Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Large deviations for Independent Metropolis Hastings and Metropolis-adjusted Langevin algorithm (2403.08691v3)

Published 13 Mar 2024 in math.PR, math.ST, and stat.TH

Abstract: In this paper, we prove large deviation principles for the empirical measures associated with the Independent Metropolis Hastings (IMH) sampler and the Metropolis-adjusted Langevin Algorithm (MALA). These are the first large deviation results for empirical measures of Markov chains arising from specific Metropolis-Hastings methods on a continuous state space. Moreover, we show that the existing large deviation framework, that we developed in a previous work (Milinanni and Nyquist, 2024), does not cover the Random Walk Metropolis sampler, even in cases when the underlying Markov chain is geometrically ergodic.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (61)
  1. “An introduction to MCMC for machine learning” In Machine Learning 50 Springer, 2003, pp. 5–43
  2. “Comparison of Markov chains via weak Poincaré inequalities with application to pseudo-marginal MCMC” In The Annals of Statistics 50.6 Institute of Mathematical Statistics, 2022, pp. 3592–3618
  3. “Weak Poincaré Inequalities for Markov chains: theory and applications”, 2023 arXiv:2312.11689 [math.PR]
  4. Søren Asmussen and Peter W Glynn “Stochastic simulation: algorithms and analysis” Springer, 2007
  5. Yves F Atchadé and François Perron “On the geometric ergodicity of Metropolis-Hastings algorithms” In Statistics 41.1 Taylor & Francis, 2007, pp. 77–84
  6. Mylene Bédard and Jeffrey S Rosenthal “Optimal scaling of Metropolis algorithms: Heading toward general target distributions” In Canadian Journal of Statistics 36.4 Wiley Online Library, 2008, pp. 483–503
  7. Julian Besag “Comments on “Representations of knowledge in complex systems” by U. Grenander and MI Miller” In Journal of the Royal Statistical Society Series B 56.591-592, 1994, pp. 4
  8. Joris Bierkens “Non-reversible metropolis-hastings” In Statistics and Computing 26.6 Springer, 2016, pp. 1213–1228
  9. Joris Bierkens, Pierre Nyquist and Mikola C Schlottke “Large deviations for the empirical measure of the zig-zag process” In The Annals of Applied Probability 31.6 Institute of Mathematical Statistics, 2021, pp. 2811–2843
  10. Austin Brown and Galin L Jones “Exact convergence analysis for metropolis–hastings independence samplers in Wasserstein distances” In Journal of Applied Probability 61.1 Cambridge University Press, 2024, pp. 33–54
  11. “Analysis and approximation of rare events” In Representations and Weak Convergence Methods. Series Prob. Theory and Stoch. Modelling 94 Springer, 2019
  12. “Large Deviations for the Emprirical Measures of Reflecting Brownian Motion and Related Constrained Processes in R+subscript𝑅R_{+}italic_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT” In Electronic Journal of Probability 8.none Institute of Mathematical StatisticsBernoulli Society, 2003, pp. 1–46
  13. Ole F Christensen, Gareth O Roberts and Jeffrey S Rosenthal “Scaling limits for the transient phase of local Metropolis–Hastings algorithms” In Journal of the Royal Statistical Society Series B: Statistical Methodology 67.2 Oxford University Press, 2005, pp. 253–268
  14. A De Acosta “Moderate deviations for empirical measures of Markov chains: lower bounds” In The Annals of Probability 25.1 Institute of Mathematical Statistics, 1997, pp. 259–284
  15. “Large deviations techniques and applications” Springer Science & Business Media, 2009
  16. Persi Diaconis, Susan Holmes and Radford M Neal “Analysis of a nonreversible Markov chain sampler” In The Annals of Applied Probability 10.3 Institute of Mathematical Statistics, 2000, pp. 726–752
  17. Jim Doll, Paul Dupuis and Pierre Nyquist “A large deviations analysis of certain qualitative properties of parallel tempering and infinite swapping algorithms” In Applied Mathematics & Optimization 78 Springer, 2018, pp. 103–144
