  Markov Chain Monte Carlo (MCMC) methods have become a cornerstone of many
modern scientific analyses by providing a straightforward approach to
numerically estimate uncertainties in the parameters of a model using a
sequence of random samples. This article provides a basic introduction to MCMC
methods by establishing a strong conceptual understanding of what problems MCMC
methods are trying to solve, why we want to use them, and how they work in
theory and in practice. To develop these concepts, I outline the foundations of
Bayesian inference, discuss how posterior distributions are used in practice,
explore basic approaches to estimate posterior-based quantities, and derive
their link to Monte Carlo sampling and MCMC. Using a simple toy problem, I then
demonstrate how these concepts can be used to understand the benefits and
drawbacks of various MCMC approaches. Exercises designed to highlight various
concepts are also included throughout the article.
