PPMs with Spatial PriorsWill,Nelson,Karl

GLM-AR(p) models (+Stefan) with spatial priors on regression/AR coefficients for single-subject fMRI

β

A

q1 q2

α

W

Y

( ) ( )

( ) ( )1

1 2; ,

K

kk

k k

p p

p Ga q q

α

α α=

=

=

∏α

( ) ( )

( ) ( )( )1

11; ,

KTk

k

T T Tk k k

p p

p N α

=

−−

=

=

∏W w

w w 0 S S

u1 u2

λ

( ) ( )

( ) ( )1

1 2; ,

N

nn

n n

p p

p Ga u u

λ

λ λ=

=

=

∏λ1

1

( ) ( )

( ) ( ; , )

N

nn

n n P

p p

p N β=

−

=

=

∏A a

a a 0 I

Y=XW+E

Global prior in SPM2:

[TxN] [TxK] [KxN] [TxN]

Laplacian prior:

=S L

=S I

Variational Bayes

( , , , | ) ( | ) ( | ) ( | ) ( | )k n n nk n

q q q q qα λ = ∏ ∏W A λ α Y Y w Y a Y Y

In chosen approximate posterior, W factorises over n:

In prior, W factorises over k:

1 2 1 2( , , , ) ( | ) ( | , ) ( | ) ( | , )k k k n nk n

p p p q q p p u uα α β λ = ∏ ∏W A λ α w a

log ( )[ ( | ), ( | )]

L p YF KL q Y p Yθ θ=

= +

Update SS of approximate posterior to maximise (lower bound on) evidence

Updating approximate posterior

( )ˆˆ( ) ; ,n n n nq N=w w w Σ

[ ]( )T Tdiag= ⊗B H α S S H

1,

ˆN

n ni ii i n= ≠

= − ∑r B w

( )( ) 1

Tnn n n n

n n n nn

λ

λ−

= +

= +

w Σ b r

Σ A B

( ) ( )

( ) ( )

( )

1

1

2

; ,1 1 1

2

2

K

kk

k k k k

TT Tk k k

k

k

k k k

q q

q Ga g h

Trg q

Nh q

g h

α

α α

α

=

=

=

= + +

= +

=

∏α

Σ S S w S Sw

1

2

( ) ( ; , )

1 12

2

n n n n

n

n

n

q Ga b c

Gb u

Tc u

λ λ=

= +

= +

1

( ) ( ; , )

( )n n n n

n n n P

n n n n

q N

λ β

λ

−

=

= +

=

a a m V

V C I

m D V

Regression coefficients

Spatial precisions

Observation noise

AR coefficients

y

x t

Synthetic Data 1 : from Laplacian Prior

Iteration Number

F

y

x

y

x

Least Squares VB – Laplacian Prior

`Coefficient RESELS’ = 366Coefficients = 1024

Synthetic Data II : blobs

True Smoothing

Global prior Laplacian prior

1-Specificity

Sen

sitiv

ity

Event-related fMRI: Faces versus chequerboard

Smoothing

Global prior Laplacian Prior

Event-related fMRI: Familiar faces versus unfamiliar faces

Smoothing

Global prior Laplacian Prior

Summary

• No need to smooth data at 1st level• Greater sensitivity• Implementation in parallel to non-Bayesian

methods – spm/devel• 5-10 minutes per slice• Model comparison of signal and noise

models for single/multiple regions/slices

The future

• Laplacian priors for AR coefficients• Mixture priors • Wavelet priors• 2nd level noise models

Maps of AR(1) coefficient

β

A

q1 q2

α

W

Y

( ) ( )

( ) ( )1

1 2; ,

K

kk

k k

p p

p Ga q q

α

α α=

=

=

∏α

( ) ( )

( ) ( )( )1

11; ,

KTk

k

T T Tk k k

p p

p N α

=

−−

=

=

∏W w

w w 0 S S

u1 u2

λ

( ) ( )

( ) ( )1

1 2; ,

N

nn

n n

p p

p Ga u u

λ

λ λ=

=

=

∏λ

( )1

1 1

( ) ( )

( ) ; , ( )

P

pp

Tp p p

p p

p N β=

− −

=

=

∏A a

a a 0 S S

Y=XW+E[TxN] [TxK] [KxN] [TxN]

r1 r2

1

1 2

( ) ( )

( ) ( ; , )

P

pp

p p

p p

p Ga r r

β

β β=

=

=

∏β

0pβ →

pβ →∞

Voxel-wise AR:

Spatial pooled AR:

Updating approximate posterior

( )ˆˆ( ) ; ,n n n nq N=w w w Σ

[ ]( )T Tdiag= ⊗B H α S S H

1,

ˆN

n ni ii i n= ≠

= − ∑r B w

( )( ) 1

Tnn n n n

n n n nn

λ

λ−

= +

= +

w Σ b r

Σ A B

( ) ( )

( ) ( )

( )

1

1

2

; ,1 1 1

2

2

K

kk

k k k k

TT Tk k k

k

k

k k k

q q

q Ga g h

Trg q

Nh q

g h

α

α α

α

=

=

=

= + +

= +

=

∏α

Σ S S w S Sw

Regression coefficients

Spatial precisions for W

1

2

( ) ( ; , )

1 12

2

n n n n

n

n

n

q Ga b c

Gb u

Tc u

λ λ=

= +

= +

Observation noise

AR coefficients

( )( )

( )

1

1,

( ) ( ; , )

( )

n n n n

nn n nn

nn n n n

T T

N

n ni ii i n

q N

diag

λ

λ

−

= ≠

=

= +

= +

= ⊗

= − ∑

a a m V

V C J

m V D j

J H β S S H

j J m

( )

1

1 2

1 1

2 2

1 2

( ) ( )

( ) ( ; , )

1 1 1( )2

2

P

pp

p p p p

T T Tp p p

p

p

p pp

q q

q Ga r r

Trr r

Pr r

r r

β

β β

β

=

=

=

= + +

= +

=

∏β

V S S m S Sm

Spatial precisions for A

Gaussian smoothing

Wavelet smoothing