1932
0
  1. Home
  2. A-Z Publications
  3. Annual Review of Statistics and Its Application
  4. Volume 9, 2022
  5. Article

Annual Review of Statistics and Its Application

Volume 9, 2022

Discrete Latent Variable Models

Abstract

We review the discrete latent variable approach, which is very popular in statistics and related fields. It allows us to formulate interpretable and flexible models that can be used to analyze complex datasets in the presence of articulated dependence structures among variables. Specific models including discrete latent variables are illustrated, such as finite mixture, latent class, hidden Markov, and stochastic block models. Algorithms for maximum likelihood and Bayesian estimation of these models are reviewed, focusing, in particular, on the expectation–maximization algorithm and the Markov chain Monte Carlo method with data augmentation. Model selection, particularly concerning the number of support points of the latent distribution, is discussed. The approach is illustrated by summarizing applications available in the literature; a brief review of the main software packages to handle discrete latent variable models is also provided. Finally, some possible developments in this literature are suggested.

Keyword(s): data augmentation, expectation-maximization algorithm, finite mixture model, hidden Markov model, latent class model, model selection, stochastic block model
Loading

Article metrics loading...

/content/journals/10.1146/annurev-statistics-040220-091910
2022-03-07
2024-04-16
Loading full text...

Full text loading...

/deliver/fulltext/statistics/9/1/annurev-statistics-040220-091910.html?itemId=/content/journals/10.1146/annurev-statistics-040220-091910&mimeType=html&fmt=ahah

