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Bayesian Inference

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.

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Bayesian Inference

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

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

  3. One parameter Likelihood and Prior Posterior Likelihood Prior Posterior Relative Precision Weighting

  4. Two parameters

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

  6. Prior Prior: mi 1/a=1

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

  8. Least Squares Estimates

  9. Least Squares Estimates Error=7.10

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

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

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

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

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

  15. PPMs Advantages Disadvantages Use of shrinkage 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

  16. Photic Photic SPC SPC SPC 0.96 0.85 0.70 0.84 1.36 0.96 V1 V1 V1 0.06 -0.02 V5 0.39 0.57 V5 V5 Motion Motion 0.23 0.58 Attention Attention Photic 0.86 0.75 1.42 0.89 Attention 0.55 -0.02 0.56 Motion Comparing Dynamic Causal Models m=2 m=1 m=3 Bayesian Evidence: Bayes factors:

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