Appendix A — Notation – Hidden Markov Models

A.1 Model

Model notation
Symbol	Meaning	Introduced
\(E = \{1,\dots,K\}\)	hidden state space	Definition 3.2
\(K\)	number of states, or components in the model	Definition 3.2
\(K_+\)	number of states occupied by the data	Chapter 11
\(\mathcal{Y}\)	observation space, with dominating measure \(\nu\)	Definition 3.2
\(S_t\)	hidden state at time \(t\)	Definition 2.1
\(Y_t\)	observation at time \(t\)	Definition 3.2
\(Y_{s:t}\)	the block \((Y_s, Y_{s+1}, \dots, Y_t)\)	Chapter 1
\(T\)	length of the observed series	Definition 3.3
\(\boldsymbol{\delta}\)	initial distribution, \(\delta_i = \mathbb{P}(S_1 = i)\)	Definition 3.2
\(\boldsymbol{\Gamma}\)	transition matrix, \(\gamma_{ij} = \mathbb{P}(S_{t+1}=j \mid S_t = i)\)	Definition 3.2
\(\boldsymbol{\pi}\)	stationary distribution, \(\boldsymbol{\pi}^\top\boldsymbol{\Gamma} = \boldsymbol{\pi}^\top\)	Definition 2.2
\(f_i(\cdot)\), \(F_i\)	emission density and distribution for state \(i\)	Definition 3.2
\(\phi_i\)	emission parameter for state \(i\)	Definition 3.2
\(\mathbf{Q}(y)\)	\(\operatorname{diag}(f_1(y),\dots,f_K(y))\)	Definition 3.4
\(\theta\)	full parameter \((\boldsymbol{\delta},\boldsymbol{\Gamma},\phi_{1:K})\)	Definition 3.2
\(\Theta\)	parameter space	Chapter 3

A.2 Inference

Inference notation
Symbol	Meaning	Introduced
\(\tilde\alpha_t(i)\)	unnormalised forward variable, \(p(S_t=i, y_{1:t})\)	Definition 5.1
\(\alpha_t(i)\), \(\hat\alpha_t(i)\)	filtering probability \(\mathbb{P}(S_t = i \mid y_{1:t})\)	Definition 5.1
\(\beta_t(i)\)	backward variable, \(p(y_{t+1:T} \mid S_t = i)\)	Definition 5.2
\(c_t\)	scale factor, equal to \(p(y_t \mid y_{1:t-1})\)	Proposition 5.1
\(\gamma_t(i)\)	smoothing probability \(\mathbb{P}(S_t = i \mid y_{1:T})\)	Theorem 5.4
\(\xi_t(i,j)\)	pairwise smoothing probability	Proposition 5.2
\(m_t(j)\), \(\psi_t(j)\)	Viterbi value and backpointer	Theorem 5.5
\(L_T(\theta)\), \(\ell_T(\theta)\)	likelihood and log-likelihood	Definition 3.3
\(Q(\theta \mid \theta')\)	EM objective	Definition 6.2
\(\mathcal{F}(q,\theta)\)	evidence lower bound	Chapter 6
\(U_T(\theta)\), \(I(\theta)\)	score and Fisher information per observation	Definition 8.1

A.3 Bayesian and hierarchical

Bayesian notation
Symbol	Meaning	Introduced
\(e_0\), \(e_{jk}\)	Dirichlet hyperparameters for rows of \(\boldsymbol{\Gamma}\)	Definition 9.1
\(G_0\)	prior for emission parameters	Chapter 9
\(N_{jk}\), \(N_{j\cdot}\)	transition counts and row totals	Definition 9.2
\(\boldsymbol{\eta}\)	mixture weights	Definition 11.1
\(\gamma_K\)	Dirichlet parameter, \(\alpha\) or \(\alpha/K\)	Definition 11.2
\(\mathcal{C}\)	partition of the data into occupied components	Proposition 11.2
\(\boldsymbol{\beta}\)	shared base measure over states	Chapter 10
\(\Sigma_i\), \(\Omega_i\)	random effect covariances	Chapter 10

A.4 General

General notation
Symbol	Meaning
\(\Delta^{K-1}\)	probability simplex in \(\mathbb{R}^K\)
\(\mathbf{1}\)	column vector of ones
\(\lVert\cdot\rVert_{\mathrm{TV}}\)	total variation distance, \(\tfrac12\sum_i \lvert\mu_i-\nu_i\rvert\)
\(\mathrm{KL}(p \Vert q)\)	Kullback–Leibler divergence
\(H(\cdot)\)	Shannon entropy
\(\kappa(\boldsymbol{\Gamma})\)	Dobrushin contraction coefficient
\(\varrho\)	absolute spectral gap, \(1 - \lvert\lambda_2\rvert\)
\(\perp\!\!\!\perp\)	conditional independence
\(\mathrm{Dir}_K(\cdot)\)	Dirichlet distribution on \(\Delta^{K-1}\)
\(\mathfrak{S}_K\)	symmetric group on \(K\) elements

A.5 Conventions

Row vectors multiply transition matrices from the left, so that \(\boldsymbol{\mu}^\top\boldsymbol{\Gamma}\) is the distribution one step on. Densities are taken with respect to a fixed dominating measure, so the same formulas cover discrete and continuous observations. Indices \(i, j, k\) run over states, \(t\) over time, \(n\) over series or particles, and \(m\) over MCMC iterations.

A superscript star, \(\theta^\star\), denotes a true parameter; a circle, \(\theta^\circ\), a pseudo-true one under misspecification; a hat, \(\hat\theta\), an estimator.