Markov Chains: Mathematical Modeling of State Transitions with Practical Code Implementation

This article focuses on Markov Chains, a mathematical tool designed to model transition probabilities between states in stochastic systems. Widely applied across numerous domains including natural language processing, finance, and physics, Markov Chains operate on the fundamental principle that the current state depends solely on the immediately preceding state, independent of earlier states (memoryless property). This characteristic makes Markov Chains particularly valuable in diverse applications. From an implementation perspective, Markov Chains can be represented using transition matrices where each element P(i,j) denotes the probability of moving from state i to state j. Code implementations typically involve: 1) Defining state space and transition probabilities, 2) Creating probability matrices using numpy arrays or similar data structures, 3) Implementing state evolution through matrix multiplication. Furthermore, Markov Chains have spawned important variants and extensions such as Hidden Markov Models (HMMs) for dealing with unobserved states and Markov Decision Processes (MDPs) for reinforcement learning scenarios, enhancing their flexibility and power as modeling tools. Key functions in practical implementations often include probability normalization checks, state sequence generation algorithms, and convergence verification methods for steady-state analysis.