Table 1.

Glossary

Term	Definition	Synonyms
Conditional independence property	Assumption made for the state‐dependent process: conditional on the state at time t, the observation at time t is independent of all other observations and states
Forward algorithm	Recursive scheme for updating the likelihood and state probabilities of an HMM through time	Filtering
Forward–backward algorithm	Recursive scheme for calculating state probabilities for any point in time: $Pr(S_t=i|x_1,\dots ,x_T)$	Local state decoding; smoothing
Hidden Markov model (HMM)	A special class of state‐space model with a finite number of hidden states that typically assumes some form of the Markov property and the conditional independence property	Dependent mixture model; latent Markov model; Markov‐switching model; regime‐switching model; state‐switching model; multi‐state model
Initial distribution $\left(\mathbf{\delta }\right)$	The probability of being in any of the $N$ states at the start of the sequence: $\mathbf{\delta }=\left(Pr\left(S_1=1\right),\dots ,Pr\left(S_1=N\right)\right)$	Initial probabilities; prior probabilities
Markov property	Assumption made for the state process: $Pr(S_{t+1}|S_t,S_{t‐1},\dots )=Pr(S_{t+1}|S_t)$ (‘conditional on the present, the future is independent of the past’)	Memoryless property
Sojourn time	The amount of time spent in a state before switching to another state	Dwell time; occupancy time
State process $\left(S_t\right)$	Unobserved, serially correlated sequence of states describing how the system evolves over time: $S_t\in \left\{1,\dots ,N\right\}$ for $t=1,\dots ,T$	Hidden/latent process; system process
State transition probability $\left({\mathrm{\gamma }}_{\mathit{ij}}\right)$	The probability of switching from state $i$ at time $t$ to state $j$ at time $t+1$, ${\mathrm{\gamma }}_{\mathit{ij}}=Pr(S_{t+1}=j|S_t=i)$, usually represented as an $N\times N$ transition probability matrix $\left(\mathbf{\Gamma }\right)$
State‐dependent distribution $(f(x_t|S_t=i))$	Probability distribution of an observation $x_t$ conditional on a particular state being active at time $t$, usually from some parametric class (e.g. categorical, Poisson, normal) and represented as an $N\times N$ diagonal matrix $\mathbf{P}\left(x_t\right)=\text{diag}(f(x_t|S_t=1),\dots ,f(x_t|S_t=N))$	Emission distribution; measurement model; observation distribution; output distribution; response distribution
State‐dependent process $\left(X_t\right)$	The observed process within an HMM, which is assumed to be driven by the underlying unobserved state process	Observation process
State‐space model	A conditionally specified hierarchical model consisting of two linked stochastic processes, a latent system process model and an observation process model
Viterbi algorithm	Recursive scheme for finding the sequence of states which is most likely to have given rise to the observed sequence	Global state decoding