Bayesian Inference of Hidden Nonhomogeneous Markov Model

Abstract

Abstract: At present, heterogeneous longitudinal data are often encountered in many fields such as pedagogy, psychology, economics, and sociology. Hidden Markov model has attracted extensive attention because of its good modeling and analysis effect on heterogeneous longitudinal data. This paper introduces the non-homogeneous Markov state transition mode and combines it with multivariate normal distribution to propose a hidden nonhomogeneous Markov multivariate normal distribution model. Firstly, the mathematical definition of hidden nonhomogeneous Markov multivariate normal distribution model is introduced. Then, the derivation process of the posterior distribution of model parameters and the design of MCMC algorithm are introduced in detail, and four simulation experiments are set for comparative analysis. The results show that the Bayesian inference method is reliable. In addition, the main contribution of this paper is to show that hidden nonhomogeneous Markov model can cover homogeneous hidden Markov model, which is more widely used.

Key words: hidden nonhomogeneous Markov model, multivariate normal distribution, Bayesian inference, MCMC algorithm

CLC Number:

O212.8

SHEN Xiujuan, GENG Mi. Bayesian Inference of Hidden Nonhomogeneous Markov Model[J]. Journal of Qujing Normal University, 2023, 42(3): 1-8.

References

[1]BAUM L E, PETRIE T.Statistical inference for probabilistic functions of finite state Markov chains[J].Annals of Mathematical Statistics,1966,37(1):1554-1563.
[2]RABINER L R.A tutorial on hidden Markov model and selected applications in speech recognition system[J].Proc IEEE,1989,77(2):259-286.
[3]ALTMAN R M.Mixed hidden Markov models[J].Journal of the American Statiastical Association,2007(102):201-210.
[4]Robert C P,Celeux G,Diebolt J. Bayesian estimation of hidden Markov chains: a stochastic implementation[J]. Statistics Probability Letters,1993,16(1):77-83.
[5]GASSIAT E,ROUSSEAU J. About the posterior distribution in hidden Markov models with unknown number of states[J]. Bernoulli,2014,20(4):2039-2075.
[6]张波,刘鹤飞,王坤.隐马尔可夫多元正态分布参数的极大似然估计[J].应用概率统计,2020,36(2):162-172.
[7]刘鹤飞,王坤,蒋成飞.隐状态个数未知的隐马尔可夫多元正态分布的贝叶斯推断[J].统计研究,2017,34(12):119-125.
[8]王坤,刘鹤飞,蒋成飞.隐马尔可夫结构方程及其贝叶斯估计[J].数理统计与管理,2018,37(2):273-279.
[9]楼振凯,侯福均,楼旭明.部分状态可见的隐马尔可夫模型状态序列的估计方法[J].统计研究,2019,36(6):107-114.
[10]朱慧明,黄仁存,游万海.贝叶斯隐马尔科夫异质面板模型及应用研究[J]. 统计研究,2014,14(7):97-104.
[11]王坤,沈秀娟,刘鹤飞.隐非齐次马尔可夫多元线性回归模型及其贝叶斯估计[J]. 统计与决策,2022(2):16-20.
[12]SONG X Y,KANG K,OU Y M,et al. Bayesian analysis of semiparametric hidden Markov models with latent variable[J].Structural Equation Modeling: A Multidisciplinary Journal, 2018,25(1):1-20.
[13]LIU H F, SONG X Y, TANG Y L, et al. Bayesian quantile nonhomogenous hidden Markov models [J].Statistical Methods in Medical Research,2021, 30(1):112-128.
[14]LIU H F, SONG X Y,ZHNAG B X. Varying-coefficient hidden Markov models with zero-effect regions[J].Computational Statistics and Data,2022(173):1-17.
[15]LIU H F, SONG X Y. Bayesian analysis of hidden Markov structural equation models with an unknown number of hidden states [J]. Econometrics and Statistics,2021, 18(C):29-43.
[16]刘鹤飞,王坤,宗凤喜. 混合个数未知的多元正态混合模型的贝叶斯推断[J]. 统计与决策,2019(2):26-29.