论文标题
立体马尔可夫链蒙特卡洛
Stereographic Markov Chain Monte Carlo
论文作者
论文摘要
对于现成的MCMC采样器而言，高维分布，尤其是那些具有沉重的尾巴的分布：无限状态空间的组合，减少梯度信息和本地移动导致经验上观察到的“粘性”和差的理论混合特性（缺乏理论混合特性） - 缺乏质量的数量阶段性。在本文中，我们介绍了一类新的MCMC采样器，该样本将欧几里得空间中原始的高维问题映射到一个领域，并纠正这些臭名昭著的混合问题。特别是，我们开发了随机步行大都市型算法以及有弹性粒子采样器的版本，这些粒子采样器对于大量的轻型和重尾分布均匀地存在，并且在高维度中也表现出快速的收敛。在最佳情况下，提议的采样器可以享受``维度的祝福''，即收敛在更高维度的速度更快。
High-dimensional distributions, especially those with heavy tails, are notoriously difficult for off-the-shelf MCMC samplers: the combination of unbounded state spaces, diminishing gradient information, and local moves results in empirically observed ``stickiness'' and poor theoretical mixing properties -- lack of geometric ergodicity. In this paper, we introduce a new class of MCMC samplers that map the original high-dimensional problem in Euclidean space onto a sphere and remedy these notorious mixing problems. In particular, we develop random-walk Metropolis type algorithms as well as versions of the Bouncy Particle Sampler that are uniformly ergodic for a large class of light and heavy-tailed distributions and also empirically exhibit rapid convergence in high dimensions. In the best scenario, the proposed samplers can enjoy the ``blessings of dimensionality'' that the convergence is faster in higher dimensions.
