Stereographic Markov Chain Monte Carlo

由于贝叶斯统计的日益普及，马尔可夫链蒙特卡罗（MCMC）方法已经发展到解决近似后验概率分布的问题。通过根据马尔可夫过程生成一个序列，我们用样本的经验分布近似这些复杂的分布。尽管存在许多MCMC算法，但它们通常对重尾或高维的目标分布表现不佳。为了解决这些问题，Yang、Latuszynski和Roberts（2022）开发了利用立体投影的MCMC算法：这些方法通过将密度转换到d维球体来瞄准欧几里得空间上支持的分布。这压缩了空间，极大地改善了收敛结果。现有的两种算法分别是基于随机游走大都会（RWM）的立体投影采样器（SPS）和基于分段确定性马尔可夫过程（PDMP）的立体弹性粒子采样器（SBPS）。在一个大的（可能是重尾的）目标分布上，两者都被证明是均匀遍历的，包括具有d个自由度的多元t分布。提出了各种启发式和模拟研究，这些研究都表明这些算法即使在重尾、高维设置中也表现良好，并且在某些情况下实际上可以受益于“维数的祝福”。均匀遍历性允许样本路径在快速返回到固定相位之前进入分布的尾部。这些算法的性能受到用于立体投影的球体的中心和半径/形状参数的选择的强烈影响。为了解决这个问题，我们设计了一种自适应SBPS算法，该算法在探索目标分布时自动更新这些参数，灵感来自Chimisov， Latuszynski和Roberts（2018）的AIR MCMC框架。我们模拟了本文的证明，得到了自适应SBPS的WLLN、SLLN和CLT。这些定理需要与Yang， Latuszynski和Roberts（2022）的一致遍历性结果相同的目标正则性条件，以及参数空间紧性的假设。为此，我们引入了一种研究一致遍历连续时间马尔可夫过程再生的新方法：通过将（连续时间）SBPS路径划分为固定长度的段，我们在càdlàg函数空间中得到一个“段链”。这个新链是一个离散时间马尔可夫链，继承了我们原始过程的一致遍历性，并且可以以这样一种方式“分裂”，将样本路径划分为一个相同分布的序列，1相关的短途旅行。我们现在已经在Julia编程语言中获得了自适应SBPS算法的工作实现。在玩具示例上的运行表明，即使在高维、重尾设置下，即使在算法的初始参数没有明确规定的情况下，该算法也表现良好。需要进行更多的模拟研究，特别是探索算法在目标坐标中遇到异质尺度时的行为，以及我们的方法与更传统的算法（即HMC）相比有什么优点/缺点。展望未来，有两种可能的研究途径似乎特别相关。首先，尽管有强有力的启发式证据来指导我们选择SBPS的最佳中心和尺度参数，但即使在简单的例子中，也没有明确的理论结果来证实这些选择。也没有任何直觉来指导玩家选择补充速度。因此，研究这一方向以帮助从业者向前发展是有兴趣的。其次，我们可以研究其他传统的（欧几里得）算法，如HMC，可以从立体投影中受益。

Motivated by the increasing popularity of Bayesian statistics, Markov Chain Monte Carlo (MCMC) methods have been developed to tackle the problem of approximating posterior probability distributions. By generating a sequence according to a Markov process, we approximate these complicated distributions with the empirical distribution of the sample. Although many MCMC algorithms exist, they often perform poorly against target distributions that are heavy tailed or high dimensional.To address these issues, Yang, Latuszynski and Roberts (2022) develop MCMC algorithms that utilise the stereographic projection: these methods target distributions supported on Euclidean space by transforming the density onto the d-dimensional sphere. This compactifies the space and allows for greatly improved convergence results. The two existing algorithms are the Stereographic Projection Sampler (SPS), a Random-Walk Metropolis (RWM) based method, and the Stereographic Bouncy Particle Sampler (SBPS), a Piecewise Deterministic Markov Process (PDMP). Both are shown to be uniformly ergodic on a large class of (potentially heavy tailed) target distributions, including multivariate t-distributions with d degrees of freedom. Various heuristics and simulation studies are presented which all suggest that these algorithms behave well even in heavy tailed, high dimensional settings, and can in fact benefit from a "blessing of dimensionality" in some cases. Uniform ergodicity allows the sample paths to make excursions into the tails of distributions before quickly returning to the stationary phase.The performance of these algorithms is strongly influenced by the choice of the centre and radius/shape parameters of the sphere used in the stereographic projection. To address this, we design an Adaptive SBPS algorithm which automatically updates these parameters as we explore the target distribution, inspired by the AIR MCMC framework from Chimisov, Latuszynski and Roberts (2018). We mimic the proofs in this paper to obtain a WLLN, SLLN and a CLT for the adaptive SBPS. These theorems require the same regularity conditions on the target as the uniform ergodicity results from Yang, Latuszynski and Roberts (2022), as well as an assumption of compactness of the parameter space. In doing so, we introduce a novel method for the study of regenerations in uniformly ergodic, continuous time Markov processes: by dividing the (continuous time) SBPS paths into segments of fixed length, we obtain a "segment chain" in the space of càdlàg functions. This new chain is a discrete time Markov chain, and inherits the uniform ergodicity of the properties from our original process, and can be "split" in such a way to divide the sample paths into a sequence of identically distributed, 1-dependent excursions.We have now got a working implementation of the Adaptive SBPS algorithm in the Julia programming language. Runs on toy examples suggest that the algorithm performs well even in high dimensional, heavy tailed settings, even when the initial parameters of the algorithm are poorly specified. Many more simulation studies are required, exploring in particular how the algorithm behaves when encountering heterogenous scales in the coordinates of the target, and what advantages/disadvantages our methods have over more traditional algorithms (namely HMC).Looking forward, there are two possible avenues of research that seem particularly relevant. Firstly, although there is strong heuristic evidence to guide our choice of optimal centre and scale parameters for the SBPS, there are yet to be definitive theoretical results to confirm these choices, even in simple examples. There is also no intuition whatsoever to guide the choice of refreshment rate. It would therefore be of interest to investigate this direction to assist practitioners going forward. Secondly, we could investigate what other traditional (Euclidean) algorithms, such as HMC, could benefit from the stereographic projection.