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Bayesian inference and generative models
Klaas Enno Stephan
Lecture as part of "Methods & Models for fMRI data analysis",
University of Zurich & ETH Zurich, 22 November 2016
With slides from and many thanks to:
Kay Brodersen,
Will Penny,
Sudhir Shankar Raman
Slides with a yellow title were not covered in detail in the
lecture and will not be part of the exam.
Why should I know about Bayesian inference?
Because Bayesian principles are fundamental for
• statistical inference in general
• system identification
• translational neuromodeling ("computational assays")
– computational psychiatry
– computational neurology
• contemporary theories of brain function (the "Bayesian brain")
– predictive coding
– free energy principle
– active inference
Why should I know about Bayesian inference?
Because Bayesian principles are fundamental for
• statistical inference in general
• system identification
• translational neuromodeling ("computational assays")
– computational psychiatry
– computational neurology
• contemporary theories of brain function (the "Bayesian brain")
– predictive coding
– free energy principle
– active inference
Bayes‘ Theorem
Reverend Thomas Bayes
1702 - 1761
“Bayes‘ Theorem describes, how an ideally rational person
processes information."
Wikipedia
( | ) ( )( | )
( )
p y pP y
p y
likelihood prior
evidence
posterior
Bayesian inference: an animation
Generative models
• specify a joint probability distribution over all variables (observations and
parameters)
• require a likelihood function and a prior:
• can be used to randomly generate synthetic data (observations) by sampling
from the prior
– we can check in advance whether the model can explain certain
phenomena at all
• model comparison based on the model evidence
( , | ) ( | , ) ( | ) ( | , )p y m p y m p m p y m
( | ) ( | , ) ( | )p y m p y m p m d
Observation of data
Formulation of a generative model
Model inversion – updating one's beliefs
( | ) ( | ) ( )p y p y p
Model
likelihood function p(y|)
prior distribution p()
Measurement data y
maximum a posteriori
(MAP) estimates
model evidence
Principles of Bayesian inference
Priors
Priors can be of different sorts, e.g.
• empirical (previous data)
• empirical (estimated from
current data using a hierarchical
model → "empirical Bayes")
• uninformed
• principled (e.g., positivity
constraints)
• shrinkage
Example of a shrinkage prior
Advantages of generative models
• describe how observed data were generated
by hidden mechanisms
• we can check in advance whether a model
can explain certain phenomena at all
• force us to think mechanistically and be
explicit about pathophysiological theories
• formal framework for differential
diagnosis:
statistical comparison of competing
generative models, each of which provides a
different explanation of measured brain
activity or clinical symptoms
mechanism 1 mechanism N...
data y
A generative modelling framework for fMRI & EEG:
Dynamic causal modeling (DCM)
Friston et al. 2003, NeuroImage
( , , )dx
f x udt
),(xgy
Model inversion:
Estimating neuronal
mechanisms
EEG, MEG fMRI
Forward model:
Predicting measured
activity
dwMRI
Stephan et al. 2009, NeuroImage
endogenous
connectivity
direct inputs
modulation of
connectivity
Neural state equation CuxBuAx j
j )( )(
u
xC
x
x
uB
x
xA
j
j
)(
hemodynamic
modelλ
x
y
integration
BOLDyyy
activity
x1(t)
activity
x2(t) activity
x3(t)
neuronal
