Bayesian Model Comparison Will Penny London-Marseille Joint Meeting, Institut de Neurosciences Cognitive de la Mediterranee, Marseille, September 28-29, 2009 V1V5SPC V1 V5 SPC Wellcome Centre for Neuroimaging, UCL, UK.
Overview Priors, likelihoods and posteriors Model selection using evidence Model selection for groups Comparing model families
Overview Priors, likelihoods and posteriors Model selection using evidence Model selection for groups Comparing model families
Bayesian Paradigm: priors and likelihood Model:
Bayesian Paradigm: priors and likelihood Model: Prior:
Sample curves from prior (before observing any data) Mean curve Bayesian Paradigm: priors and likelihood Model: Prior:
Bayesian Paradigm: priors and likelihood Model: Prior: Likelihood:
Bayesian Paradigm: priors and likelihood Model: Prior: Likelihood:
Bayesian Paradigm: priors and likelihood Model: Prior: Likelihood:
Bayesian Paradigm: posterior Model: Prior: Likelihood: Bayes Rule: Posterior:
Bayesian Paradigm: posterior Model: Prior: Likelihood: Bayes Rule: Posterior:
Bayesian Paradigm: posterior Model: Prior: Likelihood: Bayes Rule: Posterior:
Overview Priors, likelihoods and posteriors Model selection using evidence Model selection for groups Comparing model families
Model Selection Cost function Bayes Rule: normalizing constant Model evidence:
V1 V5 SPC Model, m Parameters: Prior Posterior Likelihood Prior Posterior Evidence Parameter Model Second level of Bayesian Inference
Bayes Factors V1 V5 SPC Model, m=i V1 V5 SPC Model, m=j Model Evidences: Bayes factor: 1 to 3: Weak 3 to 20: Positive 20 to 100: Strong >100: Very Strong
Overview Priors, likelihoods and posteriors Model selection using evidence Dynamic Causal Models Model selection for groups Comparing model families
Single region u2u2 u1u1 z1z1 z2z2 z1z1 u1u1 a 11 c
Multiple regions u2u2 u1u1 z1z1 z2z2 z1z1 z2z2 u1u1 a 11 a 22 c a 21
Modulatory inputs u2u2 u1u1 z1z1 z2z2 u2u2 z1z1 z2z2 u1u1 a 11 a 22 c a 21 b 21
Reciprocal connections u2u2 u1u1 z1z1 z2z2 u2u2 z1z1 z2z2 u1u1 a 11 a 22 c a 12 a 21 b 21
Overview Priors, likelihoods and posteriors Model selection using evidence Dynamic Causal Models Model selection for groups Comparing model families
x1x1 x2x2 u1u1 x3x3 u2u2 x1x1 x2x2 u1u1 x3x3 u2u2 incorrect model (m 2 )correct model (m 1 ) Figure 2 m2m2 m1m1
MOG LG RVF stim. LVF stim. FG LD|RVF LD|LVF LD MOG LG RVF stim. LVF stim. FG LD LD|RVFLD|LVF MOG m2m2 m1m1 Models from Klaas Stephan
Random Effects Inference Different subjects can use different models. is the probability that model m is used in the population at large. We wish to make an inference about this.
Overview Priors, likelihoods and posteriors Model selection using evidence Model selection for groups Comparing model families
F A P DCM of Auditory Word Processing: Data from an fMRI study by Alex Leff and Tom Schofield P: Posterior STS A: Anterior STS F: 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 connectivity 7*64=448 possible networks 26*448=11,648 models in group of 26 subjects
Input families: Where does the input go ?
AFPAFPAPFPAF
F A P F A P F A P F A P (a) (d) (c) (b) Four of the top 16 models:
Bayesian Model Averaging
Same but now for RFX model probs p(m|Y)
F A P DCM of Auditory Word Processing: Data from an fMRI study by Alex Leff and Tom Schofield P: Posterior STS A: Anterior STS F: 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
Summary First and second levels of Bayesian inference Model selection for groups Comparing model families DCM for EEG-fMRI Thank-you !