# DCM precision

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```					            Bayesian models for fMRI data

Klaas Enno Stephan
Laboratory for Social and Neural Systems Research
Institute for Empirical Research in Economics
University of Zurich

Functional Imaging Laboratory (FIL)
Wellcome Trust Centre for Neuroimaging
University College London

With many thanks for slides & images to:            The Reverend Thomas Bayes
(1702-1761)
FIL Methods group,
particularly Guillaume Flandin

Methods & models for fMRI data analysis
19 November 2008
Why do I need to learn 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
Bayesian segmentation    Spatial priors          Posterior probability       Dynamic Causal
and normalisation   on activation extent         maps (PPMs)                 Modelling
Image time-series
Statistical parametric map (SPM)
Kernel          Design matrix

Realignment        Smoothing        General linear model

Statistical        Gaussian
inference         field theory
Normalisation

Template                                              p <0.05
Parameter estimates
Problems of classical (frequentist) statistics
p-value: probability of getting the observed data in the effect’s
absence. If small, reject null hypothesis that there is no effect.
H0 :   0        Probability of observing the data y,
p( y | H 0 )      given no effect ( = 0).
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

p( | y )    Probability of the effect,
given the observed data
Overview of topics

• Bayes' rule
• Bayesian update rules for Gaussian densities
• Bayesian analyses in SPM5
– Segmentation & spatial normalisation
– Posterior probability maps (PPMs)
• 1st level: specific spatial priors
• 2nd level: global spatial priors

– Bayesian Model Selection (BMS)
Bayes in motion - an animation
Bayes’ rule
Given data y and parameters , the conditional probabilities are:
p( y, )                           p( y, )
p( | y )                        p( y |  ) 
p( y )                             p( )
Eliminating p(y,) gives Bayes’ rule:

Likelihood           Prior

p( y |  ) p( )
Posterior         P( | y ) 
p( y )
Evidence
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

p( | y )  p( y |  ) p( )
Posterior mean & variance of univariate Gaussians
Likelihood & Prior
y  
p ( y |  )  N ( y; ,  )
2
e

p( )  N ( ;  p ,  p )
2



Posterior
Posterior: p( | y)  N ( ; , )
2

1   1  1                                                   Likelihood
 2 2
 2 e  p                              p
Prior
 1            
 2   12  p 
   2

e    p     
Posterior mean =
variance-weighted combination
of prior mean and data mean
Same thing – but expressed as precision weighting
Likelihood & prior
y  
p ( y |  )  N ( y; ,  )
1
e

p( )  N ( ;  p , 1 )
p


Posterior
Posterior: p( | y )  N ( ;  , 1 )
Likelihood
  e   p                                p
Prior
e  p
    p
   

Relative precision weighting
Same thing – but explicit hierarchical perspective
Likelihood & Prior                              y   (1)   (1)
p( y |  )  N ( y; ,1 /  )
(1)               (1)      (1)      (1)   ( 2)   ( 2)
p( (1) )  N ( (1) ; ( 2 ) ,1 / ( 2 ) )


(1)

Posterior
Posterior
Likelihood
p (   (1)
| y )  N ( ;  ,1 /  )
(1)

  (1)  ( 2 )
 ( 2)
Prior
(1) (1) ( 2 ) ( 2 )
              
        

Relative precision weighting
Bayesian GLM: univariate case
Normal densities
Univariate
p( )  N ( ; p , p )
2            linear       y  x  e
model

 | y
p ( y |  )  N ( y;x,  e2 )
x

p( | y)  N ( ; | y ,2| y )
p
1         x2          1
           
   2
|y          2
e       p
2

 x     1    
 | y       2 y  2 p 
2

|y
p 
 e          

Relative precision weighting
Bayesian GLM: multivariate case
Normal densities                           General
Linear    y  Xθ  e
p (θ)  N (θ; η p , C p )                    Model

p (y | θ)  N (y; Xθ, Ce )

p (θ | y )  N (θ; η | y , C | y )
2
1          1         1
C | y  XT Ce X  C p
               1
η | y  C | y XT Ce y  C p η p    
One step if Ce is known.
1
Otherwise iterative estimation
with EM.
An intuitive example

10

5
2

0

-5
Prior
Likelihood
-10           Posterior
-10        -5        0    5   10
1
Less intuitive

10

5
2

0

-5
Prior
Likelihood
-10         Posterior
-10      -5        0    5   10
1
Even less intuitive

