Concepts in Monte Carlo sampling

TL;DR

This paper explores advanced Monte Carlo algorithms using a one-dimensional anharmonic oscillator as a simplified model, highlighting connections to molecular dynamics, non-reversible Markov chains, and practical sampling techniques.

Contribution

It introduces and explains modern Monte Carlo methods, including lifted non-reversible chains and thinning, with clear examples and Python code, in a self-contained manner.

Findings

Connections between Monte Carlo and molecular dynamics

Implementation of Metropolis and factorized algorithms

Illustration of thinning with bounding potentials

Abstract

We discuss modern ideas in Monte Carlo algorithms in the simplified setting of the one-dimensional anharmonic oscillator. After reviewing the connection between molecular dynamics and Monte Carlo, we introduce to the Metropolis and the factorized Metropolis algorithms and to lifted non-reversible Markov chains. We furthermore illustrate the concept of thinning, where moves are accepted by simple bounding potentials rather than, in our case, the harmonic and quartic constituents of the anharmonic oscillator. We point out the multiple connections of our example algorithms with real-world sampling problems. The paper is fully self-contained and Python implementations are provided.