Statistical Inference for Markov and Hidden Markov Models

I am concerned here with inferring the parameters and/or the structure of the model, not with the estimation of the hidden state (in th HMM case); that problem falls under filtering.

Parameter inference (what machine learning types would call "learning") given a known, fixed structure. Determining the structure ("discovery"). Order estimation as a particular case of discovery, and of model selection.

Patrick Billingsley, Statistical Inference for Markov Chains
Andrew Fraser, Hidden Markov Models and Dynamical Systems [Review: The Statistics of Moving Shadows]
Peter Guttorp, Stochastic Modelling of Scientific Data

David Blackwell and Lambert Koopmans, "On the Identifiability Problem for Functions of Finite Markov Chains", The Annals of Mathematical Statistics 28 (1957): 1011--1015 [An old, but very clear, paper on the problems presented by what we now call hidden Markov models]
Peter Bühlmann and Abraham J. Wyner, "Variable Length Markov Chains", The Annals of Statistics 27 (1999): 480--513 [Preprint available as Berkeley statistics department technical report 479]
Olivier Cappé "Online EM Algorithm for Hidden Markov Models", arxiv:0908.2359
George Cybenko and Valentino Crespi, "Learning Hidden Markov Models using Non-Negative Matrix Factorization", arxiv:0809.4086 [Though it contains an error about the capacities of our CSSR algorithm]
Subhashis Ghosal and Yongqiang Tang, "Bayesian Consistency for Markov Processes", Sankhya 68 (2006): 227--239
Subhashis Ghosal and Aad van der Vaart, "Convergence Rates of Posterior Distributions for Non-IID Observations", Annals of Statistics 35 (2007): 192--223 = arxiv:0708.0491
Gusztav Morvai and Benjamin Weiss
- "Estimating the Lengths of Memory Words", IEEE Transactions on Information Theory 54 (2008): 3804--3807
- "On estimating the memory for finitarily Markovian processes", Ann. I. H. Poincaré-PR 43 (2007): 15--30 arxiv:0712.0105
F. Papangelou, "Large Deviations and the Bayesian Estimation of Higher-Order Markov Transition Functions ", Journal of Applied Probability 33 (1996): 18--27 [JSTOR]
Yuval Peres and Paul Shields, "Two new Markov order estimators", math.ST/0506080 [Very nice.]
Christopher C. Strelioff, James P. Crutchfield, Alfred W. Hubler, "Inferring Markov Chains: Bayesian Estimation, Model Comparison, Entropy Rate, and Out-of-class Modeling", math.ST/0703715
Daniel R. Upper, Theory and Algorithms for Hidden Markov Models and Generalized Hidden Markov Models [Ph.D. thesis, math dept., Berkeley, 1997; online]

quite

E. Racca, F. Laio, D. Poggi and L. Ridolfi, "Test to determine the Markov order of a time series", Physical Review E 75 (2007): 011126 [The test is to linearly regress x(t+1) on x(t), x(t-1), etc., out to some finite order, and see how far back you have to go before the regression coefficients are insignificantly different from zero. This is not crazy as a first cut idea, but it's not generally valid, and in fact fails for the logistic map.]

Definitely

S. S. Melnyk, O. V. Usatenko, V. A. Yampol'skii and V. A. Golick, "Competition between Two Kinds of Correlations in Literary Texts", physics/0402042 [This has surely got to be one of the ugliest ways of parameterizing a Markov chain I have seen; it's a miracle they don't get probabilities greater than 1, if indeed they don't.]

CRS, "Dynamics of Bayesian Updating with Dependent Data and Misspecified Models", arxiv:0901.1342
CRS and Kristina Lisa Klinkner, "Blind Construction of Optimal Nonlinear Recursive Predictors for Discrete Sequences", cs.LG/0406011 [CSSR]

