Post on 21-Dec-2015
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)