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data is always complementary in these projections and comparable to the posterior probability distribution and not the posterior distribution itself. This fact may be of interest in practical situations as the posterior probability is far less spread out than the posterior density.

Discussion
==========

We provide a general statistical methodology for analysis of multi-subject fMRI data. We suggested to analyze fMRI data with a vector autoregressive model from which some series will be regressed out. We provide a solution to decompose a variance into different components depending on the fMRI process in the data, and we use that decomposition in a straightforward way to build a general methodology for analysis of fMRI data.

The main advantage of this approach is that the regression coefficients are interpretable in the sense that they estimate the strength of the statistical association between time series. We can estimate or test relationships between fMRI time series in a single calculation, thus avoiding any type of multi-step estimation procedure. Moreover, by assuming a variance decomposition that includes a statistical term and an anatomical term, the approach provides a model-based statistical inference framework that is strictly connected to variance decomposition. In addition, this decomposition can be easily extended to control for anatomical regression in the approach.

The model described here for multi-subject fMRI is surely not the only good candidate for statistical analysis of fMRI data; a variety of other models exist. A review of these models can be found in e.g. [@WalterSachs10]. The choice of a model can obviously depend on the research question and type of fMRI data. The approaches for variance decomposition that we proposed here could be extended to general models in a straightforward way, e.g. it would be possible to model serial dependence in a different way than the variance decomposition.

The variance decomposition explained in Section [sec:Method] is a decomposition that takes into account *only* the serial correlation structure of the fMRI series. The approach also allows to apply a correction for the preprocessing steps, that includes other effects, such as head motion or slice acquisition, provided that this process is also regressed out. Unfortunately, in the present settings, this cannot be extended to general models as, as we argued, these decompositions are not the direct generalization of the fMRI variance decomposition.

The decomposition presented here assumes stationarity in a univariate sense, i.e. that it describes how one subject influences another subject.

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