Hidden Markov Models (HMMs)
From Probability Foundations to Advanced Theory
Welcome to the Hidden Markov Models Course.
This site presents a calm, rigorous, graduate-level treatment of HMMs:
- From measure-theoretic probability and Markov chains
- Through inference algorithms (forward–backward, Viterbi, EM)
- To asymptotic theory and advanced variants (switching, nonparametric, state-space models)
Start with foundations View full curriculum
This course is designed for students who want proofs and derivations, not just code snippets. Use the buttons above to either dive straight into the notes or skim the overall structure first.
Course modules at a glance
Module A – Foundations (Sections 0–1)
Mathematical background for a rigorous HMM course:
- Probability spaces, conditional expectations, basic measure-theoretic language.
- Stochastic matrices, Perron–Frobenius theory, spectral gap and mixing.
- Finite-state Markov chains: ergodicity, stationary laws, reversibility, and non-homogeneous chains.
Module B – Building HMMs (Sections 2–3)
Construction of HMMs as probabilistic graphical models:
- Conditional independence structure and joint factorization of \(S_{1:T}, Y_{1:T}\).
- Observation models: discrete, continuous, and exponential-family emissions.
- Formal HMM definition \((\boldsymbol{\delta}, \boldsymbol{\Gamma}, f_1,\dots,f_K)\) and likelihood in matrix form.
Module C – Inference Algorithms (Section 4)
Core algorithms for posterior computation and decoding:
- Forward filtering and numerically stable log / scaled implementations.
- Forward–backward smoothing and pairwise state probabilities.
- Viterbi decoding, dynamic programming optimality, and max-product semiring.
Module D – Parameter Estimation & Identifiability (Section 5)
How to fit HMMs from data:
- Maximum likelihood estimation and non-convex log-likelihood geometry.
- EM / Baum–Welch as coordinate ascent on an evidence lower bound.
- Identifiability up to label switching and structural pathologies.
Module E – Statistical Theory & Advanced Models (Sections 6–9)
Asymptotics and extensions beyond basic finite-state HMMs:
- Consistency and asymptotic normality of the MLE; Fisher information for dependent data.
- Continuous-state / state-space models and the Kalman filter as an HMM.
- Nonparametric and infinite-state HMMs; online and decision-theoretic perspectives.
Module F – Applications & Proof-Based Problems (Sections 10–11)
Connecting theory to practice and consolidating understanding:
- Applications in speech recognition, bioinformatics, finance, epidemiology, and more.
- Proof-based problem sets covering Markov chains, inference algorithms, EM, identifiability, and asymptotics.
Get Started with HMMs Today!
Start with foundations View full curriculum
Use the sidebar to jump directly to individual sections (0–11), or read them linearly as a graduate course. Mathematics is rendered directly in the browser, and all derivations are written to be compatible with the notation in Zucchini, MacDonald & Langrock and the more theoretical treatments of Capp'e–Moulines–Ryd'en and Douc–Moulines–Stoffer.