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Solving Bayesian inverse problems via Fisher adaptive Metropolis adjusted Langevin algorithm

Li-Li Wang, Guang-Hui Zheng

TL;DR

The paper tackles efficient sampling for Bayesian inverse problems by introducing Fisher adaptive MALA, which uses the inverse Fisher information matrix as the optimal preconditioner in Langevin-based proposals. It provides a convergence analysis showing the online Fisher information estimate converges at a rate of $\mathcal{O}(1/n)$ with a learning rate $\gamma_n=1/n$, and demonstrates the method on linear and nonlinear inverse problems. Across three numerical experiments, FisherMALA significantly outperforms standard AdaMALA and pCN, achieving higher effective sample sizes and faster convergence, especially in high-dimensional settings. The work integrates stochastic approximation with gradient-based MCMC to leverage posterior geometry, and suggests future extensions to adjoint-based derivatives and richer priors.

Abstract

The preconditioned Metropolis adjusted Langevin algorithm (MALA) is a widely used method in statistical applications, where the choice of the preconditioning matrix plays a critical role. Recently, Titsias \cite{Titsias2024} demonstrated that the inverse Fisher information matrix is the optimal preconditioner by minimizing the expected squared jump distance and proposed an adaptive scheme to estimate the Fisher matrix using the sampling history. In this paper, we apply the Fisher adaptive Metropolis adjusted Langevin algorithm (MALA) to Bayesian inverse problems. Moreover, we provide a rigorous convergence rate analysis for the adaptive scheme used to estimate the Fisher matrix. To evaluate its performance, we use this algorithm to sample from posterior distributions in several Bayesian inverse problems. And compare its constructions with the standard adaptive Metropolis adjusted Langevin algorithm (which employs the empirical covariance matrix of the posterior distribution as the preconditioner) and the preconditioned Crank-Nicolson (pCN) algorithm. Our numerical results demonstrate show that the Fisher adaptive MALA is highly effective for Bayesian inversion, and significantly outperforms other sampling methods, particularly in high-dimensional settings.

Solving Bayesian inverse problems via Fisher adaptive Metropolis adjusted Langevin algorithm

TL;DR

The paper tackles efficient sampling for Bayesian inverse problems by introducing Fisher adaptive MALA, which uses the inverse Fisher information matrix as the optimal preconditioner in Langevin-based proposals. It provides a convergence analysis showing the online Fisher information estimate converges at a rate of with a learning rate , and demonstrates the method on linear and nonlinear inverse problems. Across three numerical experiments, FisherMALA significantly outperforms standard AdaMALA and pCN, achieving higher effective sample sizes and faster convergence, especially in high-dimensional settings. The work integrates stochastic approximation with gradient-based MCMC to leverage posterior geometry, and suggests future extensions to adjoint-based derivatives and richer priors.

Abstract

The preconditioned Metropolis adjusted Langevin algorithm (MALA) is a widely used method in statistical applications, where the choice of the preconditioning matrix plays a critical role. Recently, Titsias \cite{Titsias2024} demonstrated that the inverse Fisher information matrix is the optimal preconditioner by minimizing the expected squared jump distance and proposed an adaptive scheme to estimate the Fisher matrix using the sampling history. In this paper, we apply the Fisher adaptive Metropolis adjusted Langevin algorithm (MALA) to Bayesian inverse problems. Moreover, we provide a rigorous convergence rate analysis for the adaptive scheme used to estimate the Fisher matrix. To evaluate its performance, we use this algorithm to sample from posterior distributions in several Bayesian inverse problems. And compare its constructions with the standard adaptive Metropolis adjusted Langevin algorithm (which employs the empirical covariance matrix of the posterior distribution as the preconditioner) and the preconditioned Crank-Nicolson (pCN) algorithm. Our numerical results demonstrate show that the Fisher adaptive MALA is highly effective for Bayesian inversion, and significantly outperforms other sampling methods, particularly in high-dimensional settings.

Paper Structure

This paper contains 12 sections, 1 theorem, 52 equations, 13 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Fisher Adaptive MALA in Bayesian inversion
  3. Bayesian inversion framework
  4. Fisher adaptive MALA
  5. Numerical Examples
  6. Inverse source problem
  7. The parameter identification problem
  8. An advection-diffusion-reaction (ADR) inverse problem
  9. Conclusion
  10. Acknowledgements
  11. The initialization phase
  12. Autocorrelation function (ACF) and Effective sample size (ESS)

Key Result

Proposition 2.1

Let $\pi(x)$ be the target distribution, and define $s(x)=\nabla\log\pi(x)$ and $x\thicksim \pi(x)$. Denote $s_n=\nabla\log\pi(x_n)$ where $x_n\thicksim \pi(x_n)$. Assume fourth moment of the score function $E[\|s_ns_n^\top\|^2_{F}]$ is bounded. If the learning rates is set as $\gamma_n=1/n$, then t

Figures (13)

  • Figure 3.1: The exact solution and inversion results.
  • Figure 3.2: The inversion results of ten-time sampling after burn-in phase for all sampler.
  • Figure 3.3: The ACF for the point of state space $x_p=0.5$ for unknown function $f(x)$ with $lags=500$.
  • Figure 3.4: Point-wise ESS for unknown function $f(x)$ with $lags=500$.
  • Figure 3.5: The increase curves of ESS with respect to time at fixed $lags=500$.
  • ...and 8 more figures

Theorems & Definitions (2)

  • Proposition 2.1
  • proof