1. Bayesian models for fMRI data
Klaas Enno Stephan
Translational Neuromodeling Unit (TNU) Institute for Biomedical Engineering, University of Zurich & ETH Zurich
Laboratory for Social & Neural Systems Research (SNS), University of Zurich
Wellcome Trust Centre for Neuroimaging, University College London
With many thanks for slides & images to:
FIL Methods group,
particularly Guillaume Flandin and Jean Daunizeau
The Reverend Thomas Bayes
( )
SPM Course Zurich
13-15 February 2013

3. Bayes‘ Theorem Posterior Likelihood Prior Evidence
Reverend Thomas Bayes
“Bayes‘ Theorem describes, how an ideally rational person processes information."
Wikipedia

4. Bayes’ Theorem
Given data y and parameters , the joint probability is:
Eliminating p(y,) gives Bayes’ rule:
Likelihood
Prior
Posterior
Evidence

5. Bayesian inference: an animation

6. Principles of Bayesian inference
Formulation of a generative model
likelihood p(y|)
prior distribution p()
Observation of data y
Update of beliefs based upon observations, given a prior state of knowledge

7. Posterior mean & variance of univariate Gaussians
Likelihood & Prior
Posterior
Posterior:
Likelihood
Prior
Posterior mean = variance-weighted combination of prior mean and data mean

8. Same thing – but expressed as precision weighting
Likelihood & prior
Posterior
Posterior:
Likelihood
Prior
Relative precision weighting

9. Same thing – but explicit hierarchical perspective
Likelihood & Prior
Posterior
Posterior
Likelihood
Prior
Relative precision weighting

10.  Why should I know about Bayesian stats?
Because Bayesian principles are fundamental for
statistical inference in general
sophisticated analyses of (neuronal) systems
contemporary theories of brain function

11. Problems of classical (frequentist) statistics
p-value: probability of observing data in the effect’s absence
Limitations:
One can never accept the null hypothesis
Given enough data, one can always demonstrate a significant effect
Correction for multiple comparisons necessary
Solution: infer posterior probability of the effect

12. posterior distribution
Generative models: Forward and inverse problems
forward problem
likelihood  prior
inverse problem
posterior distribution

13. Dynamic causal modeling (DCM)
EEG, MEG
fMRI
Forward model:
Predicting measured activity given a putative neuronal state
Model inversion:
Estimating neuronal mechanisms from brain activity measures
Friston et al. (2003) NeuroImage

14. sensations – predictions
The Bayesian brain hypothesis & free-energy principle
sensations – predictions
Prediction error
Change sensory input
Change
predictions
Action
Perception
Maximizing the evidence (of the brain's generative model)
= minimizing the surprise about the data (sensory inputs).
Friston et al. 2006,
J Physiol Paris

15. Individual hierarchical Bayesian learning
volatility
associations
events in the world
sensory stimuli
Mathys et al. 2011, Front. Hum. Neurosci.

16. Aberrant Bayesian message passing in schizophrenia:
abnormal (precision-weighted) prediction errors 
abnormal modulation of NMDAR-dependent synaptic plasticity at forward connections of cortical hierarchies
Backward & lateral
input
Forward & lateral
g: generative model
: expectation of approximate recognition density
: parameters of generative model (= connection strengths between levels)
: hyperparameters (= parameters encoding the uncertainty of the approximate recognition model)
: prediction error
Forward recognition effects
De-correlating lateral interactions
Backward generation effects
Lateral interactions
mediating priors
Stephan et al. 2006, Biol. Psychiatry

17.  Why should I know about Bayesian stats?
Because SPM is getting more and more Bayesian:
Segmentation & spatial normalisation
Posterior probability maps (PPMs)
1st level: specific spatial priors
2nd level: global spatial priors
Dynamic Causal Modelling (DCM)
Bayesian Model Selection (BMS)
EEG: source reconstruction

18. Bayesian segmentation Posterior probability
and normalisation
Spatial priors
on activation extent
Posterior probability
maps (PPMs)
Dynamic Causal
Modelling
Image time-series
Statistical parametric map (SPM)
Kernel
Design matrix
Realignment
Smoothing
General linear model
Statistical
inference
Gaussian
field theory
Normalisation
p <0.05
Template
Parameter estimates

19. Spatial normalisation: Bayesian regularisation
Deformations consist of a linear combination of smooth basis functions (3D DCT).
Find maximum a posteriori (MAP) estimates:
Deformation parameters
MAP:
“Difference” between template and source image
Squared distance between parameters and their expected values (regularisation)

20. Spatial normalisation: overfitting
Affine registration.
(2 = 472.1)
Template
image
Non-linear
registration
without
regularisation.
(2 = 287.3)
Non-linear
registration
using
regularisation.
(2 = 302.7)

