EEG source localization based on a structured sparsity prior and a partially collapsed Gibbs sampler

Abstract

In this paper, we propose a hierarchical Bayesian model approximating the ℓ20 mixed-norm regularization by a multivariate Bernoulli Laplace prior to solve the EEG inverse problem by promoting spatial structured sparsity. The posterior distribution of this model is too complex to derive closed-form expressions of the standard Bayesian estimators. An MCMC method is proposed to sample this posterior and estimate the model parameters from the generated samples. The algorithm is based on a partially collapsed Gibbs sampler and a dual dipole random shift proposal for the non-zero positions. The brain activity and all other model parameters are jointly estimated in a completely unsupervised framework. The results obtained on synthetic data with controlled ground truth show the good performance of the proposed method when compared to the ℓ21 approach in different scenarios, and its capacity to estimate pointlike source activity.

A partially collapsed Gibbs sampler with accelerated convergence for EEG source localization

“…We have observed that the standard PCGS developed in [8] gets sometimes stuck around local maxima of the posterior 8. This section studies two kinds of moves which allow the sampler to escape from these local maxima, and thus ensure a faster convergence of the PCGS.…”

“…To estimate the model parameters, we proposed in [8] to draw samples from (8) using a PCGS, sampling z i and x i jointly. The corresponding conditional distributions are summarized in Table I, where GIG, IG and Be are the generalized inverse Gaussian, inverse gamma and beta distributions (see also [8]). Note that X −i denotes the matrix X whose i-th row has been set to zero and that the following notations have been used…”

“…The main contribution of this paper is to study some specific schemes that allow the convergence of the PCGS to be accelerated. For this, we consider a Bayesian model introduced in [8] to solve an EEG ill-posed inverse problem. The model approximates an ℓ 2,0 mixed norm regularization using a multivariate Bernoulli Laplacian prior.…”

Bayesian sparse regularization for parallel MRI reconstruction using complex Bernoulli–Laplace mixture priors

“…The use of this effect has demonstrated its ability to provide flexible and adaptive regularizations for the resolution of ill-posed inverse problems. Generally, the use of a Bernoulli and continuous distributions mixture priors like multivariate Bernoulli-Laplace [24,25], Bernoulli-Gaussian [26], Bernoulli-Exponential [20] and Bernoulli-generalized Gaussian-Laplace [27] allows promoting the desired image sparsity levels directly in the original space. These Bayesian Bernoullibased models enable to estimate the regularization parameters and hyperparameters directly from the observed data, which is not possible with other variational methods [6,13,17,28].…”