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Large deviations for Independent Metropolis Hastings and Metropolis-adjusted Langevin algorithm

Federica Milinanni, Pierre Nyquist

TL;DR

The paper addresses the problem of establishing large deviation principles for the empirical measures of Metropolis-Hastings-based samplers on continuous state spaces, focusing on IMH and MALA and clarifying limitations for RWM. The authors leverage a Lyapunov-function–based framework from prior work to derive LDPs with speed $n$ and rate function $I( u) = \inf_{\gamma\in A(\nu)} R(\gamma \parallel \nu \otimes K)$, proving precise conditions under which IMH and MALA admit LDPs. They show that IMH admits an LDP when the target and proposal tails satisfy specific relations (e.g., $\alpha=\beta$ with $\eta>\gamma$ or $\alpha>\beta$), and that MALA admits an LDP in the regimes $\beta=2$, $\varepsilon\gamma<2$ or $1<\beta<2$, with explicit Lyapunov functions; conversely, for RWM no Lyapunov function of the required form exists, underscoring the limits of the current framework. These results mark the first LDPs for concrete MH dynamics on uncountable spaces and illuminate the link between geometric ergodicity and large deviations, highlighting directions for extending LDP theory to broader MH schemes.

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.

Large deviations for Independent Metropolis Hastings and Metropolis-adjusted Langevin algorithm

TL;DR

The paper addresses the problem of establishing large deviation principles for the empirical measures of Metropolis-Hastings-based samplers on continuous state spaces, focusing on IMH and MALA and clarifying limitations for RWM. The authors leverage a Lyapunov-function–based framework from prior work to derive LDPs with speed and rate function , proving precise conditions under which IMH and MALA admit LDPs. They show that IMH admits an LDP when the target and proposal tails satisfy specific relations (e.g., with or ), and that MALA admits an LDP in the regimes , or , with explicit Lyapunov functions; conversely, for RWM no Lyapunov function of the required form exists, underscoring the limits of the current framework. These results mark the first LDPs for concrete MH dynamics on uncountable spaces and illuminate the link between geometric ergodicity and large deviations, highlighting directions for extending LDP theory to broader MH schemes.

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.
Paper Structure (12 sections, 15 theorems, 114 equations, 1 table)

This paper contains 12 sections, 15 theorems, 114 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions
  4. Convergence and ergodicity of Markov chains
  5. The Metropolis-Hastings algorithm
  6. The Metropolis-Hastings transition kernel.
  7. Large deviation principle for the empirical measures of MH chains
  8. An equivalent asymptotic formulation of (A.3)
  9. Independent Metropolis-Hastings algorithm
  10. Metropolis-adjusted Langevin algorithm
  11. Random Walk Metropolis algorithm
  12. LDP for MH chains, Lyapunov condition and geometric ergodicity: Summary and discussion

Key Result

Theorem 1

Let $\{X_i\}_{i\ge 0}$ be the MH chain from Section sec:MH and $K(x,dy)$ the associated MH kernel. Let $\{L^n\}_{n\ge 1}\subset\mathcal{P}(S)$ be the corresponding sequence of empirical measures. Under Assumptions ass:targetAbsContLambda-ass:compactSpaceOrLyapunov, $\{L^n\}_{n\ge 1}$ satisfies an LD

Theorems & Definitions (28)

  • Definition 1
  • Theorem 1: Theorem 4.1 in milinanni2024large
  • Lemma 1
  • proof : Proof of Lemma \ref{['lem:rto0']}
  • Theorem 2
  • Proposition 1
  • proof : Proof of Proposition \ref{['prop:LyapunovForIMH']}
  • Theorem 3
  • Proposition 2
  • Lemma 2
  • ...and 18 more