states
t
driving
input u1(t)
modulatory
input u2(t)
t
endogenous
connectivity
direct inputs
modulation of
connectivity
Neuronal state equationModulatory input
t
u2(t)
t
Driving input
u1(t)
𝑨 =𝝏 𝒙
𝝏𝒙
𝑩(𝒋) =𝝏
𝝏𝒖𝒋
𝝏 𝒙
𝝏𝒙
𝑪 =𝝏 𝒙
𝝏𝒖
𝒙 = 𝑨 + 𝒖𝒋𝑩𝒋 𝒙 + 𝑪𝒖
Hemodynamic model
𝝂𝒊(𝒕) and 𝒒𝒊(𝒕)
Neuronal states
𝒙𝒊(𝒕)
𝒙𝟏(𝒕)𝒙𝟑(𝒕)
𝒙𝟐(𝒕)
BOLD signal change equation
𝒚 = 𝑽𝟎 𝒌𝟏 𝟏 − 𝒒 + 𝒌𝟐 𝟏 −𝒒
𝝂+ 𝒌𝟑 𝟏 − 𝝂 + 𝒆
with 𝒌𝟏 = 𝟒. 𝟑𝝑𝟎𝑬𝟎𝑻𝑬, 𝒌𝟐 = 𝜺𝒓𝟎𝑬𝟎𝑻𝑬, 𝒌𝟑 = 𝟏 − 𝜺
𝝉 𝝂 = 𝒇 − 𝝂𝟏/𝜶
𝝉 𝒒 = 𝒇𝑬(𝒇, 𝑬𝟎)/𝑬𝟎 − 𝝂𝟏/𝜶𝒒/𝝂
Local hemodynamic
state equations
Changes in volume (𝝂)
and dHb (𝒒)
𝒇Balloon model
𝒔 = 𝒙 − 𝜿𝒔 − 𝜸 𝒇 − 𝟏
𝒇 = 𝒔
vasodilatory
signal and flow
induction (rCBF)
BOLD signal
y(t)Stephan et al. 2015,
Neuron
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
Neural population activity
0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
0 10 20 30 40 50 60 70 80 90 100-1
0
1
2
3
4
0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
fMRI signal change (%)
x1 x2
x3
CuxDxBuAdt
dx n
j
j
j
m
i
i
i
1
)(
1
)(
Nonlinear Dynamic Causal Model for fMRI
Stephan et al. 2008, NeuroImage
u1
u2
Why should I know about Bayesian inference?
Because Bayesian principles are fundamental for
• statistical inference in general
• system identification
• translational neuromodeling ("computational assays")
– computational psychiatry
– computational neurology
• contemporary theories of brain function (the "Bayesian brain")
– predictive coding
– free energy principle
– active inference
Generative models as "computational assays"
( | , )p y m
( | , )p y m ( | )p m
( | , )p y m
( | , )p y m ( | )p m
Differential diagnosis based on generative models of
disease symptoms
SYMPTOM
(behaviour
or physiology)
HYPOTHETICAL
MECHANISM...
( | )kp m y( | , )kp y m
y
1m Kmkm ...
( | ) ( )( | y)
( | ) ( )
k kk
k k
k
p y m p mp m
p y m p m
Stephan et al. 2016, NeuroImage
Application to brain activity and
behaviour of individual patients
Computational assays:
Models of disease mechanisms
Detecting physiological subgroups
(based on inferred mechanisms)
Translational Neuromodeling
Individual treatment prediction
disease mechanism A
disease mechanism B
disease mechanism C
( , , )dx
f x udt
Stephan et al. 2015, Neuron
Perception = inversion of a hierarchical generative model
environm. states
others' mental states
bodily states
( | , )p x y m
( | , ) ( | )p y x m p x mforward model
perception
neuronal states
Example: free-energy principle and active inference
Change
sensory input
sensations – predictions
Prediction error
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
How is the posterior computed =
how is a generative model inverted?
Bayesian Inference
Approximate Inference
Variational Bayes
MCMC Sampling
Analytical solutions
How is the posterior computed =
how is a generative model inverted?
• compute the posterior analytically
– requires conjugate priors
– even then often difficult to derive an analytical solution
• variational Bayes (VB)
– often hard work to derive, but fast to compute
– cave: local minima, potentially inaccurate approximations
• sampling methods (MCMC)
– guaranteed to be accurate in theory (for infinite computation time)
– but may require very long run time in practice
– convergence difficult to prove
Conjugate priors
If the posterior p(θ|x) is in the same family as the prior p(θ), the prior and
posterior are called "conjugate distributions", and the prior is called a "conjugate
prior" for the likelihood function.
