Markov Chains and Monte Carlo Methods

Master markov chains and monte carlo methods with applications in probability and combinatorics.

27 min read
Advanced

Introduction

Learning Objectives:

  • Understand Markov chains
  • Compute stationary distributions
  • Apply MCMC methods

Markov Chain

Sequence where: P(Xn+1=j∣X0,...,Xn)=P(Xn+1=j∣Xn)P(X_{n+1} = j | X_0, ..., X_n) = P(X_{n+1} = j | X_n)

Stationary distribution pipi: piP=pipi P = pi (eigenvector of transition matrix)

Applications

Apply these concepts to solve real-world problems in probability and statistics.

python
import numpy as np
import matplotlib.pyplot as plt

# Example implementation
print("Apply concepts from Markov Chains and Monte Carlo Methods")

Key Takeaways

Master these advanced concepts to complete your probability and combinatorics journey!

🎲

Combinatorics and Probability: From Counting to Inference

Master the mathematical foundations of counting, probability, and randomness. This comprehensive course bridges pure combinatorics with computational probability, covering everything from basic counting principles to advanced probabilistic methods, limit theorems, and real-world applications in algorithms, finance, and data science.