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Determining the best model or models for a particular data set, a process known as Bayesian model comparison, is a critical part of probabilistic inference. Typically, this process assumes a fixed model-space (that is, a fixed set of candidate models). However, it is also possible to perform Bayesian inference over model-spaces themselves, thus determining which spaces provide the best explanation for observed data. Model-space inference (MSI) allows the effective exclusion of poorly performing models (a process analogous to Automatic Relevance Detection), and thus mitigates against the well-known phenomenon of model dilution, resulting in posterior probability estimates that are, on average, more accurate than those produced when using a fixed model-space. We focus on model comparison in the context of multiple independent data sets (as produced, for example, by multi-subject behavioural or neuroimaging studies), and cast our proposal as a development of random-effects Bayesian Model Selection, the current state-of-the-art in the field. We demonstrate the increased accuracy of MSI using simulated behavioural and neuroimaging data, as well as by assessing predictive performance in previously-acquired empirical data. Additionally, we explore other applications of MSI, including formal testing for a diversity of models within a population, and comparison of model-spaces between populations. Our approach thus provides an important new tool for model comparison., Comment: 21 pages, 6 figures

We introduce a probabilistic (Bayesian) framework and associated software toolbox for mapping population receptive fields (pRFs) based on fMRI data. This generic approach is intended to work with stimuli of any dimension and is demonstrated and validated in the context of 2D retinotopic mapping. The framework enables the experimenter to specify generative (encoding) models of fMRI timeseries, in which experimental manipulations enter a pRF model of neural activity, which in turns drives a nonlinear model of neurovascular coupling and Blood Oxygenation Level Dependent (BOLD) response. The neuronal and haemodynamic parameters are estimated together on a voxel-by-voxel or region-of-interest basis using a Bayesian estimation algorithm (variational Laplace). This offers several novel contributions to receptive field modelling. The variance / covariance of parameters are estimated, enabling receptive fields to be plotted while properly representing uncertainty about pRF size and location. Variability in the haemodynamic response across the brain is accounted for. Furthermore, the framework introduces formal hypothesis testing to pRF analysis, enabling competing models to be evaluated based on their model evidence (approximated by the variational free energy), which represents the optimal tradeoff between accuracy and complexity. Using simulations and empirical data, we found that parameters typically used to represent pRF size and neuronal scaling are strongly correlated, which should be taken into account when making inferences. We used the framework to compare the evidence for six variants of pRF model using 7T functional MRI data and we found a circular Difference of Gaussians (DoG) model to be the best explanation for our data overall. We hope this framework will prove useful for mapping stimulus spaces with any number of dimensions onto the anatomy of the brain., Comment: 30 pages, 10 figures. Code available at https://github.com/pzeidman/BayespRF