Download - Bayesian Model Comparison
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Bayesian Model Comparison
Will Penny
London-Marseille Joint Meeting,Institut de Neurosciences Cognitive de la Mediterranee,
Marseille, September 28-29, 2009
V1
V5
SPC
V1
V5
SPC
Wellcome Centre for Neuroimaging, UCL, UK.
![Page 2: Bayesian Model Comparison](https://reader036.vdocuments.net/reader036/viewer/2022062304/568132a2550346895d9943e6/html5/thumbnails/2.jpg)
Overview
• Priors, likelihoods and posteriors
• Model selection using evidence
• Model selection for groups
• Comparing model families
![Page 3: Bayesian Model Comparison](https://reader036.vdocuments.net/reader036/viewer/2022062304/568132a2550346895d9943e6/html5/thumbnails/3.jpg)
Overview
• Priors, likelihoods and posteriors
• Model selection using evidence
• Model selection for groups
• Comparing model families
![Page 4: Bayesian Model Comparison](https://reader036.vdocuments.net/reader036/viewer/2022062304/568132a2550346895d9943e6/html5/thumbnails/4.jpg)
Bayesian Paradigm:priors and likelihood
eZy Model:
Z
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Bayesian Paradigm:priors and likelihood
1
2
eZy Model:
Prior:
)2/exp(
),0()(
2
122
T
kk INp
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Sample curves from prior (before observing any data)
Mean curve
x
Z
Bayesian Paradigm:priors and likelihood
1
2
eZy Model:
Prior:
)2/exp(
),0()(
2
122
T
kk INp
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1
2
Bayesian Paradigm:priors and likelihood
1
2
)2/)(exp(
),(),(
),|(),(
21
111
1
111
ii
ii
N
ii
Zy
ZNyp
ypyp
eZy Model:
Prior:
)2/exp(
),0()(
2
122
T
kk INp
Likelihood:
x
Z
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Bayesian Paradigm:priors and likelihood
1
2
eZy Model:
Prior:
)2/exp(
),0()(
2
122
T
kk INp
Likelihood:
)2/)(exp(
),(),(
),|(),(
21
111
1
111
ii
ii
N
ii
Zy
ZNyp
ypyp
x
Z
![Page 9: Bayesian Model Comparison](https://reader036.vdocuments.net/reader036/viewer/2022062304/568132a2550346895d9943e6/html5/thumbnails/9.jpg)
Bayesian Paradigm:priors and likelihood
1
2
eZy Model:
Prior:
)2/exp(
),0()(
2
122
T
kk INp
Likelihood:
)2/)(exp(
),(),(
),|(),(
21
111
1
111
ii
ii
N
ii
Zy
ZNyp
ypyp
x
Z
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Bayesian Paradigm: posterior
yCZ
IZZC
CNyp
T
kT
1
1
21
, ,|
x
Z
eZy Model:
Prior:
)2/exp(
),0()(
2
122
T
kk INp
Likelihood:
Bayes Rule:
)|(),|(),( pypyp
Posterior:
N
iiypyp
111 ),|(),(
1
2
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Bayesian Paradigm: posterior
1
2
x
Z
yCZ
IZZC
CNyp
T
kT
1
1
21
, ,|
eZy Model:
Prior:
)2/exp(
),0()(
2
122
T
kk INp
Likelihood:
Bayes Rule:
)|(),|(),( pypyp
Posterior:
N
iiypyp
111 ),|(),(
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Bayesian Paradigm: posterior
1
2
x
Z
yCZ
IZZC
CNyp
T
kT
1
1
21
, ,|
eZy Model:
Prior:
)2/exp(
),0()(
2
122
T
kk INp
Likelihood:
Bayes Rule:
)|(),|(),( pypyp
Posterior:
N
iiypyp
111 ),|(),(
![Page 13: Bayesian Model Comparison](https://reader036.vdocuments.net/reader036/viewer/2022062304/568132a2550346895d9943e6/html5/thumbnails/13.jpg)
Overview
• Priors, likelihoods and posteriors
• Model selection using evidence
• Model selection for groups
• Comparing model families
![Page 14: Bayesian Model Comparison](https://reader036.vdocuments.net/reader036/viewer/2022062304/568132a2550346895d9943e6/html5/thumbnails/14.jpg)
Model Selection
|log myp
2
olsZy
Cos
t fu
nct
ion
Bayes Rule:
)(
)|(),|(),(
myp
mpmypmyp
normalizing constant
dmpmypmyp )|(),|()(
)()(
)|(log
mcomplexitymaccuracy
myp
Model evidence:
constkmcomplexity
constZymaccuracy
2
21
2
2
1
log)(
)(
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{ , , , }θ A B C h
( | , ) ( | )( | , )
( | )
p m p mp m
p m
y θ θθ y
y
V1
V5
SPC
Model, mParameters:
PriorPosterior Likelihood
( | ) ( )( | )
( )
p m p mp m
p
yy
y
PriorPosterior Evidence
Parameter Parameter
Model Model
Second level of Bayesian Inference
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Bayes Factors
V1
V5
SPC
( | ) ( | , ) ( | )p m i p m i p m i d y y θ θ θ
( | ) ( | , ) ( | )p m j p m j p m j d y y θ θ θ
Model, m=i
V1
V5
SPC
Model, m=j
Model Evidences:
Bayes factor:( | )
( | )ij
p m iB
p m j
y
y
1 to 3: Weak3 to 20: Positive20 to 100: Strong>100: Very Strong
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Overview
• Priors, likelihoods and posteriors
• Model selection using evidence
• Dynamic Causal Models
