Bayesian comparison of spatially regularised general linear models

Abstract

In previous work (Penny et al., [2005]: Neuroimage 24:350-362) we have developed a spatially regularised General Linear Model for the analysis of functional magnetic resonance imaging data that allows for the characterisation of regionally specific effects using Posterior Probability Maps (PPMs). In this paper we show how it also provides an approximation to the model evidence. This is important as it is the basis of Bayesian model comparison and provides a unified framework for Bayesian Analysis of Variance, Cluster of Interest analyses and the principled selection of signal and noise models. We also provide extensions that implement spatial and anatomical regularisation of noise process parameters.

Figure 1

Generative model: The figure shows the probabilistic dependencies underlying our generative model for fMRI data. The quantities in square brackets are constants and those in circles are random variables. The spatial regularisation coefficients α constrain the regression coefficients W. The parameters λ and A define the autoregressive error processes that contribute to the measurements. The spatial regularisation coefficients β constrain the AR coefficients A. The graph shows that the joint probability of parameters and data can be written p(Y, W, A, λ, α, β) = p(Y|W, A, λ)p(W|α)p(A|β)p(λ|u ₁, u ₂)p(α|q ₁, q ₂)p(β|r ₁, r ₂), where the first term is the likelihood and the other terms are the priors. The likelihood is given in Eq. (11) and the priors are defined in greater detail in Appendices APPENDIX A: PRECISIONS, APPENDIX B: REGRESSION COEFFICIENTS, APPENDIX C: AR COEFFICIENTS.

Figure 2

Approximate posteriors: The full approximate posterior distribution is q(W, A, λ, α, β) = [∏${}_{\mathit{\text{n}}\text{=1}}^{\mathit{\text{N}}}$ q(w_n)q(a_n)q(λ_n)] q(α)q(β). The boxes in the figure show each component of the approximate posterior along with update equations for their sufficient statistics.

Figure 3

Face paradigm: (a) Experimental stimuli and (b) time series of stimuli presentation. [Colour figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 4

AR(1) images for synthetic data: Each row in this figure corresponds to analysis of a different data set. The leftmost column shows the AR(1) profile used to generate the data. The second, third and 4th columns show AR profiles as estimated by models with tissue‐type priors having 1, 2 and 3 (known) discrete levels, respectively. The fifth column shows the estimated profiles from models with spatial GMRF priors. The final column shows bar plots of the posterior model probabilities. The first three bars correspond to models with tissue‐type priors having 1, 2 and 3 levels and the fourth bar corresponds to the spatial GMRF model. These results show that our approximation to the model evidence can correctly detect the type of structure in the coefficients.

Figure 5

COI analysis for a uniformly activated region: The figure shows of a plot of sensitivity versus number of voxels in the cluster, N, for models using a spatial prior (circles and solid line) and a shrinkage prior (crosses and dotted line).

Figure 6

COI analysis for a non‐uniformly activated region: The figure shows of a plot of sensitivity versus number of voxels in the cluster, N, for a model using a spatial prior (circles and solid line), ROI analysis using the mean voxel time series (crosses and dotted line) and an ROI analysis using the principal component time series (plusses and dashed line).

Figure 7

Noise‐free time series: From type 2 data (thin line), generated from an FIR model, and type 1 data (thick line), generated from a best fitting Informed basis set model. These data were used to compare the sensitivity of nested versus non‐nested model comparison, in the context of selecting an optimal hemodynamic basis set.

Figure 8

Nested versus non‐nested: The figure shows of a plot of sensitivity versus number of voxels in the cluster, N, for non‐nested model comparison (circles and solid line) versus nested model comparison (crosses and dotted line).

Figure 9

AR(1) images for face data: The top row shows estimated profiles from a tissue‐type prior (smoothed CSF versus other, prior (iii)) and the bottom row shows the estimated profiles from models with spatial GMRF priors. Columns in this figure show results for slices z = −27, 3, 33 and 63 mm.

Figure 10

Average effect of faces: The top row shows maps of the difference in contributions to the log evidence, U_n(2)−U_n(1), for slices z = −24, −21, −18 and −15 mm. The bottom row shows the same map but thresholded so that only effects with a posterior probability greater than 0.999 (difference in log evidence = 4.6) survive.

Figure 11

Main effect of repetition: The top row shows maps of the difference in contributions to the log evidence, U_n(3)−U_n(2), for slices z = −24, −21, −18 and −15 mm. The bottom row shows the same maps but thresholded so that only effects with a posterior probability greater than 0.999 (difference in log evidence = 4.6) survive.

Figure 12

Comparing hemodynamic basis sets: The bar plots show the log‐evidence for each hemodynamic basis set for COIs in (a) LOC x = −45, y = −60, z = −24 mm, (b) ROC x = 45, y = −66, z = −24 mm and (c) Sensorimotor Cortex x = 36, y = −9, z = 66 mm. The evidence values have been normalised (by subtraction) so that the minimal log‐evidence is zero. [Colour figure can be viewed in the online issue, which is available at www. interscience.wiley.com.]