Literature Cited

  1. Abbe E 2017. Community detection and stochastic block models: recent developments. J. Mach. Learn. Res. 18:6446–531
     [Google Scholar]
  2. Akai N, Hirayama T, Morales LY, Akagi Y, Liu H, Murase H. 2019. Driving behavior modeling based on hidden Markov models with driver's eye-gaze measurement and ego-vehicle localization. 2019 IEEE Intelligent Vehicles Symposium (IV)949–56 New York: IEEE
  3. Akaike H 1973. Information theory and an extension of the maximum likelihood principle. Second International Symposium on Information Theory F Csáki, BN Petrov 267–81 Budapest: Akad. Kiado
  4. Altman RM. 2007. Mixed hidden Markov models: an extension of the hidden Markov model to the longitudinal data setting. J. Am. Stat. Assoc. 102:201–10
     [Google Scholar]
  5. Bacci S, Pandolfi S, Pennoni F 2014. A comparison of some criteria for states selection in the latent Markov model for longitudinal data. Adv. Data Anal. Classif. 8:125–45
     [Google Scholar]
  6. Bandeen-Roche K, Miglioretti DL, Zeger SL, Rathouz P. 1997. Latent variable regression for multiple discrete outcomes. J. Am. Stat. Assoc. 92:1375–86
     [Google Scholar]
  7. Bartholomew D, Knott M, Moustaki I. 2011. Latent Variable Models and Factor Analysis: A Unified Approach Chichester, UK: Arnold. , 3rd ed..
  8. Bartolucci F. 2006. Likelihood inference for a class of latent Markov models under linear hypotheses on the transition probabilities. J. R. Stat. Soc. Ser. B 68:155–78
     [Google Scholar]
  9. Bartolucci F, Bacci S, Gnaldi M. 2014a. multiLCIRT: an R package for multidimensional latent class item response models. Comput. Stat. Data Anal. 71:971–85
     [Google Scholar]
  10. Bartolucci F, Bacci S, Gnaldi M. 2015. Statistical Analysis of Questionnaires: A Unified Approach Based on R and Stata Boca Raton, FL: CRC Press
  11. Bartolucci F, Bacci S, Mira A 2018a. On the role of latent variable models in the era of big data. Stat. Probab. Lett. 136:165–69
     [Google Scholar]
  12. Bartolucci F, Bacci S, Pennoni F. 2014b. Longitudinal analysis of self-reported health status by mixture latent auto-regressive models. J. R. Stat. Soc. Ser. C 63:267–88
     [Google Scholar]
  13. Bartolucci F, Chiaromonte F, Don PK, Lindsay BG. 2017a. Composite likelihood inference in a discrete latent variable model for two-way clustering-by-segmentation problems. J. Comput. Graph. Stat. 26:388–402
     [Google Scholar]
  14. Bartolucci F, De Luca G. 2003. Likelihood-based inference for asymmetric stochastic volatility models. Comput. Stat. Data Anal. 42:445–49
     [Google Scholar]
  15. Bartolucci F, Farcomeni A. 2009. A multivariate extension of the dynamic logit model for longitudinal data based on a latent Markov heterogeneity structure. J. Am. Stat. Assoc. 104:816–31
     [Google Scholar]
  16. Bartolucci F, Farcomeni A. 2015. A discrete time event-history approach to informative drop-out in mixed latent Markov models with covariates. Biometrics 71:80–89
     [Google Scholar]
  17. Bartolucci F, Farcomeni A. 2021. A spatio-temporal model based on discrete latent variables for the analysis of COVID-19 incidence. Spat. Stat. In press. https://doi.org/10.1016/j.spasta.2021.100504
    [Crossref] [Google Scholar]
  18. Bartolucci F, Farcomeni A, Pennoni F 2013. Latent Markov Models for Longitudinal Data Boca Raton, FL: Chapman and Hall/CRC
  19. Bartolucci F, Farcomeni A, Pennoni F 2014c. Latent Markov models: a review of a general framework for the analysis of longitudinal data with covariates. TEST 23:433–65
     [Google Scholar]
  20. Bartolucci F, Lupparelli M. 2016. Pairwise likelihood inference for nested hidden Markov chain models for multilevel longitudinal data. J. Am. Stat. Assoc. 111:216–28
     [Google Scholar]
