# American Institute of Mathematical Sciences

October  2021, 15(5): 1099-1119. doi: 10.3934/ipi.2021030

## Where Bayes tweaks Gauss: Conditionally Gaussian priors for stable multi-dipole estimation

 1 Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy 2 BESA GmbH, Freihamer Str. 18, 82166 Gräfelfing, Germany

* Corresponding author: Alberto Sorrentino

Received  June 2020 Revised  December 2020 Published  October 2021 Early access  May 2021

We present a very simple yet powerful generalization of a previously described model and algorithm for estimation of multiple dipoles from magneto/electro-encephalographic data. Specifically, the generalization consists in the introduction of a log-uniform hyperprior on the standard deviation of a set of conditionally linear/Gaussian variables. We use numerical simulations and an experimental dataset to show that the approximation to the posterior distribution remains extremely stable under a wide range of values of the hyperparameter, virtually removing the dependence on the hyperparameter.

Citation: Alessandro Viani, Gianvittorio Luria, Alberto Sorrentino, Harald Bornfleth. Where Bayes tweaks Gauss: Conditionally Gaussian priors for stable multi-dipole estimation. Inverse Problems & Imaging, 2021, 15 (5) : 1099-1119. doi: 10.3934/ipi.2021030
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An example of simulated EEG (top) and MEG (bottom) recordings
Confusion matrices for the estimated number of dipoles, for three different values of the prior scale factor $k$: in the top panel results obtained with simulated EEG data, in the bottom panel results obtained with simulated MEG data
Boxplots of the OSPA metric, quantifying the distance between the true and estimated dipole configurations, for three different values of the prior scale factor $k$: in red the SESAME results, in blue the h-SESAME results; in the top panel results obtained with simulated EEG data, in the bottom panel results obtained with simulated MEG data
Variance of the posterior probability map with respect to different values of the prior scale factor $k$, as defined in eq. (16): in red the SESAME results, in blue the h-SESAME results
Estimated value of the prior width $\sigma_q$, for different values of the prior scale factor $k$
Experimental MEG data: averaged response to auditory stimuli; data taken from the sample open dataset within the MNE–Python package
with different values of the prior scale factor: $k = 0.1$ (top row), $k = 1$ (middle row) and $k = 10$ (bottom row)">Figure 7.  Posterior probability maps (left) and estimated source time courses (right) obtained by SESAME when applied to the experimental data shown in Figure 6 with different values of the prior scale factor: $k = 0.1$ (top row), $k = 1$ (middle row) and $k = 10$ (bottom row)
with different values of the prior scale factor: $k = 0.1$ (top row), $k = 1$ (middle row) and $k = 10$ (bottom row)">Figure 8.  Posterior probability maps (left) and estimated source time courses (right) obtained by h-SESAME when applied to the experimental data shown in Figure 6 with different values of the prior scale factor: $k = 0.1$ (top row), $k = 1$ (middle row) and $k = 10$ (bottom row)
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