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
Relative Precision Weighting

4. Two parameters

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

16. Comparing Dynamic Causal ModelsV1 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: