# Markov Chain Monte Carlo on Finite State Spaces

## TL;DR

This paper provides a clear, mathematically rigorous explanation of MCMC methods on finite state spaces, including convergence proofs and fundamental algorithms like Gibbs and Metropolis-Hastings.

## Contribution

It offers a simple, coherent presentation of MCMC theory with proofs of key theorems and discussion of core sampling methods, accessible with basic mathematical knowledge.

## Key findings

- Proves a convergence theorem for finite Markov chains
- Introduces a minimal Perron-Frobenius theorem version
- Discusses fundamental MCMC algorithms: Gibbs and Metropolis-Hastings

## Abstract

We elaborate the idea behind Markov chain Monte Carlo (MCMC) methods in a mathematically coherent, yet simple and understandable way. To this end, we proof a pivotal convergence theorem for finite Markov chains and a minimal version of the Perron-Frobenius theorem. Subsequently, we briefly discuss two fundamental MCMC methods, the Gibbs and Metropolis-Hastings sampler. Only very basic knowledge about matrices, convergence of real sequences and probability theory is required.