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Variational Bayesian Inference for fMRI time series

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

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Variational Bayesian Inference for fMRI time series

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

  2. 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.

  3. 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)

  4. 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

  5. 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).

  6. 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 !

  7. Synthetic Data Autoregressive coefficient posteriors: Exact p(a|Y), VB q(a|Y) Regression coefficient posteriors: Exact p(w|Y), VB q(w|Y) Noise variance posteriors: Exact p(σ2|Y), VB q(σ2|Y )

  8. fMRI Data 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. Modelling Parameters Y=Xw+e 9 regressors AR(6) model for the errors VB model fitting: 4 seconds Gibbs sampling: much longer ! Design Matrix, X

  9. fMRI Data Posterior distributions of two of the regression coefficients

  10. 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)

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