• Model selection for groups
• Comparing model families
![Page 18: Bayesian Model Comparison](https://reader036.vdocuments.net/reader036/viewer/2022062304/568132a2550346895d9943e6/html5/thumbnails/18.jpg)
Single region
1 11 1 1z a z cu
u2
u1
z1
z2
z1
u1
a11c
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Multiple regions
1 11 1 1
2 21 22 2 2
0
0
z a z uc
z a a z u
u2
u1
z1
z2
z1
z2
u1
a11
a22
c
a21
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Modulatory inputs
1 11 1 1 12
2 21 22 2 21 2 2
0 0 0
0 0
z a z z ucu
z a a z b z u
u2
u1
z1
z2
u2
z1
z2
u1
a11
a22
c
a21
b21
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Reciprocal connections
1 11 12 1 1 12
2 21 22 2 21 2 2
0 0
0 0
z a a z z ucu
z a a z b z u
u2
u1
z1
z2
u2
z1
z2
u1
a11
a22
c
a1
2
a21
b21
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Overview
• Priors, likelihoods and posteriors
• Model selection using evidence
• Dynamic Causal Models
• Model selection for groups
• Comparing model families
![Page 23: Bayesian Model Comparison](https://reader036.vdocuments.net/reader036/viewer/2022062304/568132a2550346895d9943e6/html5/thumbnails/23.jpg)
-5 -4 -3 -2 -1 0 1 2 3 4 5
Sim
ulat
ed d
ata
sets
Log model evidence differences
x1 x2u1
x3
u2
x1 x2u1
x3
u2
incorrect model (m2) correct model (m1)
Figure 2
m2 m1
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-35 -30 -25 -20 -15 -10 -5 0 5
Sub
ject
s
Log model evidence differences
MOG
LG LG
RVFstim.
LVFstim.
FGFG
LD|RVF
LD|LVF
LD LD
MOGMOG
LG LG
RVFstim.
LVFstim.
FGFG
LD
LD
LD|RVF LD|LVF
MOG
m2 m1
Models from Klaas Stephan
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)|(~ 111 mypy)|(~ 111 mypy
)|(~ 222 mypy)|(~ 111 mypy
)|(~ pmpm kk
);(~ rDirr
)|(~ pmpm kk2 2~ ( | )m p m p
),1;(~1 rmMultm
Random Effects Inference
Different subjects can use different models.
is the probability that model m is usedin the population at large.
We wish to make an inference aboutthis.
mr
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
r1
p(r 1
|y)
p(r1>0.5 | y) = 0.997
157.0,843.0
194.2,806.11
21
21
rr
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Overview
• Priors, likelihoods and posteriors
• Model selection using evidence
• Model selection for groups
• Comparing model families
![Page 28: Bayesian Model Comparison](https://reader036.vdocuments.net/reader036/viewer/2022062304/568132a2550346895d9943e6/html5/thumbnails/28.jpg)
F
A
P
DCM of Auditory Word Processing: Data from an fMRI study by Alex Leff and Tom Schofield
P: Posterior STSA: Anterior STSF: Inferior Frontal Gyrus
How does processing change for speech versus reversed speech input ?
2^6=64 possible patterns of ‘modulation’.2^3=8-1=7 possible patterns of input connectivity7*64=448 possible networks26*448=11,648 models in group of 26 subjects
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0 0.5 1
A
sk
p(s k|y
0 0.5 1
F
sk
0 0.5 1
P
sk
0 0.5 1
AF
sk
0 0.5 1
PA
sk
0 0.5 1
PF
sk
0 0.5 1
PAF
sk
Input families: Where does the input go ?
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A F P AF PA PF PAF
0 50 100 150 200 250 300 350 400 4500
0.02
0.04
0.06
0.08
0.1p(
m|Y
)
m
0 50 100 150 200 250 300 350 400 4500
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
E[r
m|Y
]
m
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F
A
P
F
A
P
F
A
P
F
A
P
(a)
(d)(c)
(b)
Four of the top 16 models:
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-1 -0.5 0 0.5 1
A to P
-1 -0.5 0 0.5 1
F to P
-1 -0.5 0 0.5 1
P to A
-1 -0.5 0 0.5 1
F to A
-1 -0.5 0 0.5 1
P to F
-1 -0.5 0 0.5 1
A to F
)|(),|()|( ympmycpycp
Bayesian
Model
Averaging
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-0.2 0 0.2 0.4
A to P
-0.2 0 0.2 0.4
F to P
-0.2 0 0.2 0.4
P to A
-0.2 0 0.2 0.4
F to A
-0.2 0 0.2 0.4
P to F
-0.2 0 0.2 0.4
A to F
Same but now for RFX model probs p(m|Y)
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F
A
P
DCM of Auditory Word Processing: Data from an fMRI study by Alex Leff and Tom Schofield
P: Posterior STSA: Anterior STSF: Inferior Frontal Gyrus
How does processing change for speech versus reversed speech input ?
(1) Input goes to P.
(2) Connections from P to F, and P to A, are increased for speech versus reversed speech
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Summary
• First and second levels of Bayesian inference
• Model selection for groups
• Comparing model families
• DCM for EEG-fMRI
• Thank-you !