  21. Bartolucci F, Lupparelli M, Montanari GE. 2009. Latent Markov models for longitudinal binary data: an application to the performance evaluation of nursing homes. Ann. Appl. Stat. 3:611–36
     [Google Scholar]
  22. Bartolucci F, Marino M, Pandolfi S. 2018b. Dealing with reciprocity in dynamic stochastic block models. Comput. Stat. Data Anal. 123:86–100
     [Google Scholar]
  23. Bartolucci F, Pandolfi S. 2020. An exact algorithm for time-dependent variational inference for the dynamic stochastic block model. Pattern Recognit. Lett. 138:362–69
     [Google Scholar]
  24. Bartolucci F, Pandolfi S, Pennoni F 2017b. LMest: an R package for latent Markov models for longitudinal categorical data. J. Stat. Softw. 81:1–38
     [Google Scholar]
  25. Bartolucci F, Pennoni F. 2007. A class of latent Markov models for capture-recapture data allowing for time, heterogeneity and behavior effects. Biometrics 63:568–78
     [Google Scholar]
  26. Bartolucci F, Pennoni F, Francis B 2007. A latent Markov model for detecting patterns of criminal activity. J. R. Stat. Soc. Ser. A 170:151–32
     [Google Scholar]
  27. Bartolucci F, Pennoni F, Vittadini G 2011. Assessment of school performance through a multilevel latent Markov Rasch model. J. Educ. Behav. Stat. 36:491–522
     [Google Scholar]
  28. Bartolucci F, Pennoni F, Vittadini G 2016. Causal latent Markov model for the comparison of multiple treatments in observational longitudinal studies. J. Educ. Behav. Stat. 41:146–79
     [Google Scholar]
  29. Baum LE, Petrie T, Soules G, Weiss N. 1970. A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Stat. 41:164–71
     [Google Scholar]
  30. Benaglia T, Chauveau D, Hunter DR, Young D. 2009. mixtools: an R package for analyzing finite mixture models. J. Stat. Softw. 32:1–29
     [Google Scholar]
  31. Berchtold A. 2004. Optimization of mixture models: comparison of different strategies. Comput. Stat. 19:385–406
     [Google Scholar]
  32. Bertoletti M, Friel N, Rastelli R. 2015. Choosing the number of clusters in a finite mixture model using an exact integrated completed likelihood criterion. METRON 73:177–99
     [Google Scholar]
  33. Besag J. 1974. Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B 36:192–236
     [Google Scholar]
  34. Biernacki C, Celeux G, Govaert G 2000. Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Trans. Pattern Anal. Mach. Intel. 22:719–25
     [Google Scholar]
  35. Biernacki C, Celeux G, Govaert G 2010. Exact and Monte Carlo calculations of integrated likelihoods for the latent class model. J. Stat. Plann. Inference 140:2991–3002
     [Google Scholar]
  36. Biernacki C, Govaert G. 1999. Choosing models in model-based clustering and discriminant analysis. J. Stat. Comput. Simul. 64:49–71
     [Google Scholar]
  37. Boer P, Huisman M, Snijders TAB, Steglich C, Wichers LHV, Zeggelink EPH. 2006. StOCNET: an open software system for the advanced statistical analysis of social networks. Statistical Software version 1.7 .
     [Google Scholar]
  38. Bouveyron C, Celeux G, Murphy TB, Raftery AE 2019. Model-Based Clustering and Classification for Data Science, with Applications in R Cambridge, UK: Cambridge Univ. Press
  39. Browne RP, McNicholas PD. 2012. Model-based clustering, classification, and discriminant analysis of data with mixed type. J. Stat. Plann. Inference 142:2976–84
     [Google Scholar]
  40. Bulla J, Bulla I, Nenadić O. 2010. hsmm—an R package for analyzing hidden semi-Markov models. Comput. Stat. Data Anal. 54:611–19
     [Google Scholar]
  41. Cai L, Choi K, Hansen M, Harrell L. 2016. Item response theory. Annu. Rev. Stat. Appl. 3:297–321
     [Google Scholar]
  42. Celeux G, Durand JB. 2008. Selecting hidden Markov model state number with cross-validated likelihood. Comput. Stat. 23:541–64
     [Google Scholar]