)(
)( )|()|(
yp
pypyp
same form analytical expression for posterior
examples (likelihood-prior):
• Normal-Normal
• Normal-inverse Gamma
• Binomial-Beta
• Multinomial-Dirichlet
Likelihood & Prior
Posterior:
Posterior mean =
variance-weighted combination of
prior mean and data mean
Prior
Likelihood
Posterior
y
Posterior mean & variance of univariate Gaussians
p
2
2
( | ) ( , )
( ) ( , )
e
p p
p y N
p N
2( | ) ( , )p y N
p
pe
pe
22
2
222
11
111
Likelihood & prior
Posterior:
Prior
Likelihood
Posterior
Same thing – but expressed as precision weighting
p
1
1
( | ) ( , )
( ) ( , )
e
p p
p y N
p N
1( | ) ( , )p y N
p
pe
pe
Relative precision weighting
y
Variational Bayes (VB)
best proxy
𝑞 𝜃
trueposterior
𝑝 𝜃 𝑦
hypothesisclass
divergence
KL 𝑞||𝑝
Idea: find an approximate density 𝑞(𝜃) that is maximally similar to the true
posterior 𝑝 𝜃 𝑦 .
This is often done by assuming a particular form for 𝑞 (fixed form VB) and
then optimizing its sufficient statistics.
Kullback–Leibler (KL) divergence
• non-symmetric measure
of the difference
between two probability
distributions P and Q
• DKL(P‖Q) = a measure of
the information lost when
Q is used to approximate
P: the expected number
of extra bits required to
code samples from P
when using a code
optimized for Q, rather
than using the true code
optimized for P.
Variational calculus
Standard calculusNewton, Leibniz, and
others
• functions
𝑓: 𝑥 ↦ 𝑓 𝑥
• derivatives d𝑓d𝑥
Example: maximize
the likelihood
expression 𝑝 𝑦 𝜃w.r.t. 𝜃
Variational
calculusEuler, Lagrange, and
others
• functionals
𝐹: 𝑓 ↦ 𝐹 𝑓
• derivatives d𝐹d𝑓
Example: maximize
the entropy 𝐻 𝑝w.r.t. a probability
distribution 𝑝 𝑥
Leonhard Euler(1707 – 1783)
Swiss mathematician, ‘Elementa Calculi
Variationum’
Variational Bayes
𝐹 𝑞 is a functional wrt. the
approximate posterior 𝑞 𝜃 .
Maximizing 𝐹 𝑞, 𝑦 is equivalent to:
• minimizing KL[𝑞| 𝑝
• tightening 𝐹 𝑞, 𝑦 as a lower
bound to the log model evidence
When 𝐹 𝑞, 𝑦 is maximized, 𝑞 𝜃 is
our best estimate of the posterior.
ln𝑝(𝑦) = KL[𝑞| 𝑝 + 𝐹 𝑞, 𝑦
divergence ≥ 0
(unknown)
neg. free energy
(easy to evaluate for a given 𝑞)
KL[𝑞| 𝑝
ln 𝑝 𝑦 ∗
𝐹 𝑞, 𝑦
KL[𝑞| 𝑝
ln 𝑝 𝑦
𝐹 𝑞, 𝑦
initialization …
… convergence
Derivation of the (negative) free energy approximation
• See whiteboard!
• (or Appendix to Stephan et al. 2007, NeuroImage 38: 387-401)
Mean field assumption
Factorize the approximate
posterior 𝑞 𝜃 into independent
partitions:
𝑞 𝜃 =
𝑖
𝑞𝑖 𝜃𝑖
where 𝑞𝑖 𝜃𝑖 is the approximate
posterior for the 𝑖th subset of
parameters.
