A Bayesian approach to quantifying uncertainty in Tikhonov solutions for the inverse problem of electrocardiography

This study aimed to quantify differences in uncertainty in Tikhonov solutions arising from mesh discretization, conductivity, and zeroth, first, and second order Tikhonov (ZOT, FOT, and SOT) solutions for the inverse problem of electrocardiography. We indirectly analyzed levels of uncertainty in Tikhonov solutions through deriving their equivalent Bayesian maximum a posteriori (MAP) estimates, and then performing regularized sampling from the Bayesian posterior distributions to form credible intervals (CIs). We calculated the percentage of the true heart voltages that fell between the 95% CIs. For all noise levels, the 95% mean CIs for FOT and SOT always captured 11% to 42% more of the true heart voltages than ZOT, suggesting that regularization with FOT and SOT may provide a greater level of certainty in reconstructing heart voltages. In summary, we provide a methodology for quantifying uncertainty in Tikhonov solutions, and use it to study different regularization techniques.