  43. Celeux G, Forbes F, Robert CP, Titterington DM. 2006. Deviance information criteria for missing data models. Bayesian Anal. 1:651–73
     [Google Scholar]
  44. Celeux G, Soromenho G. 1996. An entropy criterion for assessing the number of clusters in a mixture model. J. Classif. 13:195–212
     [Google Scholar]
  45. Chib S. 1995. Marginal likelihood from the Gibbs output. J. Am. Stat. Assoc. 90:1313–21
     [Google Scholar]
  46. Chib S. 1996. Calculating posterior distributions and modal estimates in Markov mixture models. J. Econom. 75:79–97
     [Google Scholar]
  47. Chib S, Jeliazkov I. 2001. Marginal likelihood from the Metropolis–Hastings output. J. Am. Stat. Assoc. 96:270–81
     [Google Scholar]
  48. Chiquet J, Donnet S, Barbillon P. 2020. sbm: stochastic blockmodels. R Package version 0.2.2
    [Google Scholar]
  49. Clogg CC. 1995. Latent class models. Handbook of Statistical Modeling for the Social and Behavioral Sciences G Arminger, CC Clogg, M Sobel 311–59 New York: Plenum
     [Google Scholar]
  50. Collins L, Lanza S. 2010. Latent Class and Latent Transition Analysis with Applications in the Social, Behavioral, and Health Sciences New York: Wiley
  51. Congdon P. 2006. Bayesian model choice based on Monte Carlo estimates of posterior model probabilities. Comput. Stat. Data Anal. 50:346–57
     [Google Scholar]
  52. Daudin JJ, Picard F, Robin S. 2008. A mixture model for random graphs. Stat. Comput. 18:173–83
     [Google Scholar]
  53. Davison AC, Hinkley DV. 1997. Bootstrap Methods and their Application Cambridge, UK: Cambridge Univ. Press
  54. Dayton CM, Macready GB. 1988. Concomitant-variable latent-class models. J. Am. Stat. Assoc. 83:173–78
     [Google Scholar]
  55. Dempster AP, Laird NM, Rubin DB 1977. Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. R. Stat. Soc. Ser. B 39:1–22
     [Google Scholar]
  56. Dias J 2006. Model selection for the binary latent class model: a Monte Carlo simulation. Data Science and Classification V Batagelj, HH Bock, A Ferligoj, A Žiberna 91–99 New York: Springer
     [Google Scholar]
  57. Diebolt J, Robert CP. 1994. Estimation of finite mixture distributions through Bayesian sampling. J. R. Stat. Soc. Ser. B 56:363–75
     [Google Scholar]
  58. Diggle PJ. 2013. Statistical Analysis of Spatial and Spatio-Temporal Point Patterns Boca Raton, FL: Chapman and Hall/CRC
  59. Diggle PJ, Heagerty P, Liang KY, Zeger S. 2002. Analysis of Longitudinal Data Oxford, UK: Oxford Univ. Press
  60. Everitt BS. 1984. An Introduction to Latent Variable Models Boca Raton, FL: Chapman and Hall/CRC
  61. Everitt BS, Landau S, Leese M, Stahl D 2011. Cluster Analysis New York: Wiley. , 5th ed..
  62. Feng ZD, McCulloch CE. 1996. Using bootstrap likelihood ratios in finite mixture models. J. R. Stat. Soc. Ser. B 58:609–17
     [Google Scholar]
  63. Finch WH, French BF. 2015. Latent Variable Modeling with R New York: Routledge
  64. Forcina A. 2017. A Fisher-scoring algorithm for fitting latent class models with individual covariates. Econom. Stat. 3:132–40
     [Google Scholar]
  65. Formann AK. 1995. Linear logistic latent class analysis and the Rasch model. In Rasch Models GH Fischer, IW Molenaar 239–55 New York: Springer
     [Google Scholar]
  66. Fortunato S. 2010. Community detection in graphs. Phys. Rep. 486:75–174
     [Google Scholar]
  67. Frühwirth-Schnatter S. 2006. Finite Mixture and Markov Switching Models New York: Springer
  68. Frühwirth-Schnatter S, Celeux G, Robert CP 2019. Handbook of Mixture Analysis Boca Raton, FL: Chapman and Hall/CRC
  69. Geman S, Geman D. 1984. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intel. 6:721–41
     [Google Scholar]
  70. Ghiringhelli C, Bartolucci F, Mira A, Arbia G 2021. Modelling nonstationary spatial lag models with hidden Markov random fields. Spatial Stat. 44:100522
     [Google Scholar]