For example, split parameters
and hyperparameters:𝜃1
𝜃2
𝑞 𝜃1 𝑞 𝜃2
Jean Daunizeau, www.fil.ion.ucl.ac.uk/ ~jdaunize/presentations/Bayes2.pdf , | ,p y q q q
VB in a nutshell (under mean-field approximation)
, | ,p y q q q
( )
( )
exp exp ln , ,
exp exp ln , ,
q
q
q I p y
q I p y
Iterative updating of sufficient statistics of approx. posteriors by
gradient ascent.
ln | , , , |
ln , , , , , |q
p y m F KL q p y
F p y KL q p m
Mean field approx.
Neg. free-energy
approx. to model
evidence.
Maximise neg. free
energy wrt. q =
minimise divergence,
by maximising
variational energies
VB (under mean-field assumption) in more detail
VB (under mean-field assumption) in more detail
Model comparison and selection
Given competing hypotheses
on structure & functional
mechanisms of a system, which
model is the best?
For which model m does p(y|m)
become maximal?
Which model represents the
best balance between model
fit and model complexity?
Pitt & Miyung (2002) TICS
( | ) ( | , ) ( | ) p y m p y m p m d
Model evidence (marginal likelihood):
Various approximations, e.g.:
- negative free energy, AIC, BIC
Bayesian model selection (BMS)
accounts for both accuracy
and complexity of the model
“If I randomly sampled from my
prior and plugged the resulting
value into the likelihood
function, how close would the
predicted data be – on average
– to my observed data?”
all possible datasets
y
p(y|m)
Gharamani, 2004
McKay 1992, Neural Comput.
Penny et al. 2004a, NeuroImage
Model space M is defined by prior on models.
Usual choice: flat prior over a small set of models.
Model space (hypothesis set) M
1 1
( | ) ( ) ( | )( | )
( | ) ( ) ( | )
i i ii M M
j j j
j j
p y m p m p y mp m y
p y m p m p y m
1/ if ( )
0 if
M m Mp m
m M
In this case, the posterior probability of model i is:
Differential diagnosis based on generative models of
disease symptoms
SYMPTOM
(behaviour
or physiology)
HYPOTHETICAL
MECHANISM...
( | )kp m y( | , )kp y m
y
1m Kmkm ...
( | ) ( )( | y)
( | ) ( )
k kk
k k
k
p y m p mp m
p y m p m
Stephan et al. 2016, NeuroImage
pmypAIC ),|(log
Logarithm is a
monotonic function
Maximizing log model evidence
= Maximizing model evidence
)(),|(log
)()( )|(log
mcomplexitymyp
mcomplexitymaccuracymyp
Np
mypBIC log2
),|(log
Akaike Information Criterion:
Bayesian Information Criterion:
Log model evidence = balance between fit and complexity
Approximations to the model evidence
No. of
parameters
No. of
data points
The (negative) free energy approximation F
log ( | , ) , |
accuracy complexity
F p y m KL q p m
KL[𝑞| 𝑝
ln 𝑝 𝑦|𝑚
𝐹 𝑞, 𝑦 log ( | ) , | ,p y m F KL q p y m
Like AIC/BIC, F is an accuracy/complexity tradeoff:
F is a lower bound on the log model evidence:
The (negative) free energy approximation
• Log evidence is thus expected log likelihood (wrt. q) plus 2 KL's:
log ( | )
log ( | , ) , | , | ,
p y m
p y m KL q p m KL q p y m
log ( | ) , | ,
log ( | , ) , |
accuracy complexity
F p y m KL q p y m
p y m KL q p m
The complexity term in F
• In contrast to AIC & BIC, the complexity term of the negative free energy F accounts for parameter interdependencies. UnderGaussian assumptions about the posterior (Laplace approximation):
• 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
y
T
yy CCC
mpqKL
|
1
||2
1ln
2
1ln
2
1
)|(),(
Bayes factors
)|(
)|(
2
112
myp
mypB
positive value, [0;[
But: the log evidence is just some number – not very intuitive!
A more intuitive interpretation of model comparisons is made
possible by Bayes factors:
To compare two models, we could just compare their log
evidences.
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.
