Variational Bayesian Inference for fMRI time series Will Penny, Stefan Kiebel and Karl Friston Wellcome Department of Imaging Neuroscience, University College, London, UK.

Generalised Linear Model A central concern in fMRI is that the errors from scan n-1 to scan n are serially correlated We use Generalised Linear Models (GLMs) with autoregressive error processes of order p y n = x n w + e n e n = ∑ a k e n-k + z n where k=1..p. The errors z n are zero mean Gaussian with variance σ 2.

Variational Bayes We use Bayesian estimation and inference The true posterior p(w,a,σ 2 |Y) can be approximated using sampling methods. But these are computationally demanding. We use Variational Bayes (VB) which uses an approximate posterior that factorises over parameters q(w,a,σ 2 |Y) = q(w|Y) q(a|Y) q(σ 2 |Y)

Variational Bayes Estimation takes place by minimizing the Kullback-Liebler divergence between the true and approximate posteriors. The optimal form for the approximate posteriors is then seen to be q(w|Y)=N(m,S), q(a|Y)=N(v,R) and q(1/σ 2 |Y)=Ga(b,c) The parameters m,S,v,R,b and c are then updated in an iterative optimisation scheme

Synthetic Data Generate data from y n = x w + e n e n = a e n-1 + z n where x=1, w=2.7, a=0.3, σ 2 =4 Compare VB results with exact posterior (which is expensive to compute).

Synthetic data True posterior, p(a,w|Y) VB’s approximate posterior, q(a,w|Y) VB assumes a factorized form for the posterior. For small ‘a’ the width of p(w|Y) will be overestimated, for large ‘a’ it will be underestimated. But on average, VB gets it right !

Synthetic Data Regression coefficient posteriors: Exact p(w|Y), VB q(w|Y) Noise variance posteriors: Exact p(σ 2 |Y), VB q(σ 2 |Y ) Autoregressive coefficient posteriors: Exact p(a|Y), VB q(a|Y)

fMRI Data Design Matrix, X Modelling Parameters Y=Xw+e 9 regressors AR(6) model for the errors VB model fitting: 4 seconds Gibbs sampling: much longer ! Event-related data from a visual-gustatory conditioning experiment. 680 volumes acquired at 2Tesla every 2.5 seconds. We analyse just a single voxel from x = 66 mm, y = -39 mm, z = 6 mm (Talairach). We compare the VB results with a Bayesian analysis using Gibbs sampling.

fMRI Data Posterior distributions of two of the regression coefficients

Summary Exact Bayesian inference in GLMs with AR error processes is intractable VB approximates the true posterior with a factorised density VB takes into account the uncertainty of the hyperparameters Its much less computationally demanding than sampling methods It allows for model order selection (not shown)