  18. Monroe D Donsker and SR Srinivasa Varadhan “Asymptotic evaluation of certain Markov process expectations for large time—III” In Communications on Pure and Applied Mathematics 29.4 Wiley Online Library, 1976, pp. 389–461
  19. Monroe D Donsker and SR Srinivasa Varadhan “Asymptotic evaluation of certain Markov process expectations for large time, I” In Communications on Pure and Applied Mathematics 28.1 Wiley Online Library, 1975, pp. 1–47
  20. Monroe D Donsker and SR Srinivasa Varadhan “Asymptotic evaluation of certain Markov process expectations for large time, II” In Communications on Pure and Applied Mathematics 28.2 Wiley Online Library, 1975, pp. 279–301
  21. Randal Douc, Arnaud Guillin and Eric Moulines “Bounds on regeneration times and limit theorems for subgeometric Markov chains” In Annales de l’IHP Probabilités et statistiques 44.2, 2008, pp. 239–257
  22. “A weak convergence approach to the theory of large deviations” In A weak convergence approach to the theory of large deviations, Wiley series in probability and mathematical statistics New York: Wiley, 1997
  23. “On the large deviation rate function for the empirical measures of reversible jump Markov processes” In The Annals of Probability 43.3 Institute of Mathematical Statistics, 2015, pp. 1121–1156
  24. “Analysis and Optimization of Certain Parallel Monte Carlo Methods in the Low Temperature Limit” In Multiscale Modeling & Simulation 20.1, 2022, pp. 220–249
  25. “On the infinite swapping limit for parallel tempering” In Multiscale Modeling & Simulation 10.3 SIAM, 2012, pp. 986–1022
  26. Jin Feng and Thomas G Kurtz “Large deviations for stochastic processes” American Mathematical Soc., 2006
  27. “The behavior of the spectral gap under growing drift” In Transactions of the American Mathematical Society 362.3, 2010, pp. 1325–1350
  28. “Convergence rates of the Gibbs sampler, the Metropolis algorithm and other single-site updating dynamics” In Journal of the Royal Statistical Society Series B: Statistical Methodology 55.1 Oxford University Press, 1993, pp. 205–219
  29. Andrew Gelman, Walter R Gilks and Gareth O Roberts “Weak convergence and optimal scaling of random walk Metropolis algorithms” In The Annals of Applied Probability 7.1 Institute of Mathematical Statistics, 1997, pp. 110–120
  30. Nathan Glatt-Holtz, Justin Krometis and Cecilia Mondaini “On the accept–reject mechanism for Metropolis–Hastings algorithms” In The Annals of Applied Probability 33.6B Institute of Mathematical Statistics, 2023, pp. 5279–5333
  31. Martin Hairer, Andrew M Stuart and Sebastian J Vollmer “Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions” In The Annals of Applied Probability 24.6 Institute of Mathematical Statistics, 2014, pp. 2455–2490
  32. W Keith Hastings “Monte Carlo sampling methods using Markov chains and their applications” Oxford University Press, 1970
  33. Chii-Ruey Hwang, Shu-Yin Hwang-Ma and Shuenn-Jyi Sheu “Accelerating diffusions”, 2005
  34. Søren Fiig Jarner and Ernst Hansen “Geometric ergodicity of Metropolis algorithms” In Stochastic Processes and their Applications 85.2 Elsevier, 2000, pp. 341–361
  35. Benjamin Jourdain, Tony Lelièvre and Błażej Miasojedow “Optimal scaling for the transient phase of Metropolis Hastings algorithms: The longtime behavior” In Bernoulli 20.4 [Bernoulli Society for Mathematical StatisticsProbability, International Statistical Institute (ISI)], 2014, pp. 1930–1978
  36. “Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes” In Electronic Journal of Probability 10.none Institute of Mathematical StatisticsBernoulli Society, 2005, pp. 61–123
  37. “Spectral theory and limit theorems for geometrically ergodic Markov processes” In The Annals of Applied Probability 13.1 Institute of Mathematical Statistics, 2003, pp. 304–362
  38. Samuel Livingstone “Geometric ergodicity of the random walk Metropolis with position-dependent proposal covariance” In Mathematics 9.4 MDPI, 2021, pp. 341