  71. Goldstein H. 2011. Multilevel Statistical Models New York: Wiley. , 4th ed..
  72. Goodman LA. 1974. Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika 61:215–31
     [Google Scholar]
  73. Goodman LA 2002. Latent class analysis: the empirical study of latent types, latent variables, and latent structures. Applied Latent Class Analysis JA Hagenaars, AL McCutcheon 3–55 Cambridge, UK: Cambridge Univ. Press
     [Google Scholar]
  74. Goodman LA. 2007. On the assignment of individuals to latent classes. Sociol. Methodol. 37:1–22
     [Google Scholar]
  75. Green PJ. 1995. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82:711–32
     [Google Scholar]
  76. Grün B, Leisch F. 2008. FlexMix version 2: finite mixtures with concomitant variables and varying and constant parameters. J. Stat. Softw. 28:1–35
     [Google Scholar]
  77. Hambleton RK, Shavelson RJ, Webb NM, Swaminathan H, Rogers HJ. 1991. Fundamentals of Item Response Theory, Vol. 2 Thousand Oaks, CA: SAGE
  78. Hancock GR, Harring JR, Macready GB. 2019. Advances in Latent Class Analysis: A Festschrift in Honor of C. Mitchell Dayton Charlotte, NC: IAP
  79. Hansell S. 1984. Cooperative groups, weak ties, and the integration of peer friendships. Soc. Psychol. Q. 47:316–28
     [Google Scholar]
  80. Harte D. 2021. HiddenMarkov: hidden Markov models. R Package version 1.8–13
    [Google Scholar]
  81. Heinen T. 1996. Latent Class and Discrete Latent Trait Models: Similarities and Differences Thousand Oaks, CA: SAGE
  82. Helske S, Helske J. 2019. Mixture hidden Markov models for sequence data: the seqHMM package in R. J. Stat. Softw. 88:1–32
     [Google Scholar]
  83. Himmelmann L. 2010. HMM: hidden Markov models. R Package version 1.0
    [Google Scholar]
  84. Holland PW, Laskey KB, Leinhardt S. 1983. Stochastic blockmodels: first steps. Soc. Netw. 5:109–37
     [Google Scholar]
  85. Jasra A, Holmes CC, Stephens DA. 2005. MCMC and the label switching problem in Bayesian mixture models. Stat. Sci. 20:50–67
     [Google Scholar]
  86. Jeffreys H. 1961. Theory of Probability Oxford, UK: Oxford Univ. Press. , 3rd ed..
  87. Jones BL, Nagin DS, Roeder K. 2001. A SAS procedure based on mixture models for estimating developmental trajectories. Sociol. Methods Res. 29:374–93
     [Google Scholar]
  88. Juang B, Rabiner L. 1991. Hidden Markov models for speech recognition. Technometrics 33:251–72
     [Google Scholar]
  89. Kass RE, Raftery AE. 1995. Bayes factors. J. Am. Stat. Assoc. 90:773–95
     [Google Scholar]
  90. Lange JM, Hubbard RA, Inoue LYT, Minin VN. 2015. A joint model for multistate disease processes and random informative observation times, with applications to electronic medical records data. Biometrics 71:90–101
     [Google Scholar]
  91. Langrock R, MacDonald IL, Zucchini W. 2012. Some nonstandard stochastic volatility models and their estimation using structured hidden Markov models. J. Empir. Finance 19:147–61
     [Google Scholar]
  92. Lanza ST, Coffman DL, Xu S. 2013. Causal inference in latent class analysis. Struct. Equ. Model. 20:361–83
     [Google Scholar]
  93. Lazarsfeld PF, Henry NW. 1968. Latent Structure Analysis Boston: Houghton Mifflin
  94. Lee C, Wilkinson DJ. 2019. A review of stochastic block models and extensions for graph clustering. Appl. Netw. Sci. 4:122
     [Google Scholar]
  95. Leger JB, Barbillon P, Chiquet J. 2020. blockmodels: latent and stochastic block model estimation by a “V-EM. .” R Package version 1.1.4
    [Google Scholar]
  96. Leisch F. 2004. FlexMix: a general framework for finite mixture models and latent class regression in R. J. Stat. Softw. 11:1–18
     [Google Scholar]
  97. Lindsay B, Clogg CC, Grego J. 1991. Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis. J. Am. Stat. Assoc. 86:96–107
     [Google Scholar]
