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Notes of this length contain errors. Reports are welcome.

Reporting an error

The most useful reports say where and what:

• the chapter and the numbered environment or equation — “Theorem 5.3, second line of the proof” is enough to locate anything;
• what is wrong, and if you know it, what it should say.

Mathematical errors, missing hypotheses and unclear proofs are the most valuable. Notation inconsistencies between chapters are worth reporting too, since they are easy to introduce and hard to spot from inside.

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Known limitations

Stated plainly, so that nobody spends time discovering them independently.

The HMM telescoping sampler is derived, not cited. Frühwirth-Schnatter et al. (2021) develop and validate the method for finite mixtures. The extension to HMMs in Telescoping for HMMs follows the same construction and is worked out in these notes. It has not been validated against a published HMM implementation, and 11  The telescoping sampler says so. Validate on simulated data before relying on it.

Some proofs are sketched. The identifiability theorem of Allman et al. (2009), the asymptotic normality result of Bickel et al. (1998), and the ergodic theorem for finite chains are stated with the argument outlined rather than given in full. Each is flagged where it appears, with a reference to the complete proof.

Continuous-time models are not covered. 14  Applications notes that irregular observation times in epidemiology require a continuous-time chain, but the notes do not develop the theory.

No code. By design, but it does mean the implementation exercises in 15  Exercises are unsupported.

Reuse

The notes are intended to be used and adapted. If you build a course on them, attribution to the primary references in Appendix B — Reading guide matters more than attribution here — Zucchini et al. (2016), Cappé et al. (2005), Frühwirth-Schnatter (2006) and Douc et al. (2014) did the work these notes organise.

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Allman, Elizabeth S., Catherine Matias, and John A. Rhodes. 2009. “Identifiability of Parameters in Latent Structure Models with Many Observed Variables.” The Annals of Statistics 37 (6A): 3099–132. https://doi.org/10.1214/09-AOS689.

Bickel, Peter J., Ya’acov Ritov, and Tobias Rydén. 1998. “Asymptotic Normality of the Maximum-Likelihood Estimator for General Hidden Markov Models.” The Annals of Statistics 26 (4): 1614–35. https://doi.org/10.1214/aos/1024691255.

Cappé, Olivier, Éric Moulines, and Tobias Rydén. 2005. Inference in Hidden Markov Models. Springer Series in Statistics. Springer. https://doi.org/10.1007/0-387-28982-8.

Douc, Randal, Éric Moulines, and David S. Stoffer. 2014. Nonlinear Time Series: Theory, Methods and Applications with R Examples. Chapman; Hall/CRC. https://doi.org/10.1201/b16331.

Frühwirth-Schnatter, Sylvia. 2006. Finite Mixture and Markov Switching Models. Springer Series in Statistics. Springer. https://doi.org/10.1007/978-0-387-35768-3.

Frühwirth-Schnatter, Sylvia, Gertraud Malsiner-Walli, and Bettina Grün. 2021. “Generalized Mixtures of Finite Mixtures and Telescoping Sampling.” Bayesian Analysis 16 (4): 1279–307. https://doi.org/10.1214/21-BA1294.

Zucchini, Walter, Iain L. MacDonald, and Roland Langrock. 2016. Hidden Markov Models for Time Series: An Introduction Using R. 2nd ed. Chapman; Hall/CRC. https://doi.org/10.1201/b20790.