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52256 results in Statistics and Probability

Page 19 of 2613

  • 3 - Stochastic Gradient MCMC Algorithms

  • 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 62-102

  • Summary

    This chapter introduces stochastic gradient MCMC (SG-MCMC) algorithms, designed to scale Bayesian inference to large datasets. Beginning with the unadjusted Langevin algorithm (ULA), it extends to more sophisticated methods such as stochastic gradient Langevin dynamics (SGLD). The chapter emphasises controlling the stochasticity in gradient estimators and explores the role of control variates in reducing variance. Convergence properties of SG-MCMC methods are analysed, with experiments demonstrating their performance in logistic regression and Bayesian neural networks. It concludes by outlining a general framework for SG-MCMC and offering practical guidance for efficient, scalable Bayesian learning.

  • Cutoff for the logistic SIS epidemic model with self-infection

  • Part of
  • Roxanne He, Malwina Luczak, Nathan Ross
  • Journal:
    Advances in Applied Probability , First View
    Published online by Cambridge University Press:
    04 June 2025, pp. 1-28

  • We study a variant of the classical Markovian logistic SIS epidemic model on a complete graph, which has the additional feature that healthy individuals can become infected without contacting an infected member of the population. This additional ‘self-infection’ is used to model situations where there is an unknown source of infection or an external disease reservoir, such as an animal carrier population. In contrast to the classical logistic SIS epidemic model, the version with self-infection has a non-degenerate stationary distribution, and we derive precise asymptotics for the time to converge to stationarity (mixing time) as the population size becomes large. It turns out that the chain exhibits the cutoff phenomenon, which is a sharp transition in time from one to zero of the total variation distance to stationarity. We obtain the exact leading constant for the cutoff time and show that the window size is of constant (optimal) order. While this result is interesting in its own right, an additional contribution of this work is that the proof illustrates a recently formalised methodology of Barbour, Brightwell and Luczak (2022), ‘Long-term concentration of measure and cut-off’, Stochastic Processes and their Applications 152, 378–423, which can be used to show cutoff via a combination of concentration-of-measure inequalities for the trajectory of the chain and coupling techniques.

Page 19 of 2613