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52379 results in Statistics and Probability

Page 25 of 2619

  • 1 - Background

  • Paul Fearnhead, Lancaster University, Christopher Nemeth, Newcastle University, Chris J. Oates, University of Newcastle upon Tyne, Chris Sherlock, Lancaster University
  • Book:
    Scalable Monte Carlo for Bayesian Learning
    Published online:
    16 May 2025
    Print publication:
    05 June 2025, pp 1-38
    • Chapter
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  • Summary

    This chapter provides a comprehensive overview of the foundational concepts essential for scalable Bayesian learning and Monte Carlo methods. It introduces Monte Carlo integration and its relevance to Bayesian statistics, focusing on techniques such as importance sampling and control variates. The chapter outlines key applications, including logistic regression, Bayesian matrix factorization, and Bayesian neural networks, which serve as illustrative examples throughout the book. It also offers a primer on Markov chains and stochastic differential equations, which are critical for understanding the advanced methods discussed in later chapters. Additionally, the chapter introduces kernel methods in preparation for their application in scalable Markov Chain Monte Carlo (MCMC) diagnostics.

  • 5 - Continuous-Time MCMC

  • Paul Fearnhead, Lancaster University, Christopher Nemeth, Newcastle University, Chris J. Oates, University of Newcastle upon Tyne, Chris Sherlock, Lancaster University
  • Book:
    Scalable Monte Carlo for Bayesian Learning
    Published online:
    16 May 2025
    Print publication:
    05 June 2025, pp 123-188

  • Summary

    This chapter focuses on continuous-time MCMC algorithms, particularly those based on piecewise deterministic Markov processes (PDMPs). It introduces PDMPs as a scalable alternative to traditional MCMC, with a detailed explanation of their simulation, invariant distribution, and limiting processes. Various continuous-time samplers, including the bouncy particle sampler and zig-zag process, are compared in terms of efficiency and performance. The chapter also addresses practical aspects of simulating PDMPs, including techniques for exploiting model sparsity and data subsampling. Extensions to these methods, such as handling discontinuous target distributions or distributions defined on spaces of different dimensions, are discussed.

  • 6 - Assessing and Improving MCMC

  • Paul Fearnhead, Lancaster University, Christopher Nemeth, Newcastle University, Chris J. Oates, University of Newcastle upon Tyne, Chris Sherlock, Lancaster University
  • Book:
    Scalable Monte Carlo for Bayesian Learning
    Published online:
    16 May 2025
    Print publication:
    05 June 2025, pp 189-224

  • Summary

    The development of more sophisticated and, especially, approximate sampling algorithms aimed at improving scalability in one or more of the senses already discussed in this book raises important considerations about how a suitable algorithm should be selected for a given task, how its tuning parameters should be determined, and how its convergence should be as- sessed. This chapter presents recent solutions to the above problems, whose starting point is to derive explicit upper bounds on an appropriate distance between the posterior and the approximation produced by MCMC. Further, we explain how these same tools can be adapted to provide powerful post-processing methods that can be used retrospectively to improve approximations produced using scalable MCMC.

Page 25 of 2619