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Perturbation theory for killed Markov processes and quasi-stationary distributions

Daniel Rudolf, Andi Q. Wang

TL;DR

The paper analyzes the stability of quasi-stationary distributions for killed Markov processes under perturbations of the generator. It develops a Hilbert-space perturbation framework distinguishing bounded self-adjoint perturbations and unbounded truncations of the killing rate, and provides explicit bounds on the change in the bottom eigenfunction and the resulting $\mathcal{L}^1$ stability of the quasi-stationary distributions. Key results include a main bound $\|\varphi-\widehat{\varphi}\| \le \dfrac{\|H\varphi\|}{\nu-2\|H\|}$ for bounded perturbations and an exponential-in-$M$ bound on the eigenfunction difference under killing-rate truncation in an OU setting, together with an $\mathcal{L}^1$ stability bound for the corresponding densities. The results underpin the robustness of quasi-stationary Monte Carlo methods to data perturbations and to practical truncations, offering quantitative guidance for implementing QSMC in high-dimensional Bayesian inference.

Abstract

Motivated by recent developments of quasi-stationary Monte Carlo methods, we investigate the stability of quasi-stationary distributions of killed Markov processes under perturbations of the generator. We first consider a general bounded self-adjoint perturbation operator, and after that, study a particular unbounded perturbation corresponding to truncation of the killing rate. In both scenarios, we quantify the difference between eigenfunctions of the smallest eigenvalue of the perturbed and unperturbed generators in a Hilbert space norm. As a consequence, L1 norm estimates of the difference of the resulting quasi-stationary distributions in terms of the perturbation are provided.

Perturbation theory for killed Markov processes and quasi-stationary distributions

TL;DR

The paper analyzes the stability of quasi-stationary distributions for killed Markov processes under perturbations of the generator. It develops a Hilbert-space perturbation framework distinguishing bounded self-adjoint perturbations and unbounded truncations of the killing rate, and provides explicit bounds on the change in the bottom eigenfunction and the resulting stability of the quasi-stationary distributions. Key results include a main bound for bounded perturbations and an exponential-in- bound on the eigenfunction difference under killing-rate truncation in an OU setting, together with an stability bound for the corresponding densities. The results underpin the robustness of quasi-stationary Monte Carlo methods to data perturbations and to practical truncations, offering quantitative guidance for implementing QSMC in high-dimensional Bayesian inference.

Abstract

Motivated by recent developments of quasi-stationary Monte Carlo methods, we investigate the stability of quasi-stationary distributions of killed Markov processes under perturbations of the generator. We first consider a general bounded self-adjoint perturbation operator, and after that, study a particular unbounded perturbation corresponding to truncation of the killing rate. In both scenarios, we quantify the difference between eigenfunctions of the smallest eigenvalue of the perturbed and unperturbed generators in a Hilbert space norm. As a consequence, L1 norm estimates of the difference of the resulting quasi-stationary distributions in terms of the perturbation are provided.

Paper Structure

This paper contains 19 sections, 25 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related results and literature review
  3. Result for discrete-time Markov chains
  4. Background to quasi-stationary Monte Carlo
  5. Outline
  6. Bounded perturbations
  7. Assumptions and first consequences
  8. Main estimate
  9. Additional auxiliary estimates
  10. Example: logistic regression
  11. Unbounded perturbation: truncation
  12. Projection notation
  13. Further notation and results
  14. Example: unbounded case
  15. Deferred proofs
  16. ...and 4 more sections

Key Result

Proposition 1

Assume that there exist $\alpha,\widetilde{\alpha}\in (0,1)$ and $c_\ell,\widetilde{c}_\ell,c_u,\widetilde{c}_u\colon S_0 \to (0,\infty)$, such that for any $n\in\mathbb{N}$ we have Then where and $\Vert \cdot \Vert_{\text{\rm tv}}$ denotes the total variation distance.

Figures (1)

  • Figure 1: Plots of the function $\lambda\mapsto s_M(\lambda)$, as defined in \ref{['eq:s_M_def']}, for $M=2,10,40$.

Theorems & Definitions (35)

  • Proposition 1
  • Lemma 1: Weyl's lemma
  • Lemma 2
  • Remark 1
  • Theorem 1
  • Remark 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Proposition 2
  • ...and 25 more