21. Bayesian segmentation with empirical priors
Goal: for each voxel, compute probability that it belongs to a particular tissue type, given its intensity
Likelihood: Intensities are modelled by a mixture of Gaussian distributions representing different tissue classes (e.g. GM, WM, CSF).
Priors: obtained from tissue probability maps (segmented images of 151 subjects).
p (tissue | intensity)
p (intensity | tissue) ∙ p (tissue)
Ashburner & Friston 2005, NeuroImage

22. Bayesian fMRI analyses
General Linear Model:
with
What are the priors?
In “classical” SPM, no priors (= “flat” priors)
Full Bayes: priors are predefined
Empirical Bayes: priors are estimated from the data, assuming a hierarchical generative model
Parameters of one level = priors for distribution of parameters at lower level
Parameters and hyperparameters at each level can be estimated using EM

23. Posterior Probability Maps (PPMs)
Posterior distribution: probability of the effect given the data
mean: size of effect precision: variability
Posterior probability map: images of the probability that an activation exceeds some specified threshold, given the data y
Two thresholds:
activation threshold : percentage of whole brain mean signal
probability  that voxels must exceed to be displayed (e.g. 95%)

24. 2nd level PPMs with global priors
1st level (GLM):
2nd level (shrinkage prior):
Heuristically: use the variance of mean-corrected activity over voxels as prior variance of  at any particular voxel.
(1) reflects regionally specific effects  assume that it is zero on average over voxels
 variance of this prior is implicitly estimated by estimating (2)
In the absence of evidence
to the contrary, parameters
will shrink to zero.

25. 2nd level PPMs with global priors
1st level (GLM):
voxel-specific
2nd level (shrinkage prior):
global 
pooled estimate
over voxels
Compute Cε and C via ReML/EM, and apply the usual rule for computing posterior mean & covariance for Gaussians:
Friston & Penny 2003, NeuroImage

26. PPMs vs. SPMs PPMs Posterior Likelihood Prior SPMs Bayesian test:
Classical t-test:

27. PPMs and multiple comparisons
Friston & Penny (2003): No need to correct for multiple comparisons:
Thresholding a PPM at 95% confidence: in every voxel, the posterior probability of an activation  is  95%.
At most, 5% of the voxels identified could have activations less than .
Independent of the search volume, thresholding a PPM thus puts an upper bound on the false discovery rate.
NB: being debated

28. PPMs vs.SPMs PPMs: Show activations greater than a given size
SPMs: Show voxels with non-zero activations

29. PPMs: pros and cons Advantages Disadvantages
One can infer that a cause did not elicit a response
Inference is independent of search volume
do not conflate effect-size and effect-variability
Estimating priors over voxels is computationally demanding
Practical benefits are yet to be established
Thresholds other than zero require justification

30. Model comparison and selection
Given competing hypotheses on structure & functional mechanisms of a system, which model is the best?
Pitt & Miyung (2002) TICS
Which model represents the best balance between model fit and model complexity?
For which model m does p(y|m) become maximal?

31. Bayesian model selection (BMS)
Model evidence:
Gharamani, 2004
p(y|m) y
all possible datasets
accounts for both accuracy and complexity of the model
Various approximations, e.g.:
negative free energy, AIC, BIC
a measure of generalizability
McKay 1992, Neural Comput.
Penny et al. 2004a, NeuroImage

32. Approximations to the model evidence
Logarithm is a monotonic function
Maximizing log model evidence
= Maximizing model evidence
Log model evidence = balance between fit and complexity
No. of
parameters
In SPM2 & SPM5, interface offers 2 approximations:
No. of
data points
Akaike Information Criterion:
Bayesian Information Criterion:
Penny et al. 2004a, NeuroImage

33. The (negative) free energy approximation
Under Gaussian assumptions about the posterior (Laplace approximation):

34. The complexity term in F
In contrast to AIC & BIC, the complexity term of the negative free energy F accounts for parameter interdependencies.
The complexity term of F is higher
the more independent the prior parameters ( effective DFs)
the more dependent the posterior parameters
the more the posterior mean deviates from the prior mean
NB: Since SPM8, only F is used for model selection !

35. Bayes factors
To compare two models, we could just compare their log evidences.
But: the log evidence is just some number – not very intuitive!
A more intuitive interpretation of model comparisons is made possible by Bayes factors:
positive value, [0;[
B12
p(m1|y)
Evidence
1 to 3
50-75%
weak
3 to 20
75-95%
positive
20 to 150
95-99%
strong
 150
 99%
Very strong
Kass & Raftery classification:
Kass & Raftery 1995, J. Am. Stat. Assoc.

36. BMS in SPM8: an example M1 M2 M3 M4 attention PPC PPC BF 2966
M2 better than M1
attention
stim V1 V5
stim V1 V5 M1 M2 M3 M4 V1 V5
stim
PPC M3
attention
M3 better than M2
BF  12
F = 2.450
Posterior model probability in lower plot is a normalised probability:
p(m_i|y) = p(y|m_i)/sum(p(y|m_i))
Note that under flat model priors p(m_i|y) = p(y|m_i) V1 V5
stim
PPC M4
attention
M4 better than M3
BF  23
F = 3.144

37. Thank you