Fixed effects BMS at group level
Group Bayes factor (GBF) for 1...K subjects:
Average Bayes factor (ABF):
Problems:
- blind with regard to group heterogeneity
- sensitive to outliers
k
k
ijij BFGBF )(
( )kKij ij
k
ABF BF
)|(~ 111 mypy)|(~ 111 mypy
)|(~ 222 mypy)|(~ 111 mypy
)|(~ pmpm kk
);(~ rDirr
)|(~ pmpm kk)|(~ pmpm kk
),1;(~1 rmMultm
Random effects BMS for heterogeneous groups
Dirichlet parameters
= “occurrences” of models in the population
Dirichlet distribution of model probabilities r
Multinomial distribution of model labels m
Measured data y
Model inversion
by Variational
Bayes or MCMC
Stephan et al. 2009, NeuroImage
)|(~ 111 mypy)|(~ 111 mypy
)|(~ 222 mypy)|(~ 111 mypy
)|(~ pmpm kk
);(~ rDirr
)|(~ pmpm kk)|(~ pmpm kk
),1;(~1 rmMultm
Random effects BMS
Stephan et al. 2009, NeuroImage
11( | ) ,
( )
k
k
k
k k
kk
p r Dir r rZ
Z
( | ) nkm
n k
k
p m r r
( | ) ( | ) ( | )n nk nkp y m p y p m d
0
0
0
0
1
0
1
0
, , | | ( | )
( | ) | |
1|
1
k nk
nkk
n n n
n
m
k n n k
k n k
m
n nk k k
n k
p y r m p y m p m r p r
p r p y m p m r
r p y m rZ
p y m r rZ
Write down joint probability
and take the log
0 0ln , , ln 1 ln log | lnk k nk n nk k
n k
p y r m Z r m p y m r
Maximise neg. free
energy wrt. q =
minimise divergence,
by maximising
variational energies
Mean field approx. ( , ) ( ) ( )q r m q r q m
( )
( )
( ) exp
( ) exp
log , ,
log , ,
q m
q r
q r I r
q m I m
I r p y r m
I m p y r m
Iterative updating of sufficient statistics of approx.
posteriors
Until convergence
end
0 0 [1, ,1]
0
exp ln ( | )nk n nk k k
k
nknk
nk
k
k nk
n
u p y m
ug
u
g
( 1)nk nkg q m our (normalized)
posterior belief that
model k generated the
data from subject n
k nk
n
g expected number of
subjects whose data we
believe were generated
by model k
Four equivalent options for reporting model ranking by
random effects BMS
1. Dirichlet parameter estimates
2. expected posterior probability of
obtaining the k-th model for any
randomly selected subject
3. exceedance probability that a
particular model k is more likely than
any other model (of the K models
tested), given the group data
4. protected exceedance probability:
see below
)( 1 Kkqkr
{1... }, {1... | }:
( | ; )k k j
k K j K j k
p r r y
-35 -30 -25 -20 -15 -10 -5 0 5
Su
bje
cts
Log model evidence differences
MOG
LG LG
RVF
stim.
LVF
stim.
FGFG
LD|RVF
LD|LVF
LD LD
MOGMOG
LG LG
RVF
stim.
LVF
stim.
FGFG
LD
LD
LD|RVF LD|LVF
MOG
m2 m1
m1m2
Data: Stephan et al. 2003, Science
Models: Stephan et al. 2007, J. Neurosci.
Example: Hemispheric interactions during vision
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
m1m2
1
1
11.8
84.3%r
2
2
2.2
15.7%r
%7.9921 rrp
Stephan et al. 2009a, NeuroImage
Example: Synaesthesia
• “projectors” experience
color externally colocalized
with a presented grapheme
• “associators” report an
internally evoked
association
• across all subjects: no
evidence for either model
• but BMS results map
precisely onto projectors
(bottom-up mechanisms)
and associators (top-down)
van Leeuwen et al. 2011, J. Neurosci.
