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Solving Poisson's equation for Wasserstein contractive Markov chains

Julian Hofstadler

Abstract

We study Poisson's equation in the context of general state space Markov chains. For chains satisfying a contraction assumption w.r.t. a Wasserstein distance, we show that a solution exists for Lipschitz functions and investigate its regularity properties. If the kernel is additionally reversible we are also able to show that solutions for $L^p$ functions exist. Combining our findings with Doob's inequalities for martingales we derive maximal inequalities for contractive Markov chains. A number of examples is provided to demonstrate the applicability of our results, in particular in the context of Markov chain Monte Carlo methods.

Solving Poisson's equation for Wasserstein contractive Markov chains

Abstract

We study Poisson's equation in the context of general state space Markov chains. For chains satisfying a contraction assumption w.r.t. a Wasserstein distance, we show that a solution exists for Lipschitz functions and investigate its regularity properties. If the kernel is additionally reversible we are also able to show that solutions for functions exist. Combining our findings with Doob's inequalities for martingales we derive maximal inequalities for contractive Markov chains. A number of examples is provided to demonstrate the applicability of our results, in particular in the context of Markov chain Monte Carlo methods.
Paper Structure (15 sections, 21 theorems, 96 equations)

This paper contains 15 sections, 21 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Wasserstein contractive Markov chains
  3. Definitions and notation
  4. Examples
  5. Solving Poission's equation
  6. Solutions for Lipschitz functions
  7. Solutions for $L^p$ functions
  8. General chains with a spectral gap
  9. Contractive chains with spectral gaps
  10. Maximal inequalities
  11. Regularity of the martingale $M_n$
  12. Maximal inequalities for Lipschitz functions
  13. Bounded diameter
  14. Distributions with finite moments
  15. Path-wise stability of No-U-Turn-Samplers

Key Result

Lemma 2.8

Let $(X_n)_{n \in \mathbb{N}}$ be a uniformly ergodic Markov chain, that is for some $C\in (0, \infty)$ and $\kappa<1$. Then, $(X_n)_{n \in \mathbb{N}}$ is Wasserstein contractive for $\mathcal{W}$ based on the trivial metric $d (x,y) = 2\cdot\mathds{1}_{\{x \neq y\}}$.

Theorems & Definitions (54)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 2.5: douc2018markov
  • Example 2.6: Heat bath for the Ising model -- cf. ollivier2009ricci and Joulin2010Curvature
  • Example 2.7: Simple Slice Sampling
  • Lemma 2.8
  • Example 2.9: Independent Metropolis-Hastings
  • Example 2.10: Geodesic Slice Sampling
  • ...and 44 more