# An approximate empirical Bayesian method for large-scale linear-Gaussian inverse problems

## TL;DR

This paper introduces an efficient approximate Bayesian approach for large-scale linear-Gaussian inverse problems, leveraging low-rank and randomized methods to evaluate marginal likelihoods, enabling practical hyperparameter estimation.

## Contribution

It proposes a novel low-rank approximation technique for marginal likelihood evaluation, improving computational feasibility in large-scale Bayesian inverse problems.

## Key findings

- Accurately approximates marginal likelihoods using low-rank methods
- Demonstrates efficiency with randomized SVD and spectral approximations
- Shows good performance in numerical experiments

## Abstract

We study Bayesian inference methods for solving linear inverse problems, focusing on hierarchical formulations where the prior or the likelihood function depend on unspecified hyperparameters. In practice, these hyperparameters are often determined via an empirical Bayesian method that maximizes the marginal likelihood function, i.e., the probability density of the data conditional on the hyperparameters. Evaluating the marginal likelihood, however, is computationally challenging for large-scale problems. In this work, we present a method to approximately evaluate marginal likelihood functions, based on a low-rank approximation of the update from the prior covariance to the posterior covariance. We show that this approximation is optimal in a minimax sense. Moreover, we provide an efficient algorithm to implement the proposed method, based on a combination of the randomized SVD and a spectral approximation method to compute square roots of the prior covariance matrix. Several numerical examples demonstrate good performance of the proposed method.