Enrique E. Alvarez, "Estimation in Stationary Markov Renewal Processes, with Application to Earthquake Forecasting in Turkey", Methodology and Computing in Applied Probability 7 (2005): 119--130
Sofia Andersson and Tobias Rydén, "Subspace estimation and prediction methods for hidden Markov models", Annals of Statistics 37 (2009): 4131--4152
Ana Arribas-Gil, Elisabeth Gassiat and Catherine Matias, "Parameter estimation in pair hidden Markov models", math.ST/0509280
Patrice Bertail and Stéphan Clémençon, "Edgeworth expansions of suitably normalized sample mean statistics for atomic Markov chains", Probability Theory and Related Fields 130 (2004): 388--414 [I need to learn more about Edgeworth expansions anyway]
P. J. Bickel and Y. Ritov, "Inference in Hidden Markov Models I: Local Asymptotic Normality in the Stationary Case", UCB Statistics Technical Report 383 [link]
Jose Borges and Mark Levene, "Evaluating Variable Length Markov Chain Models for Analysis of User Web Navigation Sessions", cs.AI/0606115
J. Borwanker, G. Kallianpur and B. L. S. Prakasa Rao, "The Bernstein-von Mises Theorem for Markov Processes", The Annals of Mathematical Statistics 42 (1971): 1241--1253 [JSTOR]
Olivier Cappé Eric Moulines and Tobias Ryden, Inference in Hidden Markov Models [This is a superb book, treating all the main statistical problems connected with HMMs in a rigorous manner. There is a chapter/appendix which reminds the reader who has forgetten about Markov chains on general state spaces, but remembers measure-theoretic probability very well, about their properties. (The idea that the reader of a book on HMMs may not know what the Viterbi algorithm is, but will definitely recall the Hahn-Jordan decomposition, strikes me as very much a product of the French school of probability theory — from which I have learned much! But it's not so far gone as to make the whole thing an exercise in the "general theory of processes".) Listed down here until I finish the last chapter. Blurb]
Pavel Chigansky, "Maximum Likelihood Estimator for Hidden Markov Models in continuous time", arxiv:0707.0271
C. C. Y. Dorea and L. C. Zhao, "Nonparametric Density Estimation in Hidden Markov Models", Statistical Inference for Stochastic Processes 5 (2002): 55--64
Randal Douc, "Non singularity of the asymptotic Fisher information matrix in hidden Markov models", math.ST/0511631
P. Dupont, F. Denis and Y. Esposito, "Links between probabilistic automata and hidden Markov models: probability distributions, learning models and induction algorithms", Pattern Recognition 38 (2005): 1349--1371
Farzad Eskandari and Mohammad R. Meshkani, "Empirical Bayes analysis of log-linear models for a generalized finite stationary Markov chain", Metrika 59 (2004): 173--191 [abstract]
Florence Forbes and Nathalie Peyrard, "Hidden Markov Random Field Model Selection Criteria Based on Mean Field-Like Approximations", IEEE Transactions on Pattern Analysis and Machine Intelligence 25 (2003): 1089--1101 [PostScript preprint]
Cheng-Der Fuh, "Asymptotic operating characteristics of an optimal change point detection in hidden Markov models", Annals of Statistics 32 (2004): 2305--2339 = math.ST/0503682
Antonio Galves, Charlotte Galves, Nancy L. Garcia, Florencia Leonardi, "Context tree selection and linguistic rhythm retrieval from written texts", arxiv:0902.3619
Antonio Galves, Florencia Leonardi, "Exponential inequalities for empirical unbounded context trees", arxiv:0710.5900
H. Ito, S.-I. Amari and K. Kobayashi, "Identifiability of Hidden Markov Information Sources and Their Minimum Degrees of Freedom", IEEE Transactions on Information Theory 38 (1992): 324--333
Hans R. Künsch, "State Space and Hidden Markov Models", pp. 109--173 in Ole E. Barndorff-Nielsen, David R. Cox and Claudia Klüppelberg (eds.), Complex Stochastic Systems
J. Lember, A. Koloydenko, "Adjusted Viterbi training for hidden Markov models", arxiv:0709.2317
Florencia Leonardi, "Some upper bounds for the rate of convergence of penalized likelihood context tree estimators", arxiv:0701810
E. Locherbach, "Likelihood Ratio Processes for Markovian Particle Systems with Killing and Jumps", Statistical Inference for Stochastic Processes 5 (2002): 153--177
Philipp Metzner, Frank Noe and Christof Schutte, "Estimating the sampling error: Distribution of transition matrices and functions of transition matrices for given trajectory data", Physical Review E 80 (2009): 021106 [I presume they have a good reason for not just using the delta method, and/or bootstrapping, but I'll have to read it to see what that is]
G. Morvai and B. Weiss, "Order Estimation of Markov Chains", IEEE Transactions on Information Theory 51 (2005): 1496--1497
Adam Paszkiewicz, "When transition count for a Markov chains is a complete sufficient statistic", Statistics and Probability Letters 76 (2006): 757--763 [When the initial and final states are fixed, for example]
Spiridon Penev, Hanxiang Peng, Atnon Schick and Wolfgang Wefelmeyer, "Efficient estimators for functionals of Markov chains with parametric marginals", Statistics and Probability Letters 66 (2004): 335--345
Amr Sadek and Nikolaos Limnios, "Nonparametric estimation of reliability and survival function for continuous-time finite Markov processes", Journal of Statistical Planning and Inference 133 (2005): 1--21
Anton Schick and Wolfgang Wefelmeyer, "Estimating Joint Distributions of Markov Chains", Statistical Inference for Stochastic Processes 5 (2002): 1--22
Iuliana Teodorescu, "Maximum Likelihood Estimation for Markov Chains", arxiv:0905.4131
M. J. van der Heyden et al., "Testing the Order of Discrete Markov Chains Using Surrogate Data", Physica D 117 (1998): 299--313
Ramon van Handel, "On the minimal penalty for Markov order estimation", arxiv:0908.3666
Martin J. Wainwright, "Inconsistent parameter estimation in Markov random fields: Benefits in the computation-limited setting", cs.LG/0602092
L. C. Zhao, C. C. Y. Dorea and C. R. Gonçalves, "On Determination of the Order of a Markov Chain", Statistical Inference for Stochastic Processes 4 (2001): 273--282

Notebooks

Statistical Inference for Markov and Hidden Markov Models