  39. Jonathan C Mattingly, Natesh S Pillai and Andrew M Stuart “Diffusion limits of the random walk Metropolis algorithm in high dimensions” In The Annals of Applied Probability 22.3 Institute of Mathematical Statistics, 2012, pp. 881–930
  40. Kerrie L Mengersen and Richard L Tweedie “Rates of convergence of the Hastings and Metropolis algorithms” In The Annals of Statistics 24.1 Institute of Mathematical Statistics, 1996, pp. 101–121
  41. “Equation of state calculations by fast computing machines” In The Journal of Chemical Physics 21.6 American Institute of Physics, 1953, pp. 1087–1092
  42. Sean Meyn and Richard L Tweedie “Markov Chains and Stochastic Stability” Cambridge University Press, 2009
  43. “A large deviation principle for the empirical measures of Metropolis–Hastings chains” In Stochastic Processes and their Applications 170 Elsevier, 2024, pp. 104293
  44. “An infinite swapping approach to the rare-event sampling problem.” In The Journal of Chemical Physics 135.13, 2011, pp. 134111
  45. “Weak Poincaré inequality comparisons for ideal and hybrid slice sampling”, 2024 arXiv:2402.13678 [stat.CO]
  46. “Improving the convergence of reversible samplers” In Journal of Statistical Physics 164 Springer, 2016, pp. 472–494
  47. “Irreversible Langevin samplers and variance reduction: a large deviations approach” In Nonlinearity 28.7 IOP Publishing, 2015, pp. 2081
  48. “Variance reduction for irreversible Langevin samplers and diffusion on graphs”, 2015
  49. “Monte Carlo Statistical Methods”, Springer Texts in Statistics New York, NY: Springer New York, 2004
  50. “Geometric ergodicity and hybrid Markov chains” In Electronic Communications in Probability 2, 1997, pp. 13–25
  51. Gareth O Roberts “Linking theory and practice of MCMC” In Oxford Statistical Science Series OXFORD UNIV PRESS, 2003, pp. 145–166
  52. Gareth O Roberts and Jeffrey S Rosenthal “Optimal scaling for various Metropolis-Hastings algorithms” In Statistical Science 16.4 Institute of Mathematical Statistics, 2001, pp. 351–367
  53. Gareth O Roberts and Jeffrey S Rosenthal “Optimal scaling of discrete approximations to Langevin diffusions” In Journal of the Royal Statistical Society: Series B (Statistical Methodology) 60.1 Wiley Online Library, 1998, pp. 255–268
  54. Gareth O Roberts and Jeffrey S Rosenthal “Quantitative Non-Geometric Convergence Bounds for Independence Samplers” In Methodology and Computing in Applied Probability 13.2 Springer, 2011, pp. 391–403
  55. Gareth O Roberts and Richard L Tweedie “Exponential convergence of Langevin distributions and their discrete approximations” In Bernoulli JSTOR, 1996, pp. 341–363
  56. Gareth O Roberts and Richard L Tweedie “Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms” In Biometrika 83.1 Oxford University Press, 1996, pp. 95–110
  57. Jeffrey S Rosenthal “Asymptotic variance and convergence rates of nearly-periodic Markov chain Monte Carlo algorithms” In Journal of the American Statistical Association 98.461 Taylor & Francis, 2003, pp. 169–177
  58. “Convergence of Position-Dependent MALA with Application to Conditional Simulation in GLMMs” In Journal of Computational and Graphical Statistics 32.2 Taylor & Francis, 2023, pp. 501–512
  59. Luke Tierney “A note on Metropolis-Hastings kernels for general state spaces” In The Annals of Applied Probability 8.1 Institute of Mathematical Statistics, 1998, pp. 1–9
  60. Luke Tierney “Markov Chains for Exploring Posterior Distributions” In The Annals of Statistics 22.4 Institute of Mathematical Statistics, 1994, pp. 1701–1728
  61. Guanyang Wang “Exact convergence analysis of the independent Metropolis-Hastings algorithms” In Bernoulli 28.3 Bernoulli Society for Mathematical StatisticsProbability, 2022, pp. 2012–2033
Citations (1)

Summary

We haven't generated a summary for this paper yet.

X Twitter Logo Streamline Icon: https://streamlinehq.com