Variational Bayesian Inference for fMRI time series

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Variational Bayesian Inferencefor fMRI time
series

• Will Penny, Stefan Kiebel and
• Karl Friston
• Wellcome Department of Imaging Neuroscience,
• University College, London, UK.

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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 k1..p. The errors zn are zero mean
• Gaussian with variance s2.

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Variational Bayes

• We use Bayesian estimation and inference
• The true posterior p(w,a,s2Y) 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,s2Y) q(wY) q(aY) q(s2Y)

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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(wY)N(m,S), q(aY)N(v,R)
  and q(1/s2Y)Ga(b,c)
• The parameters m,S,v,R,b and c are then updated
  in an iterative optimisation scheme

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Synthetic Data

• Generate data from
• yn x w en
• en a en-1 zn
• where x1, w2.7, a0.3, s24
• Compare VB results with exact posterior (which
  is expensive to compute).

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Synthetic data
True posterior, p(a,wY)
VBs approximate posterior, q(a,wY)
VB assumes a factorized form for the posterior.
For small a the width of p(wY) will be
overestimated, for large a it will be
underestimated. But on average, VB gets it right !
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Synthetic Data
Autoregressive coefficient posteriors Exact
p(aY), VB q(aY)
Regression coefficient posteriors Exact p(wY),
VB q(wY)
Noise variance posteriors Exact p(s2Y), VB
q(s2Y )
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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 YXwe 9 regressors AR(6)
model for the errors VB model fitting 4
seconds Gibbs sampling much longer !
Design Matrix, X
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fMRI Data
Posterior distributions of two of the regression
coefficients
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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)