  Mixture models with Gamma and or inverse-Gamma distributed mixture components
are useful for medical image tissue segmentation or as post-hoc models for
regression coefficients obtained from linear regression within a Generalised
Linear Modeling framework (GLM), used in this case to separate stochastic
(Gaussian) noise from some kind of positive or negative "activation" (modeled
as Gamma or inverse-Gamma distributed). To date, the most common choice in this
context it is Gaussian/Gamma mixture models learned through a maximum
likelihood (ML) approach; we recently extended such algorithm for mixture
models with inverse-Gamma components. Here, we introduce a fully analytical
Variational Bayes (VB) learning framework for both Gamma and/or inverse-Gamma
components. We use synthetic and resting state fMRI data to compare the
performance of the ML and VB algorithms in terms of area under the curve and
computational cost. We observed that the ML Gaussian/Gamma model is very
expensive specially when considering high resolution images; furthermore, these
solutions are highly variable and they occasionally can overestimate the
activations severely. The Bayesian Gauss-Gamma is in general the fastest
algorithm but provides too dense solutions. The maximum likelihood
Gaussian/inverse-Gamma is also very fast but provides in general very sparse
solutions. The variational Gaussian/inverse-Gamma mixture model is the most
robust and its cost is acceptable even for high resolution images. Further, the
presented methodology represents an essential building block that can be
directly used in more complex inference tasks, specially designed to analyse
MRI-fMRI data; such models include for example analytical variational mixture
models with adaptive spatial regularization or better source models for new
spatial blind source separation approaches.
