Markov chain Monte Carlo methods explicitly defined on the manifold of probability distributions have recently been established. These methods are constructed from diffusions across the manifold and the solution of the equations describing geodesic flows in the Hamilton--Jacobi representation. This paper takes the differential geometric basis of Markov chain Monte Carlo further by considering methods to simulate from probability distributions that themselves are defined on a manifold, with common examples being classes of distributions describing directional statistics. Proposal mechanisms are developed based on the geodesic flows over the manifolds of support for the distributions and illustrative examples are provided for the hypersphere and Stiefel manifold of orthonormal matrices.
On embedded hidden Markov models and particle Markov chain Monte Carlo methods
嵌入式隐马尔可夫模型（EHMM）采样方法是一种用于非线性非高斯状态空间模型中状态推断的马尔可夫链蒙特卡罗（MCMC）技术，由Neal（2003）提出; Neal et al.（2004）和Shestopaloff and Neal（2016）扩展。Shestopaloff和Neal（2013）提出了对贝叶斯参数推断的扩展。粒子MCMC（PMCMC）方法提供了另一类解决类似推理问题的MCMC方案（Andrieu et al. 2009; 2010）。所有这些方法都依赖于引入多个状态序列的人工扩展目标分布，通过构造，这些状态序列是这样的，即根据感兴趣的后验分布一个随机索引的序列。通过将PMCMC方法框架下的梅特罗波利斯-黑斯廷斯算法应用于EHMM框架，得到了新的粒子滤波器（PF）型状态推断算法和新的MCMC参数和状态推断算法.此外，我们表明，这些算法中的大多数可以被视为一般PF和PMCMC框架的特殊情况。我们比较了各种算法在低维到高维状态空间模型上的经验性能。我们证明了Shestopaloff和Neal（2016）提出的具有“局部”MCMC移动的适当调整的条件PF在应用于高维状态空间模型时可以显着优于标准条件PF，而新的PF类型算法可以证明是标准PF的一个有趣的替代方案，用于某些低维场景中的似然估计。