  98. Lindsay BG. 1988. Composite likelihood methods. Contemp. Math. 80:221–39
     [Google Scholar]
  99. Lindsay BG. 1995. Mixture Models: Theory, Geometry and Applications Arlington, VA: Am. Stat. Assoc.
  100. Linzer DA, Lewis JB. 2011. poLCA: an R package for polytomous variable latent class analysis. J. Stat. Softw. 42:1–29
     [Google Scholar]
  101. Little RJA, Rubin DB. 2020. Statistical Analysis with Missing Data New York: Wiley
  102. Louis T. 1982. Finding the observed information matrix when using the EM algorithm. J. R. Stat. Soc. Ser. B 44:226–33
     [Google Scholar]
  103. Lukočienė O, Vermunt JK 2009. Determining the number of components in mixture models for hierarchical data. Advances in Data Analysis, Data Handling and Business Intelligence A Fink, B Lausen, W Seidel, A Ultsch 241–49 New York: Springer
     [Google Scholar]
  104. MacDonald IL, Zucchini W 2016. Hidden Markov models for discrete-valued time series. Handbook of Discrete-Valued Time Series RA Davis, SH Holan, R Lund, N Ravishanker 267–86 Boca Raton, FL: Chapman and Hall/CRC
     [Google Scholar]
  105. MacKay DJC. 1997. Ensemble learning for hidden Markov models. Tech. Rep. Cavendish Lab., Univ. Cambridge Cambridge, UK.: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.9627&rep=rep1&type=pdf
    [Google Scholar]
  106. Maruotti A. 2011. Mixed hidden Markov models for longitudinal data: an overview. Int. Stat. Rev. 79:427–54
     [Google Scholar]
  107. Matias C, Miele V 2017. Statistical clustering of temporal networks through a dynamic stochastic block model. J. R. Stat. Soc. Ser. B 79:1119–41
     [Google Scholar]
  108. McClintock BT, Michelot T. 2018. momentuHMM: R package for generalized hidden Markov models of animal movement. Methods Ecol. Evol. 9:1518–30
     [Google Scholar]
  109. McGrory CA, Titterington DM. 2009. Variational Bayesian analysis for hidden Markov models. Aust. N. Z. J. Stat. 51:227–44
     [Google Scholar]
  110. McHugh RB. 1956. Efficient estimation and local identification in latent class analysis. Psychometrika 21:331–47
     [Google Scholar]
  111. McLachlan G, Peel D. 2000. Finite Mixture Models New York: Wiley
  112. McLachlan GJ, Lee SX, Rathnayake SI. 2019. Finite mixture models. Annu. Rev. Stat. Appl. 6:355–78
     [Google Scholar]
  113. McNicholas PD, ElSherbiny A, Jampani KR, McDaid AF, Murphy TB, Banks L. 2019. pgmm: parsimonious Gaussian mixture models. R Package version 1.2.4
    [Google Scholar]
  114. McNicholas PD, Murphy TB. 2008. Parsimonious Gaussian mixture models. Stat. Comput. 18:285–96
     [Google Scholar]
  115. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. 1953. Equation of state calculations by fast computing machines. J. Chem. Phys. 21:1087–92
     [Google Scholar]
  116. Muthén B 2004. Latent variable analysis: growth mixture modeling and related techniques for longitudinal data. Handbook of Quantitative Methodology for the Social Sciences D Kaplan 345–68 Thousand Oaks, CA: SAGE
     [Google Scholar]
  117. Muthén B 2007. Latent variable hybrids: overview of old and new models. Advances in Latent Variable Mixture Models GR Hancock, KM Samuelsen 1–24 Charlotte, NC: IAP
     [Google Scholar]
  118. Muthén B, Shedden K. 1999. Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics 55:463–69
     [Google Scholar]
  119. Muthén L, Muthén B. 2013. Mplus: Statistical Analysis with Latent Variables User's Guide Los Angeles, CA: Muthén & Muthén
  120. Nagin D. 1999. Analyzing developmental trajectories: a semi-parametric, group-based approach. Psychol. Methods 4:139–57
     [Google Scholar]
  121. Nowicki K, Snijders TAB. 2001. Estimation and prediction for stochastic blockstructures. J. Am. Stat. Assoc. 96:1077–87
     [Google Scholar]
