Hyperparameter estimation using a resolution matrix for Bayesian sensing

Abstract

Bayesian image processing has progressively increased in importance in various fields of the natural sciences. It utilizes prior knowledge and forward models of the observational processes through Bayes’ theorem, enabling the accurate estimation of model parameters that represent the physical quantities of the target. Moreover, using hyperparameter estimation, we can determine the hidden physical parameters that govern the processes in and the structure of the target and sensing systems, such as the spatial continuity of the model parameters and the magnitude of the observational noise. Such a general framework, which uses Bayesian estimation to understand the essential physics of a target system, can be called ‘Bayesian sensing’. This paper discusses the physical meaning of and the mechanism underlying Bayesian sensing using the concept of resolution in the spatial-inversion problem. The spatial resolution of the model parameters can be mapped using a resolution matrix, more rigorously, a model resolution matrix defined as a linear mapping from the true model parameters to the recovered model parameters. We formulate the resolution matrix for Bayesian image processing and also show that in terms of resolution, the optimal hyperparameters are obtained from internally consistent equations that connect the estimated optimal hyperparameters with the actual ones calculated from the estimated model parameters. In addition, we show the equivalence of the internally consistent equations to the expectation-maximization (EM) algorithm and formulate the confidence intervals for the estimated hyperparameters, which indicate the general convergence of the hyperparameter estimates. We also show the effectiveness of the proposed method by performing synthetic numerical tests for two inversion-problem settings: linear travel-time seismic tomography and image deblurring. The resulting equations can contribute to understanding the hidden physical processes in and the structure of the target and observation systems for various problems.