Bayesian Inference Will Penny

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Presentation transcript:

Bayesian Inference Will Penny Wellcome Department of Imaging Neuroscience, University College London, UK SPM Course, London, May 2004

Bayesian Inference Gaussians Posterior Probability Maps (PPMs) Hemodynamic Models (HDMs) Comparing models (DCMs)

One parameter Likelihood and Prior Posterior Likelihood Prior Relative Precision Weighting

Two parameters

Posterior Probability Maps - PPMs Bayesian parameter estimation Least squares parameter estimation No Priors Shrinkage priors

Prior Prior: mi 1/a=1

Data Likelihood: yi i=1..V=20 voxels n=1..N=10 scans

Least Squares Estimates

Least Squares Estimates Error=7.10

Bayesian Estimates E-Step: M-Step: Iteration 1 Error=7.10

Bayesian Estimates E-Step: M-Step: Iteration 2 Error=2.95

Bayesian Estimates E-Step: M-Step: Iteration 3 Error=2.35

SPMs and PPMs PPMs: Show activations of a given size SPMs: show voxels with non-zero activations

Hemodynamic Models - HDMs Photic Motion Attention fMRI study of attention to visual motion

PPMs Advantages Disadvantages Use of shrinkage One can infer a cause priors over voxels is computationally demanding Utility of Bayesian approach is yet to be established One can infer a cause DID NOT elicit a response SPMs conflate effect-size and effect-variability

Comparing Dynamic Causal Models V1 V5 SPC Motion Photic Attention 0.86 0.56 -0.02 1.42 0.55 0.75 0.89 V1 V5 SPC Motion Photic Attention 0.96 0.39 0.06 0.58 m=3 V1 V5 SPC Motion Photic Attention 0.85 0.57 -0.02 1.36 0.70 0.84 0.23 Bayesian Evidence: Bayes factors: