Conditionally Gaussian Hypermodels for Cerebral Source Localization

Abstract

Bayesian modeling and analysis of the magnetoencephalography and electroencephalography modalities provide a flexible framework for introducing prior information complementary to the measured data. This prior information is often qualitative in nature, making the translation of the available information into a computational model a challenging task. We propose a generalized gamma family of hyperpriors which allows the impressed currents to be focal and we advocate a fast and efficient iterative algorithm, the iterative alternating sequential algorithm for computing maximum a posteriori (MAP) estimates. Furthermore, we show that for particular choices of the scalar parameters specifying the hyperprior, the algorithm effectively approximates popular regularization strategies such as the minimum current estimate and the minimum support estimate. The connection between priorconditioning and adaptive regularization methods is also pointed out. The posterior densities are explored by means of a Markov chain Monte Carlo strategy suitable for this family of hypermodels. The computed experiments suggest that the known preference of regularization methods for superficial sources over deep sources is a property of the MAP estimators only, and that estimation of the posterior mean in the hierarchical model is better adapted for localizing deep sources.

Sparse source travel-time tomography of a laboratory target: accuracy and robustness of anomaly detection

“…An inversion estimate can then be produced, for instance, through a robust sampling-based (e.g. Markov chain Monte Carlo) method [53,41,1]. Such an inversion procedure can be interesting, for instance, in ultrasonic detection and classification of breast lesions [28,29,30,31,32].…”

“…m = 20, and since x (i) and z (i) can be obtained by optimizing a quadratic function [41] in a straightforward manner. When β = 1.5, the gamma and inverse gamma hyperprior can be shown to result in L 1 -and minimum support type estimates (g) and (ig) of …”

“…The maximizer of the posterior was estimated via the following iterative alternating sequential (IAS) maximum a posteriori (MAP) algorithm [5,41,42,43]:…”

“…We utilized a statistical inversion approach in which a subjective posterior probability distribution was maximized via the iterative alternating sequential (IAS) algorithm [40,5,41,42,43]. A conditionally Gaussian prior was used with variance either a fixed (f) or distributed according to the gamma (g) or inverse gamma (ig) hyperprior density.…”

Electromagnetic 3D subsurface imaging with source sparsity for a synthetic object

“…In addition to (g) and (ig), Dirac's delta distribution of the form p(z) = δ θ 0 (z) was used as a hyperprior (f), resulting in a fixed prior variance given by θ 0 . A reconstruction was produced by maximizing the posterior via the following iterative alternating sequential (IAS) maximum a posteriori (MAP) algorithm [4,35,36,37]:…”