A First Course in Monte Carlo Methods

TL;DR

A First Course in Monte Carlo Methods delivers a concise, self-contained introduction to Monte Carlo methods for high-dimensional problems, unifying transformation, rejection, and Markov-chain approaches under a common framework of computing expectations $\mathcal{I}_f[h] = \int h(x) f(x)\,dx = \mathbb{E}_{X\sim f}[h(X)]$ when $f$ is known up to a normalizing constant. It explains how Bayesian statistics and statistical mechanics motivate sampling from difficult targets, and presents foundational algorithms: inverse-transform and Knothe–Rosenblatt transport, rejection and ABC, classical MC, importance sampling, MH, Gibbs, Langevin (ULA/MALA), annealing/tempering, and Hamiltonian Monte Carlo, with convergence diagnostics and practical guidance. The notes emphasize three principles: deterministic transport maps, probabilistic accept/reject rules, and augmentations via auxiliary variables and gradient-inspired proposals to improve mixing in high dimensions. The text also discusses convergence assessment, variance reduction, and practical structure for teaching and self-guided learning, with bibliographic remarks guiding further research. Overall, the work provides a principled, breadth-first tour of timeless Monte Carlo ideas that scale to complex Bayesian and statistical-physics problems.

Abstract

This is a concise mathematical introduction to Monte Carlo methods, a rich family of algorithms with far-reaching applications in science and engineering. Monte Carlo methods are an exciting subject for mathematical statisticians and computational and applied mathematicians: the design and analysis of modern algorithms are rooted in a broad mathematical toolbox that includes ergodic theory of Markov chains, Hamiltonian dynamical systems, transport maps, stochastic differential equations, information theory, optimization, Riemannian geometry, and gradient flows, among many others. These lecture notes celebrate the breadth of mathematical ideas that have led to tangible advancements in Monte Carlo methods and their applications. To accommodate a diverse audience, the level of mathematical rigor varies from chapter to chapter, giving only an intuitive treatment to the most technically demanding subjects. The aim is not to be comprehensive or encyclopedic, but rather to illustrate some key principles in the design and analysis of Monte Carlo methods through a carefully-crafted choice of topics that emphasizes timeless over timely ideas. Algorithms are presented in a way that is conducive to conceptual understanding and mathematical analysis -- clarity and intuition are favored over state-of-the-art implementations that are harder to comprehend or rely on ad-hoc heuristics. To help readers navigate the expansive landscape of Monte Carlo methods, each algorithm is accompanied by a summary of its pros and cons, and by a discussion of the type of problems for which they are most useful. The presentation is self-contained, and therefore adequate for self-guided learning or as a teaching resource. Each chapter contains a section with bibliographic remarks that will be useful for those interested in conducting research on Monte Carlo methods and their applications.

Figures (25)

• Figure 1. 1: Estimation of $\pi$ with sample size $N=10^4$. Here, $B_{10000} = 7854$ draws fell within the unit circle, leading to an estimate $\widehat{\pi}_{10000} = 3.1416.$
• Figure 2. 2: A strictly increasing c.d.f. and associated inverse c.d.f.
• Figure 2. 3: Inverse transformation method for sampling from an Exponential(1) distribution. Uniform samples (first row) are transformed into samples from the target distribution (second row).
• Figure 2. 4: Rejection sampling for a $\text{Beta}(2,2)$ distribution, see Example \ref{['ex:example']}. The target p.d.f. is given by $f(x) = 6x(1-x) \ \text{for} \ x \in (0,1).$ Proposed samples are obtained by uniformly sampling in the rectangle with base $1$ and height $M.$ We accept those samples that lie below the curve of $f.$ The first coordinate of each accepted sample is distributed according to the target.
• Figure 2. 5: ABC for a Beta-binominal model as described in Example \ref{['ABC-Binomial']}. The results illustrate the trade-off between cost and accuracy in the choice of tolerance.

Theorems & Definitions (85)

• Theorem 2.1: Transport: Uniform to Target
• Theorem 2.2: Transport: Proposal to Target
• Theorem 2.3: Transport via Triangular Maps
• Theorem 2.4: Rejection Sampling Acceptance Rate
• Remark 2.5
• Theorem 3.1: Classical Monte Carlo Error
• Theorem 3.2: Importance Sampling Error
• Theorem 3.3: Importance Sampling Optimal Proposal
• Theorem 3.4: Autonormalized Importance Sampling Error
• Remark 3.5