Convergence Speed and Approximation Accuracy of Numerical MCMC

TL;DR

This work develops two complementary perturbation analyses for numerical MCMC: ergodicity-based and spectral-gap-based frameworks. By introducing a Lyapunov-function driven $d_V$ metric and a χ^2-divergence perspective, the authors derive non-asymptotic bounds showing that small, well-controlled perturbations $\widehat{P}$ of a fast-mixing kernel $P$ yield near-equivalent convergence rates and estimator accuracy; these results extend to common MH samplers such as Random Walk MH and MALA, as well as to parallel tempering, with explicit bounds in terms of the perturbation size $\varepsilon$ and density-approximation errors. Theoretical claims are validated through Bayesian inverse problems in a predator-prey model, demonstrating that discretization-induced perturbations have diminishing impact as the discretization is refined. Overall, the paper provides a cohesive, practical framework to quantify and transfer convergence guarantees from ideal MC schemes to numerically implemented perturbations across several MCMC algorithms.

Abstract

When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how perturbation of MCMC affects the convergence speed and Monte Carlo estimation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has similar convergence speed and high approximation accuracy as well. We discuss two different analysis frameworks: ergodicity and spectral gap, both are widely used in the literature. Our results can be easily extended to obtain non-asymptotic error bounds for MCMC estimators. We also demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis and Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally we present some simple numerical examples to verify our theoretical claims.

Figures (5)

• Figure 1: Left and middle: the trajectories of $\gamma_p(t; \theta_{\rm true})$ and $\gamma_q(t; \theta_{\rm true})$ computed using the second order Runge--Kutta method with different time step size $h$. Right: the $L_2$ error of the model outputs with different time step size $h$. Here $G(\theta_{\rm true})$ is computed using $h = h_0 \times 2^{-6}$. The trajectories computed by the time step size $h = h_0 \times 2^{-6}$ is used to generate synthetic data set. The observed data sets of the prey and predator are shown as circles and squares, respectively.
• Figure 2: Marginal distributions of perturbed posteriors defined by various time step sizes.
• Figure 3: Autocorrelation time of each of the parameter Markov chains simulated by the RWM algorithm. Here different colored lines represent Markov chains targeting invariant measures defined by different time discretization steps.
• Figure 4: Autocorrelation time of each of the parameter Markov chains simulated by the MALA algorithm. Here different colored lines represent Markov chains targeting invariant measures defined by different time discretization steps.
• Figure 5: Autocorrelation time of each of the parameter Markov chains simulated by Algorithm \ref{['alg:re']}. Here different colored lines represent Markov chains targeting invariant measures defined by different time discretization steps.

Theorems & Definitions (32)

• Theorem 2.1: Theorem 3.1 in rudolf2018perturbation
• Theorem 2.2
• Proposition 2.3
• Remark 3.1
• Theorem 3.2
• Proposition 3.3
• Theorem 3.4
• Proposition 3.5
• Proposition 3.6
• Lemma 4.1