Markov Models

Markov Models: Predicting the Future based on the Present

Markov models are stochastic models used to predict the future behavior of systems by looking at their current state. They rely on the Markov property, which states that the probability of the next state depends only on the current state, not on the system's previous history.

Think of it like walking a dog on a leash. The direction you take next (turn left, go straight, etc.) only depends on where you and the dog are right now, not where you've been before.

Here are some key features of Markov models:

• States and Transitions: The system is represented by a set of possible states (sunny/rainy, healthy/sick, product A/product B) and transitions between those states (sunny transitions to rainy with a certain probability).
• Probability Matrix: The probabilities of transitions between states are captured in a matrix, called the transition matrix. This matrix allows you to calculate the likelihood of reaching any future state from the current one.

Types of Markov Models: There are different types of Markov models, depending on the complexity of the system being modeled.

• Discrete-time Markov models: Transitions happen at defined intervals (days, hours, etc.).
• Continuous-time Markov models: Transitions can happen at any time.
• Hidden Markov models: The actual states are hidden, and you only observe their outcomes (coughing, buying product A).

Applications of Markov Models:
Markov models are used in various fields for prediction and analysis:

• Finance: Forecasting stock prices, predicting customer churn in banks.
• Weather prediction: Modeling weather patterns, predicting rain/snow.
• Bioinformatics: Analyzing gene sequences, identifying protein structure.
• Natural language processing: Predicting the next word in a sentence, machine translation.
• Robot navigation: Planning robot movement paths, avoiding obstacles.

Learning more about Markov models:

If you'd like to explore further, here are some resources:Wikipedia article on Markov models: https://en.wikipedia.org/wiki/Markov_model
MIT OpenCourseware: Introduction to Probability and Statistics: https://ocw.mit.edu/courses/18-05-introduction-to-probability-and-statistics-spring-2014/
Interactive Markov model simulator: https://www.markovlab.ai/