排序（即协方差、效率、Peskun和谱间隙排序）将所提出的几何MH算法与其他MCMC算法进行比较。在多种多峰、非线性和高维示例中，几何算法相较于其他MH算法（如随机游走Metropolis、独立MH及Metropolis调整Langevin算法的变体）展现出优越性能。特别地，我们通过大量模拟和实际数据应用，比较了这些算法在混合模型、逻辑回归模型以及超高维贝叶斯变量选择模型分析中的表现。本文附有公开可用的R包。