An effective introduction to the Markov Chain Monte Carlo method

TL;DR

This paper provides an intuitive and accessible introduction to the Markov Chain Monte Carlo method, emphasizing core concepts through simple models suitable for advanced undergraduates.

Contribution

It offers a semi-rigorous, conceptual framework for understanding MCMC, covering key ideas like ergodicity and detailed balance with minimal mathematical complexity.

Findings

Clear explanation of MCMC concepts using population dynamics analogy

Accessible to senior undergraduate physics students

Includes comprehensive coverage of core MCMC principles

Abstract

We present an intuitive, conceptual, but semi-rigorous introduction to the celebrated Markov Chain Monte Carlo method using a simple model of population dynamics as our motivation and focusing on a few elementary distributions. Conceptually, the population flow between cities closely resembles the random walk of a single walker in a state space. We start from two states, then three states, and finally the setup is fully generalized to many states of both discrete and continuous distributions. Despite the mathematical simplicity, the setup remarkably includes all the essential concepts of Markov Chain Monte Carlo without loss of generality, e.g., ergodicity, global balance and detailed balance, proposal or selection probability, acceptance probability, up to the underlying stochastic matrix, and error analysis. Our teaching experience suggests that most senior undergraduate students in physics can closely follow these materials without much difficulty.