论文标题
矩阵空间上后期的MCMC算法
MCMC Algorithms for Posteriors on Matrix Spaces
论文作者
论文摘要
我们研究了马尔可夫链蒙特卡洛（MCMC）算法，用于在基质空间上定义的目标分布。如此重要的抽样问题尚未分析。我们通过开发适当的理论框架来弥补这一差距，迈出了一个重要的一步，该框架允许在这种情况下识别典型的MCMC算法的奇异性属性。除了标准的随机步行大都市（RWM）和预处理曲柄 - 尼古尔森（PCN）之外，本文在新算法的开发中的贡献称为“混合” PCN（MPCN）。对于重要的一类带有沉重的尾巴的基质分布，RWM和PCN在几何上不具有几何形式。相比之下，MPCN在具有不同尾巴行为的目标之间具有鲁棒性，并且在重尾分布类别中具有很好的经验性能。在这项工作中，MPCN的几何形状越野性尚未得到充分证明，因为由于状态空间的复杂性，某些剩余的漂移条件非常具有挑战性。但是，我们确实在证明方面取得了很大的进步，并详细介绍了未来工作的最后一步。 我们通过数值应用说明了各种算法的计算性能， 包括在财务统计中产生的具有挑战性模型的真实数据的校准。
We study Markov chain Monte Carlo (MCMC) algorithms for target distributions defined on matrix spaces. Such an important sampling problem has yet to be analytically explored. We carry out a major step in covering this gap by developing the proper theoretical framework that allows for the identification of ergodicity properties of typical MCMC algorithms, relevant in such a context. Beyond the standard Random-Walk Metropolis (RWM) and preconditioned Crank--Nicolson (pCN), a contribution of this paper in the development of a novel algorithm, termed the `Mixed' pCN (MpCN). RWM and pCN are shown not to be geometrically ergodic for an important class of matrix distributions with heavy tails. In contrast, MpCN is robust across targets with different tail behaviour and has very good empirical performance within the class of heavy-tailed distributions. Geometric ergodicity for MpCN is not fully proven in this work, as some remaining drift conditions are quite challenging to obtain owing to the complexity of the state space. We do, however, make a lot of progress towards a proof, and show in detail the last steps left for future work. We illustrate the computational performance of the various algorithms through numerical applications, including calibration on real data of a challenging model arising in financial statistics.
