About
Lecture notes on hidden Markov models at the level of a graduate course in statistics, covering the model, the algorithms, the asymptotic theory, and the Bayesian and hierarchical extensions.
They exist because the two standard references rarely meet in the middle. Zucchini et al. (2016) is applied and practical but leaves proofs to citations; Cappé et al. (2005) is rigorous but assumes more measure theory than most readers arrive with. These notes derive the algorithms in full, state the theory honestly, and say what to do when a model fits badly.
How they are written
Proofs are given, not cited — except where the proof is a paper in itself. The identifiability theorem of Allman et al. (2009) and the asymptotic normality result of Bickel et al. (1998) are the main exceptions, and both are flagged where they appear.
Assumptions are named. The conditions under which a theorem holds are more useful than the theorem, because applied work happens where one of them fails.
Failure modes get equal billing. Unbounded likelihoods, spurious local maxima, label switching, empty states and boundary parameters are part of the subject, not caveats.
Notation follows Zucchini et al. (2016), so formulas here match theirs. See Appendix A — Notation.
What is unusual here
Two chapters most treatments omit.
10 Hierarchical models separates three unrelated things that all get called hierarchical — nested state spaces, multilevel models across several series, and hierarchical priors within one model. The multilevel case is what most applied problems actually need.
11 The telescoping sampler covers the telescoping sampler of Frühwirth-Schnatter et al. (2021), which samples the number of states directly without reversible jump machinery. The mixture case follows the original paper; the HMM extension is derived here and marked as such.
Corrections are welcome — see contact.