monte carlo methods

Introductory teaching resources for Markov chain Monte Carlo (MCMC) methods—particularly Gibbs sampling—lack beginner-friendly, hands-on tools. Method: This paper introduces a reproducible pedagogical framework implemented in R, tightly integrating theoretical exposition with line-by-line executable code. Designed progressively, it covers foundational topics including probabilistic modeling, transformation of random variables, numerical integration, and Gibbs sampler implementation. Contribution/Results: Diverging from conventional lecture notes or opaque software packages, the framework pioneers a “theory–code–visualization” triadic pedagogy, supported by comprehensive case studies and an interactive learning environment. Empirical evaluation demonstrates significant improvements in beginners’ conceptual understanding and practical proficiency regarding sampling mechanisms, convergence diagnostics, and posterior inference. The framework fills a critical gap by providing a lightweight, highly accessible, and pedagogically grounded MCMC teaching tool.

This study addresses the high variance inherent in Markov chain Monte Carlo (MCMC) sampling and the limitation of control variate methods that rely on analytical solutions to the Poisson equation, which restricts their applicability to general target distributions. To overcome these challenges, this work proposes a variance reduction framework based on normalizing flows. By leveraging bijective transformations to map the target distribution into a reference latent space, the authors derive the transformed Markov kernel and an explicit solution to the corresponding Poisson equation. This approach extends exact control variates to arbitrary target distributions and unifies the theoretical frameworks of importance sampling and control variates. Experimental results demonstrate that the proposed method significantly outperforms state-of-the-art samplers and existing control variate techniques on both synthetic and real-world posterior distributions.

Unadjusted Hamiltonian Monte Carlo (HMC) and underdamped Langevin algorithms suffer from asymptotic bias in high-dimensional sampling due to numerical integration errors and lack automatic step-size adaptation. Method: This paper establishes, for the first time, a quantitative relationship between Hamiltonian energy error and asymptotic bias, and proposes the first black-box, provably bounded step-size adaptation scheme that eliminates the need for Metropolis–Hastings (MH) correction. Contribution/Results: The method rigorously controls asymptotic bias within any user-specified tolerance. Theoretical analysis—validated on Gaussian and canonical Bayesian models—confirms strong bias controllability. Empirical evaluation demonstrates several-fold speedup over MH-adjusted samplers in high dimensions, alongside markedly improved stability and overall performance. This breakthrough overcomes a key practical barrier to deploying unadjusted samplers in real-world applications.

Stochastic MALA (sMALA) suffers from posterior drift—deviating from the true Gibbs posterior toward a non-Gibbs target—due to minibatch-induced bias, thereby compromising uncertainty quantification in Bayesian neural networks. Method: We propose a stochastic Metropolis–Hastings algorithm with an explicit, computable correction term that rigorously restores sampling from the original Gibbs posterior. This is the first MH-based stochastic sampler to incorporate an analytically tractable bias correction. Contributions/Results: We establish a PAC-Bayesian theoretical foundation for deep nonparametric regression, proving that the corrected algorithm achieves optimal posterior contraction rates and yields credible sets with guaranteed high coverage probability. Numerical experiments demonstrate that its uncertainty quantification performance matches classical MALA and significantly outperforms uncorrected sMALA, validating both statistical fidelity and practical efficacy.

This work addresses the challenge of simultaneously achieving non-reversibility, low discretization error, and computational efficiency in Piecewise Deterministic Markov Process (PDMP)-based MCMC methods. We propose a numerical approximation framework grounded in operator splitting, yielding both adaptive and non-adaptive samplers—marking the first systematic incorporation of splitting schemes into PDMP design (e.g., Bouncy Particle Sampler, Zig-Zag Sampler), augmented with a non-reversible Metropolis–Hastings correction to eliminate discretization bias. Each iteration requires only a single gradient evaluation while attaining second-order weak convergence. Theoretically, we establish geometric ergodicity criteria and derive an asymptotic expansion of the invariant measure with respect to step size. Experiments on Bayesian inverse problems in imaging and interacting particle systems demonstrate substantial gains in sampling efficiency over state-of-the-art baselines, confirming the method’s rigorous theoretical foundation and practical scalability.

This work proposes a novel framework that integrates diffusion models with physics-informed sequential Monte Carlo (SMC) to generate solutions of partial differential equations (PDEs) that satisfy physical constraints. For the first time, a physics-guided mechanism based on PDE residuals is embedded directly into the stochastic sampling process of the diffusion model, enabling dynamic enforcement of multi-physics and coupled PDE constraints during generation by jointly leveraging observational data and physical laws. Experimental results across multiple benchmark PDE systems and multi-physics coupling problems demonstrate that the proposed method significantly outperforms existing state-of-the-art generative models, achieving lower numerical errors while maintaining high solution fidelity. This approach establishes a new paradigm for scalable, high-accuracy generative PDE solvers.

This work addresses the challenge of parameter estimation and uncertainty quantification for diffusion processes under low-frequency discrete observations by proposing the first martingale posterior method tailored to this setting. The approach integrates diffusion bridge simulation with Bayesian inference, circumventing the need for fine temporal discretization typically required by conventional MCMC algorithms by relying solely on numerically approximated transition densities. Theoretical analysis demonstrates that the time-discretization error of the proposed method is only of order $O(\Delta)$, substantially reducing computational complexity. Empirical results confirm that the algorithm achieves competitive statistical accuracy while offering speedups of several orders of magnitude compared to state-of-the-art MCMC methods.

This study investigates the asymptotic behavior and computational efficiency of Piecewise Deterministic Monte Carlo (PDMC) algorithms—specifically the Bouncy Particle Sampler and the Zig-Zag Process—in complex settings characterized by high dimensionality, anisotropy, and large data volumes. By modeling these algorithms as continuous-time Markov processes and leveraging scaling limit analysis together with functional central limit theorems, the work systematically examines their theoretical properties. The analysis clarifies convergence characteristics and dimension dependence across a range of challenging scenarios, elucidating how anisotropy and dataset size influence algorithmic efficiency. These findings provide a rigorous theoretical foundation to guide the selection and refinement of PDMC methods in practical applications.

This work addresses the challenge of implementing resampling steps in conditional sequential Monte Carlo (CSMC) algorithms, particularly when dealing with complex dependency structures. The authors propose a unified framework that accommodates a wide range of standard and nonstandard resampling schemes—including systematic, adaptive, and chopthin resampling—without requiring strong assumptions such as unbiasedness or exchangeability. By preserving the order of ancestor indices and avoiding random permutations, the framework not only simplifies algorithmic implementation but also significantly broadens the theoretical applicability of CSMC. This approach guarantees the validity of CSMC under substantially weaker conditions than previously required, thereby enhancing its robustness and flexibility in practical applications.

This work addresses the challenges of inefficient sampling and excessive variance in Monte Carlo estimation arising from scale disparities among modes and low-density regions in multimodal distributions. To overcome these issues, the authors propose a modular Markov chain Monte Carlo (MCMC) method that constructs parallel Markov chains confined to subsets of the target space and combines their estimates via weights derived from inter-subset transition probabilities. By innovatively integrating parallel constrained sampling with a principled weighted fusion mechanism, the approach substantially enhances sampling efficiency and reduces estimator variance, yielding more robust multimodal expectation estimates within simulated annealing frameworks. Theoretical justification is provided through central limit theorem–type results, and numerical experiments—including Bayesian sparse regression with spike-and-slab priors—demonstrate the method’s superior performance.