  122. Nylund K, Asparouhov T, Muthén B 2007. Deciding on the number of classes in latent class analysis and growth mixture modeling: a Monte Carlo simulation study. Struct. Eq. Model. 14:535–69
     [Google Scholar]
  123. O'Connell J, Højsgaard S. 2011. Hidden semi Markov models for multiple observation sequences: the mhsmm package for R. J. Stat. Softw. 39:1–22
     [Google Scholar]
  124. Pauli F, Racugno W, Ventura L. 2011. Bayesian composite marginal likelihoods. Stat. Sin. 21:149–64
     [Google Scholar]
  125. Proust-Lima C, Philipps V, Liquet B 2017. Estimation of extended mixed models using latent classes and latent processes: the R package lcmm. J. Stat. Softw. 78:1–56
     [Google Scholar]
  126. R Core Team 2021. R: A language and environment for statistical computing. Statistical Software R Found. Stat. Comput. Vienna:
     [Google Scholar]
  127. Rabe-Hesketh S, Skrondal A, Pickles A. 2004. GLLAMM manual Work. Pap., Div. Biostat., Univ. Calif. Berkeley:
  128. Rasch G 1961. On general laws and the meaning of measurement in psychology. Berkeley Symposium on Mathematical Statistics and Probability J Neyman 321–33 Berkeley, CA: Univ. Calif. Press
  129. Richardson S, Green P. 1997. On Bayesian analysis of mixture with an unknown number of components. J. R. Stat. Soc. Ser. B 59:731–92
     [Google Scholar]
  130. Rizopoulos D, Ghosh P. 2011. A Bayesian semiparametric multivariate joint model for multiple longitudinal outcomes and a time-to-event. Stat. Med. 30:1366–80
     [Google Scholar]
  131. Robert CP, Ryden T, Titterington DM 2000. Bayesian inference in hidden Markov models through the reversible jump Markov chain Monte Carlo method. J. R. Stat. Soc. Ser. B 62:57–75
     [Google Scholar]
  132. Rubin DB 2005. Causal inference using potential outcomes: design, modeling, decisions. J. Am. Stat. Assoc. 100:322–31
     [Google Scholar]
  133. Sambridge M. 2014. A parallel tempering algorithm for probabilistic sampling and multimodal optimization. Geophys. J. Int. 196:357–74
     [Google Scholar]
  134. Schlattmann P. 2009. Medical Applications of Finite Mixture Models Berlin: Springer-Verlag
  135. Schwarz G. 1978. Estimating the dimension of a model. Ann. Stat. 6:461–64
     [Google Scholar]
  136. Scrucca L, Fop M, Murphy TB, Raftery AE. 2016. mclust 5: clustering, classification and density estimation using Gaussian finite mixture models. R J. 8:289–317
     [Google Scholar]
  137. Shephard N 1996. Statistical aspects of arch and stochastic volatility. Time Series Models: In Econometric, Finance and Other Fields DR Cox, DV Hinkley 1–67 Boca Raton, FL: Chapman and Hall/CRC
     [Google Scholar]
  138. Skrondal A, Rabe-Hesketh S. 2004. Generalized Latent Variable Modelling: Multilevel, Longitudinal and Structural Equation Models Boca Raton, FL: Chapman and Hall/CRC
  139. Smyth P. 2000. Model selection for probabilistic clustering using cross-validated likelihood. Stat. Comput. 10:63–72
     [Google Scholar]
  140. Snijders TA, Nowicki K. 1997. Estimation and prediction for stochastic blockmodels for graphs with latent block structure. J. Classif. 14:75–100
     [Google Scholar]
  141. Spezia L. 2010. Bayesian analysis of multivariate Gaussian hidden Markov models with an unknown number of regimes. J. Time Ser. Anal. 31:1–11
     [Google Scholar]
  142. Spiegelhalter DJ, Best NG, Carlin BP, Van Der Linde A 2002. Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B 64:583–639
     [Google Scholar]
  143. Tabouy T, Barbillon P, Chiquet J. 2020. Variational inference for stochastic block models from sampled data. J. Am. Stat. Assoc. 115:455–66
     [Google Scholar]
  144. Tanner MA, Wong WH. 1987. The calculation of posterior distributions by data augmentation. J. Am. Stat. Assoc. 82:528–40
     [Google Scholar]
  145. Titterington DM, Smith AFM, Makov HE. 1985. Statistical Analysis of Finite Mixture Distributions New York: Wiley
