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Markov Chain
Discrete-time stochastic process where the probability of the future state depends only on the present state and not on past states (Markov property).
Hidden States
Random variables not directly observable from the system that evolve according to a Markov chain and generate the visible observations.
Transition Probabilities
Matrix defining the probabilities of transitioning from one hidden state to another at each moment, characterizing the system's dynamics.
Emission Probabilities
Conditional probability distribution that associates with each hidden state the probability of generating each possible observation.
Viterbi Algorithm
Dynamic programming algorithm that finds the most likely sequence of hidden states that generated a given observation sequence.
Baum-Welch Algorithm
Variant of the EM algorithm for estimating the parameters of an HMM from unlabeled observation sequences.
Observation Sequence
Ordered set of observable data produced by the system, serving as input for inference in HMMs.
Emission Matrix
Matrix where each element b(j,k) represents the probability of emitting symbol k from hidden state j.
Initial distribution
Probability vector defining the distribution of the hidden state at time t=0 before any observation.
Decoding problem
Fundamental question of finding the most likely sequence of hidden states that generated a given observation sequence.
Evaluation problem
Calculation of the probability that a given HMM model generated a specific observation sequence.
Learning problem
Automatic adjustment of HMM parameters (transition and emission probabilities) from training data.
Discrete HMM
Hidden Markov model where observations come from a finite set of discrete symbols with discrete emission probabilities.
Continuous HMM
HMM variant where observations are continuous variables, typically modeled by Gaussian mixtures.
HMM topology
Structure of allowed connections between hidden states, determining the possible transitions in the model.
Absorbing states
Special states in an HMM with a transition probability of 1 to themselves, preventing any exit once reached.
Parameter smoothing
Technique of adding pseudo-counts to avoid zero probabilities during HMM parameter estimation.
Posterior marginal
Posterior probability of a hidden state at a given time, calculated by marginalizing over all other hidden states.