Overfitting at the level of models
• #models risk of overfitting
• solutions:
– regularisation: definition of model
space = choosing priors p(m)
– family-level BMS
– Bayesian model averaging (BMA)
|
| , |m
p y
p y m p m y
|
| ( )
| ( )m
p m y
p y m p m
p y m p m
posterior model probability:
BMA:
Model space partitioning: comparing model families
• partitioning model space into K subsets
or families:
• pooling information over all models in
these subsets allows one to compute
the probability of a model family, given
the data
• effectively removes uncertainty about
any aspect of model structure, other
than the attribute of interest (which
defines the partition)
Stephan et al. 2009, NeuroImage
Penny et al. 2010, PLoS Comput. Biol.
1,..., KM f f
kp f
Family-level inference: fixed effects
• We wish to have a uniform prior at the
family level:
• This is related to the model level via
the sum of the priors on models:
• Hence the uniform prior at the family
level is:
• The probability of each family is then
obtained by summing the posterior
probabilities of the models it includes:
Penny et al. 2010, PLoS Comput. Biol.
1
kp fK
( )k
k
m f
p f p m
1
:k
k
m f p mK f
1.. 1..| |k
k N N
m f
p f y p m y
Family-level inference: random effects
• The frequency of a family in the
population is given by:
• In RFX-BMS, this follows a Dirichlet
distribution, with a uniform prior on the
parameters (see above).
• A uniform prior over family
probabilities can be obtained by
setting:
Stephan et al. 2009, NeuroImage
Penny et al. 2010, PLoS Comput. Biol.
k
k m
m f
s r
1: ( )k prior
k
m f mf
( )p s Dir
Family-level inference: random effects – a special case
• When the families are of equal size, one can simply sum the posterior model
probabilities within families by exploiting the agglomerative property of the
Dirichlet distribution:
Stephan et al. 2009, NeuroImage
1 2
1 2
1 2 1 2
* * *
1 2
* * *
1 2
, ,..., ~ , ,...,
, ,...,
~ , ,...,
J
J
K K
k k J k
k N k N k N
k k J k
k N k N k N
r r r Dir
r r r r r r
Dir
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) = 0.986
Model space partitioning:
comparing model families
0
2
4
6
8
10
12
14
16
alp
ha
0
2
4
6
8
10
12
alp
ha
0
20
40
60
80
Su
mm
ed
lo
g e
vid
en
ce
(re
l. t
o R
BM
L)
CBMNCBMN(ε) CBML CBML(ε)RBMN
RBMN(ε) RBML RBML(ε)
CBMNCBMN(ε) CBML CBML(ε)RBMN
RBMN(ε) RBML RBML(ε)
nonlinear models linear models
FFX
RFX
4
1
*
1
k
k
8
5
*
2
k
k
nonlinear linear
log
GBF
Model
space
partitioning
1 73.5%r 2 26.5%r
1 2 98.6%p r r
m1m2
m1m2
Stephan et al. 2009, NeuroImage
MMN
A1 A1
STG
input
STG
IFG
prediction
prediction
error
input
standardsdeviants
Mismatch negativity (MMN)
Garrido et al. 2009, Clin. Neurophysiol.
• elicited by surprising stimuli
(scales with unpredictability)
• in schizophrenic patients
• classical interpretations:
– pre-attentive change
detection
– neuronal adaptation
• current theories:
– reflection of (hierarchical)
Bayesian inference
• in schizophrenic patients
• Highly relevant for
computational assays of
glutamatergic and cholinergic
transmission:
+ NMDAR
+ ACh (nicotinic &
muscarinic)
– 5HT
– DA
standards
deviantsMismatch negativity
(MMN)
placebo
ketamine
MMN
Schmidt, Diaconescu et al. 2013, Cereb. Cortex
ERPs
MMN
Lieder et al. 2013, PLoS Comput. Biol.
Modelling Trial-by-Trial Changes of the Mismatch
Negativity (MMN)
Lieder et al. 2013, PLoS Comput. Biol.