10          Prior
Likelihood
Posterior

5
2

0

-5

-10

-10           -5   0    5     10
1
Bayesian (fixed effects) group analysis

Likelihood distributions from different                            Under Gaussian assumptions this is
subjects are independent                                           easy to compute:
 one can use the posterior from one                               group                      individual
subject as the prior for the next                                  posterior                  posterior
covariance                 covariances

p( | y1 )        p( y1 |  ) p( )                                                          N

p( | y1 , y2 )  p( y2 |  ) p( y1 |  ) p( )                    C  1
 | y1 ,..., y N     C|1 i
y
i 1
 p( y2 |  ) p( | y1 )
 N 1             
...                                                                 | y ,..., y           C | yi | yi C | y1 ,..., y N
p( | y1 ,..., y N )  p( y N |  ) p( | y N 1 )...p( | y1 )                            i 1             
1         N

group                   individual posterior
“Today’s posterior is tomorrow’s prior”
posterior               covariances and means
mean
Bayesian analyses in SPM5

• 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
Spatial normalisation: Bayesian regularisation
Deformations consist of a linear
combination of smooth basis functions
      lowest frequencies of a 3D
discrete cosine transform.

Find maximum a posteriori (MAP) estimates: simultaneously minimise
– squared difference between template and source image
– squared difference between parameters and their priors
Deformation parameters

MAP:    log p( | y)  log p( y |  )  log p( )  log p( y)

“Difference” between template         Squared distance between parameters and
and source image                 their expected values (regularisation)
Bayesian segmentation with empirical priors
• Goal: for each voxel, compute
p (tissue| intensity)
probability that it belongs to a
particular tissue type, given its   p (intensity | tissue) ∙ p (tissue)
intensity
• Likelihood model:
Intensities are modelled by a
mixture of Gaussian distributions
representing different tissue
classes (e.g. GM, WM, CSF).
• Priors are obtained from tissue
probability maps (segmented
images of 151 subjects).

Ashburner & Friston 2005, NeuroImage
Unified segmentation & normalisation
• Circular relationship between segmentation & normalisation:
– Knowing which tissue type a voxel belongs to helps normalisation.
– Knowing where a voxel is (in standard space) helps segmentation.

• Build a joint generative model:
– model how voxel intensities result from mixture of tissue type distributions
– model how tissue types of one brain have to be spatially deformed to match
those of another brain

• Using a priori knowledge about the parameters:
adopt Bayesian approach and maximise the posterior probability

Ashburner & Friston 2005, NeuroImage
Bayesian fMRI analyses
General Linear Model:
y  X         with    ~ N (0, C )

What are the priors?
• In “classical” SPM, no priors (= “flat” priors)
• Full Bayes: priors are predefined on a principled or empirical basis
• Empirical Bayes: priors are estimated from the data, assuming a
hierarchical generative model  PPMs in SPM

Parameters of one level = priors for distribution
of parameters at lower level
Parameters and hyperparameters at each
level can be estimated using EM
Posterior Probability Maps (PPMs)
Posterior distribution: probability of the effect given the data
p( | y )    mean: size of effect
precision: variability

Posterior probability map: images of the probability (confidence) that
an activation exceeds some specified threshold, given the data y

p(   | y )              p( | y )


Two thresholds:
• activation threshold : percentage of whole brain mean signal
(physiologically relevant size of effect)
• probability  that voxels must exceed to be displayed (e.g. 95%)
PPMs vs. SPMs

p( | y)  p( y |  ) p( )

PPMs
Posterior              Likelihood              Prior

SPMs

                                                      u

                                         t  f ( y)
Bayesian test: p(     | y)      Classical t-test: p(t  u |   0)  
2nd level PPMs with global priors
1st level (GLM):
y  X (1) (1)   (1)          p( )  N (0, C )

 (1)   ( 2)   ( 2)        p( )  N (0, C )
2nd level (shrinkage prior):
 0   ( 2)
Basic idea: use the variance of  over voxels
as prior variance of  at any particular voxel.                  p( )
2nd level:
(2) = average effect over voxels,
(2) = voxel-to-voxel variation.
0
(1)reflects regionally specific effects
 assume that it sums to zero over all voxels              In the absence of evidence
 shrinkage prior at the second level                      to the contrary, parameters
 variance of this prior is implicitly estimated                will shrink to zero.
by estimating (2)
Shrinkage Priors
Small & variable effect             Large & variable effect

Small but clear effect              Large & clear effect
2nd level PPMs with global priors
1st level (GLM):
y  X   (1)   p( )  N (0, C )         voxel-specific

2nd level (shrinkage prior):

  0   ( 2)   p( )  N (0, C ) global 
pooled estimate

 We are looking for the same                    Once Cε and C are known, we can
effect over multiple voxels                    apply the usual rule for computing the
posterior mean & covariance:
 Pooled estimation of C over
voxels
C | y  X T C1 X  C1 
1

Friston & Penny 2003, NeuroImage                            m | y  C | y X T C1 y
PPMs and multiple comparisons

