Exploiting locality in high-dimensional factorial hidden Markov models

作     者：Lorenzo Rimella Nick Whiteley 

作者机构：Department of Mathematics and Statistics Lancaster University Lancaster UK Institute for Statistical Science School of Mathematics University of Bristol Bristol UK and the Alan Turing Institute UK 

出 版 物：《The Journal of Machine Learning Research》 

年 卷 期：2022年第23卷第1期

页      面：134-167页

学科分类：08[工学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

主　　题：factorial hidden Markov models filtering smoothing EM algorithm high-dimensions 

摘      要：We propose algorithms for approximate filtering and smoothing in high-dimensional Factorial hidden Markov models. The approximation involves discarding, in a principled way, likelihood factors according to a notion of locality in a factor graph associated with the emission distribution. This allows the exponential-in-dimension cost of exact filtering and smoothing to be avoided. We prove that the approximation accuracy, measured in a local total variation norm, is dimension-free in the sense that as the overall dimension of the model increases the error bounds we derive do not necessarily degrade. A key step in the analysis is to quantify the error introduced by localizing the likelihood function in a Bayes rule update. The factorial structure of the likelihood function which we exploit arises naturally when data have known spatial or network structure. We demonstrate the new algorithms on synthetic examples and a London Underground passenger ow problem, where the factor graph is effectively given by the train network.