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 move an uninformed base density (e.g., a random walk proposal density) towards different global/local approximations of the target density, allowing effective exploration of the distribution simultaneously at different granular levels of the state space. 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, spatial generalized linear mixed models and ultra-high dimensional Bayesian variable selection models. A publicly available R package accompanies the article.

翻译：本文发展了一种用于马尔可夫链蒙特卡洛（MCMC）的黎曼几何框架，其中利用概率密度函数（pdf）流形上的Fisher-Rao度量，构建了Metropolis-Hastings（MH）算法的知情提议密度。我们利用了概率密度函数的平方根表示，在此表示下，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线距离，从而实现了所提出的几何MCMC方法的直接实现。与随机游走MH算法盲目地提出候选状态而不利用目标分布信息不同，几何MH算法将无信息的基密度（例如，随机游走提议密度）向目标分布的不同全局/局部近似方向移动，从而允许在状态空间的不同粒度级别上同时有效地探索分布。我们将所提出的几何MH算法与其他MCMC算法在各种马尔可夫链序关系下进行比较，即协方差序、效率序、Peskun序和谱隙序。通过多种多模态、非线性和高维示例，证明了几何算法相对于其他MH算法（如随机游走Metropolis、独立MH以及Metropolis调整Langevin算法的变体）的优越性能。特别是，我们使用广泛的模拟和实际数据应用来比较这些算法，以分析混合模型、逻辑回归模型、空间广义线性混合模型和超高维贝叶斯变量选择模型。本文附带一个公开可用的R软件包。