Variational Bayesian Inferencefor fMRI time series

Will Penny, Stefan Kiebel andKarl 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

yn = xn w + en

en= ∑ ak en-k + zn

where k=1..p. The errors zn are zero mean Gaussian with variance σ2.

Variational Bayes

• We use Bayesian estimation and inference

• The true posterior p(w,a,σ2|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,σ2|Y) = q(w|Y) q(a|Y) q(σ2|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/σ2|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

yn = x w + en

en= a en-1 + zn

where x=1, w=2.7, a=0.3, σ2=4

Compare VB results with exact posterior (which is expensive to compute).

Synthetic dataTrue 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(σ2|Y), VB q(σ2|Y )

Autoregressive coefficient posteriors:Exact p(a|Y), VB q(a|Y)

fMRI Data

Design Matrix, XModelling Parameters

Y=Xw+e

9 regressorsAR(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 analysejust 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)