MarkovModells

A Markov model is a stochastic process that satisfies the Markov property: the conditional distribution of future states depends only on the present state and not on the past. This memorylessness applies once the current state is known, meaning the history beyond the current state provides no additional information about the future.

In discrete-time Markov chains, time advances in steps and the process transitions between states in a way described by a transition matrix P, where P(i, j) is the probability of moving from state i to state j in one step. The state space can be finite or countably infinite, and an initial distribution specifies the starting state. In continuous-time Markov chains, transitions occur asynchronously and are governed by a rate or generator matrix Q, with exponential waiting times between jumps.

Hidden Markov models extend this framework by assuming that the system evolves through hidden states and observable outputs are generated according to emission probabilities. Other extensions include Markov decision processes, which add actions and rewards, and semi-Markov models that allow non-exponential waiting times.

Key tasks include estimating model parameters from data (maximum likelihood or Bayesian methods), decoding the most likely state sequence (Viterbi algorithm), and computing state probabilities with the forward–backward algorithm. For HMMs, the Baum–Welch algorithm is used for parameter estimation.

Applications span natural language processing, speech recognition, genetics and proteomics, finance and economics (for regime-switching models), queueing theory, reliability engineering, and various areas of engineering and operations research. The model’s usefulness rests on the Markov assumption and on the tractability of the state space.

Limitations include possible mis-specification of the state space, non-stationarity in real data, and data requirements for reliable estimation. Variants such as hidden Markov models, Markov decision processes, and continuous-time versus discrete-time formulations broaden the framework for modeling sequential data.