Classic Examples of Markov Chains

A Markov chain is a mathematical model that describes the evolution of a system's state over time, characterized by its "memoryless" property—the next state depends solely on the current state. This concept was first introduced by Russian mathematician Andrey Markov in 1906 and has since become one of the most important tools in probability theory and statistics.

A classic example is a weather prediction model. Suppose the weather has only two states: "sunny" and "rainy." Based on historical data statistics: If today is sunny, there is a 70% probability that tomorrow will remain sunny and a 30% chance it will turn rainy. If today is rainy, there is a 60% probability that rain will continue tomorrow and a 40% chance it will become sunny.

This simple two-state Markov chain vividly illustrates the concept of state transition probabilities. By constructing a state transition matrix—often implemented in code as a 2D array—we can predict multi-day weather probability distributions or calculate long-term steady-state ratios of sunny/rainy days. For example, in Python, the transition matrix can be represented as `P = [[0.7, 0.3], [0.6, 0.4]]`, where each row sums to 1.

Another famous application is "random text generation." By analyzing writing samples from specific authors and统计词汇转移概率 (e.g., the probability that the next word is "intelligence" when the current word is "artificial"), new text with a similar style can be generated. Early chatbots like Eliza employed this principle using n-gram models. Implementation typically involves building a dictionary mapping each word to possible subsequent words with their probabilities, then sampling from these distributions to generate sequences.

The key to understanding Markov models lies in grasping two elements: "states" and "transition probabilities." This特性 of ignoring historical paths and relying only on the current state makes Markov chains remarkably practical in fields like speech recognition (e.g., Hidden Markov Models for sequence labeling), financial modeling (stock price movements), and bioinformatics (gene sequence analysis). Although they simplify real-world complexity, they often yield satisfactory approximate results through iterative probability calculations using matrix multiplication or simulation algorithms.