  146. Turner R. 2008. Direct maximization of the likelihood of a hidden Markov model. Comput. Stat. Data Anal. 52:4147–60
     [Google Scholar]
  147. van de Pol F, Langeheine R. 1990. Mixed Markov latent class models. Sociol. Methodol. 20:213–47
     [Google Scholar]
  148. Varin C, Reid N, Firth D 2011. An overview of composite likelihood methods. Stat. Sin. 21:5–42
     [Google Scholar]
  149. Varin C, Vidoni P. 2005. A note on composite likelihood inference and model selection. Biometrika 92:519–28
     [Google Scholar]
  150. Vermunt JK. 2003. Multilevel latent class models. Sociol. Methodol. 33:213–39
     [Google Scholar]
  151. Vermunt JK 2007. Growth models for categorical response variables: standard, latent-class, and hybrid approaches. Longitudinal Models in the Behavioral and Related Sciences K van Montfort, J Oud, A Santora 139–58 Mahwah, NJ: Lawrence Erlbaum
     [Google Scholar]
  152. Vermunt JK, Langeheine R, Böckenholt U 1999. Discrete-time discrete-state latent Markov models with time-constant and time-varying covariates. J. Educ. Behav. Stat. 24:179–207
     [Google Scholar]
  153. Vermunt JK, Magidson J. 2016. Technical Guide for Latent GOLD 5.1: Basic, Advanced, and Syntax Belmont, MA: Stat. Innov. Inc.
  154. Vermunt JK, Van Dijk L. 2001. A nonparametric random-coefficients approach: the latent class regression model. Multilevel Model. Newsl. 13:6–13
     [Google Scholar]
  155. Visser I, Speekenbrink M. 2010. depmixS4: an R package for hidden Markov models. J. Stat. Softw. 36:1–21
     [Google Scholar]
  156. Viterbi A. 1967. Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Trans. Inform. Theory 13:260–69
     [Google Scholar]
  157. Wainwright MJ, Jordan MI. 2008. Graphical Models, Exponential Families, and Variational Inference Boston: Now Publ.
  158. Wang M, Bodner TE 2007. Growth mixture modeling identifying and predicting unobserved subpopulations with longitudinal data. Organ. Res. Methods 10:635–56
     [Google Scholar]
  159. Wang YJ, Wong GY. 1987. Stochastic blockmodels for directed graphs. J. Am. Stat. Assoc. 82:8–19
     [Google Scholar]
  160. Watanabe S, Opper M. 2010. Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. J. Mach. Learn. Res. 11:3571–94
     [Google Scholar]
  161. Wedel M. 2002. Concomitant variables in finite mixture models. Stat. Neerl. 56:362–75
     [Google Scholar]
  162. Welch LR. 2003. Hidden Markov models and the Baum-Welch algorithm. IEEE Inform. Theory Soc. Newsl. 53:10–13
     [Google Scholar]
  163. White A, Murphy TB. 2014. BayesLCA: an R package for Bayesian latent class analysis. J. Stat. Softw. 61:1–28
     [Google Scholar]
  164. Wiggins L. 1973. Panel Analysis: Latent Probability Models for Attitude and Behaviour Processes Amsterdam: Elsevier
  165. Wong WK. 2010. Backtesting value-at-risk based on tail losses. J. Empir. Finance 17:526–38
     [Google Scholar]
  166. Yang T, Chi Y, Zhu S, Gong Y, Jin R. 2011. Detecting communities and their evolutions in dynamic social networks—a Bayesian approach. Mach. Learn. 82:157–89
     [Google Scholar]
  167. Zhou H, Lange KL. 2010. On the bumpy road to the dominant mode. Scand. J. Stat. 37:612–31
     [Google Scholar]
  168. Zucchini W, MacDonald IL, Langrock R. 2017. Hidden Markov Models for Time Series: An Introduction Using R Boca Raton, FL: Chapman and Hall/CRC
/content/journals/10.1146/annurev-statistics-040220-091910
Loading
Discrete Latent Variable Models
Annual Review of Statistics and Its Application 9, 425 (2022); https://doi.org/10.1146/annurev-statistics-040220-091910
/content/journals/10.1146/annurev-statistics-040220-091910
/content/journals/10.1146/annurev-statistics-040220-091910
Loading

Data & Media loading...

  • Article Type: Review Article

Most Cited Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error