Bayesian Solutions and Performance Analysis in Bioelectric Inverse Problems

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Abstract
In bioelectric inverse problems, one seeks to recover bioelectric sources from remote measurements using a mathematical model that relates the sources to the measurements. Due to attenuation and spatial smoothing in the medium between the sources and the measurements, bioelectric inverse problems are generally ill-posed. Bayesian methodology has received increasing attention recently to combat this ill-posedness, since it offers a general formulation of regularization constraints and additionally provides statistical performance analysis tools. These tools include the estimation error covariance and the marginal probability density of the measurements (known as the "evidence") that allow one to predictively quantify and compare experimental designs. These performance analysis tools have been previously applied in inverse electroencephalography and magnetoencephalography, but only in relatively simple scenarios. The main motivation here was to extend the utility of Bayesian estimation techniques and performance analysis tools in bioelectric inverse problems, with a particular focus on electrocardiography. In a simulation study we first investigated whether Bayesian error covariance, computed without knowledge of the true sources and based on instead statistical assumptions, accurately predicted the actual reconstruction error. Our study showed that error variance was a reasonably reliable qualitative and quantitative predictor of estimation performance even when there was error in the prior model. We also examined whether the evidence statistic accurately predicted relative estimation performance when distinct priors were used. In a simple scenario our results support the hypothesis that the prior model that maximizes the evidence is a good choice for inverse reconstructions.