MMN model comparison
at multiple levels
Comparing
individual
models
Comparing
MMN
theories
Comparing
modeling
frameworks
Bayesian Model Averaging (BMA)
• abandons dependence of parameter
inference on a single model and takes into
account model uncertainty
• uses the entire model space considered (or
an optimal family of models)
• averages parameter estimates, weighted
by posterior model probabilities
• represents a particularly useful alternative
– when none of the models (or model
subspaces) considered clearly
outperforms all others
– when comparing groups for which the
optimal model differs
|
| , |m
p y
p y m p m y
Penny et al. 2010, PLoS Comput. Biol.
1..
1..
|
| , |
n N
n n N
m
p y
p y m p m y
NB: p(m|y1..N) can be obtained
by either FFX or RFX BMS
single-subject BMA:
group-level BMA:
Schmidt et al. 2013, JAMA Psychiatry
Prefrontal-parietal connectivity during
working memory in schizophrenia
• 17 at-risk mental
state (ARMS)
individuals
• 21 first-episode
patients
(13 non-treated)
• 20 controls
Schmidt et al. 2013, JAMA Psychiatry
BMS results for all groups
BMA results: PFC PPC connectivity
Schmidt et al. 2013, JAMA Psychiatry
17 ARMS, 21 first-episode (13 non-treated),
20 controls
Protected exceedance probability:
Using BMA to protect against chance findings
• EPs express our confidence that the posterior probabilities of models are
different – under the hypothesis H1 that models differ in probability: rk1/K
• does not account for possibility "null hypothesis" H0: rk=1/K
• Bayesian omnibus risk (BOR) of wrongly accepting H1 over H0:
• protected EP: Bayesian model averaging over H0 and H1:
Rigoux et al. 2014, NeuroImage
inference on model structure or inference on model parameters?
inference on
individual models or model space partition?
comparison of model
families using
FFX or RFX BMS
optimal model structure assumed
to be identical across subjects?
FFX BMS RFX BMS
yes no
inference on
parameters of an optimal model or parameters of all models?
BMA
definition of model space
FFX analysis of
parameter estimates
(e.g. BPA)
RFX analysis of
parameter estimates
(e.g. t-test, ANOVA)
optimal model structure assumed
to be identical across subjects?
FFX BMS
yes no
RFX BMS
Stephan et al. 2010, NeuroImage
Further reading
• Penny WD, Stephan KE, Mechelli A, Friston KJ (2004) Comparing dynamic causal models. NeuroImage
22:1157-1172.
• Penny WD, Stephan KE, Daunizeau J, Joao M, Friston K, Schofield T, Leff AP (2010) Comparing Families of
Dynamic Causal Models. PLoS Computational Biology 6: e1000709.
• Penny WD (2012) Comparing dynamic causal models using AIC, BIC and free energy. Neuroimage 59: 319-
330.
• Rigoux L, Stephan KE, Friston KJ, Daunizeau J (2014) Bayesian model selection for group studies – revisited.
NeuroImage 84: 971-985.
• Stephan KE, Weiskopf N, Drysdale PM, Robinson PA, Friston KJ (2007) Comparing hemodynamic models with
DCM. NeuroImage 38:387-401.
• Stephan KE, Penny WD, Daunizeau J, Moran RJ, Friston KJ (2009) Bayesian model selection for group
studies. NeuroImage 46:1004-1017.
• Stephan KE, Penny WD, Moran RJ, den Ouden HEM, Daunizeau J, Friston KJ (2010) Ten simple rules for
Dynamic Causal Modelling. NeuroImage 49: 3099-3109.
• Stephan KE, Iglesias S, Heinzle J, Diaconescu AO (2015) Translational Perspectives for Computational
Neuroimaging. Neuron 87: 716-732.
• Stephan KE, Schlagenhauf F, Huys QJM, Raman S, Aponte EA, Brodersen KH, Rigoux L, Moran RJ,
Daunizeau J, Dolan RJ, Friston KJ, Heinz A (2016) Computational Neuroimaging Strategies for Single Patient
Predictions. NeuroImage, in press. DOI: 10.1016/j.neuroimage.2016.06.038
Thank you
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