Markov chain

Markov chains are fundamental in probability theory and have wide applications across various fields. A Markov chain consists of a set of states and transition probabilities between those states. The key characteristic is the 'Markov property,' which states that the probability of transitioning to any particular state depends only on the current state and not on the sequence of events that preceded it. This 'memoryless' property simplifies analysis and modeling. Mathematically, a Markov chain can be represented by a transition matrix, where each element (i, j) represents the probability of transitioning from state i to state j. The sum of probabilities in each row of the transition matrix must equal 1. Markov chains can be discrete-time (where transitions occur at fixed time intervals) or continuous-time (where transitions can occur at any point in time). Applications include speech recognition (modeling sequences of phonemes), finance (modeling stock prices), genetics (modeling gene sequences), and queuing theory (modeling waiting times in systems). In AI, Markov chains are used in reinforcement learning (Markov Decision Processes), natural language processing (Hidden Markov Models), and generative models.