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.
PPMs vs.SPMs

rest [2.06]                                                                                      rest

contrast(s)                                                                               contrast(s)

<                <                                                                           <                <

3
4

SPMmip
SPMmip

1

[0, 0, 0]
1
[0, 0, 0]

4                                                                                         4
7                                                                                         7
10                                                                                        10
13                                                                                        13
16                                                                                        16
19                                                                                        19
PPM 2.06
22                                                                      SPM{T39.0}        22
25
<                              25
28                                                            <                           28
31
31                                                                                        34
34                                                                                        37
37                                                                                        40
40                                                                                        43
43
SPMresults: C:\home\spm\analysis_PET                 46                                      SPMresults: C:\home\spm\analysis_PET              46
49
49                                                                                        52
Height threshold P = 0.95                            52                                      Height threshold T = 5.50
55
55                                      Extent threshold k = 0 voxels
Extent threshold k = 0 voxels                        60                                                                                        60
1 4 7 10 13 16 19 22
1 4 7 10 13 16 19 22
Design matrix
Design matrix

PPMs: Show activations                                                                                   SPMs: Show voxels
greater than a given                                                                                      with non-zeros
size                                                                                                activations
PPMs: pros and cons

• One can infer that a     • Estimating priors over
cause did not elicit a     voxels is
response                   computationally
demanding
• Inference is
independent of search     • Practical benefits are
volume                      yet to be established
• SPMs conflate effect-    • Thresholds other than
size and effect-            zero require
variability                 justification
1st level PPMs with local spatial priors
• Neighbouring voxels often not independent
• Spatial dependencies vary across the brain
• But spatial smoothing in SPM is uniform
• Matched filter theorem: SNR maximal when
smoothing the data with a kernel which             Contrast map
matches the smoothness of the true signal
• Basic idea: estimate regional spatial
dependencies from the data and use this as
a prior in a PPM
 regionally specific smoothing
 markedly increased sensitivity
AR(1) map
Penny et al. 2005, NeuroImage
The generative spatio-temporal model
q1                q2                                r1                  r2
K
p  α    p  k                                             P

k 1
p (β)   p (  p )
p 1
p  k   Ga  k ; q1 , q2 
p (  p )  Ga (  p ; r1 , r2 )

u1           u2                                                                                             
p W   p w             
N                                         K

p  λ    p  n 
T                                         P
k                             p ( A )   p (a p )
k 1
n 1                                                                                           p 1

p  n   Ga  n ; u1 , u2                  k           
p  w T   N w T ; 0,  k1  ST S 
k
1
       p (a p )  N  a p ; 0,  p 1 (ST S) 1 

                                          W                                                    A

 = spatial precision of parameters
Y                                      = observation noise precision
 = precision of AR coefficients
Penny et al. 2005, NeuroImage                            Y=XW+E
The spatial prior
Prior for k-th parameter:

 T
k         
p w  N w ;0,    T
k
1
k    S S 
T   1

Shrinkage            Spatial precision:        Spatial
prior          determines the amount of   kernel matrix
smoothness

Different choices possible for spatial kernel matrix S.
Currently used in SPM: Laplacian prior (same as in LORETA)
Example: application to event-related fMRI data
Smoothing
Contrast maps for
familiar vs. non-familiar
faces, obtained with
- smoothing
- global spatial prior
- Laplacian prior

Global prior   Laplacian Prior
SPM5 graphical user interface
Bayesian model selection (BMS)
Given competing hypotheses
on structure & functional
mechanisms of a system, which
model is the best?

Which model represents the
best balance between model
fit and model complexity?

For which model m does
p(y|m) become maximal?
Pitt & Miyung (2002), TICS
Bayesian model selection (BMS)
p( y |  , m) p( | m)
Bayes’ rules:           p( | y, m) 
p( y | m)

Model evidence:        p( y | m)   p( y |  , m)  p( | m) d

accounts for both accuracy and complexity of the model
allows for inference about structure (generalisability) of the model

Various approximations, e.g.:            Model comparison via Bayes factor:
- negative free energy
p ( y | m1 )
- AIC                                            BF 
- BIC                                                 p( y | m2 )
Penny et al. (2004) NeuroImage
Example: BMS of dynamic causal models
attention
M1                                                             M2
 modulation of back-                                PPC                                                          PPC
ward or forward                                                M2 better than M1
attention
connection?                                                       BF = 2966
stim       V1         V5                                       stim        V1     V5

effect of attention                              PPC
on PPC?                                                                     BF = 12
M3 better than M2
stim       V1         V5

 bilinear or nonlinear                                                                        M4     attention
PPC
modulation of
BF = 23
forward connection?
M4 better than M3
stim        V1     V5
Stephan et al. (2008) NeuroImage
Thank you

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