An introduction to Bayesian inference and model comparison J. Daunizeau

Transcription

1 An introduction to Bayesian inference and model comparison J. Daunizeau ICM, Paris, France TNU, Zurich, Switzerland

2 Overview of the talk An introduction to probabilistic modelling Bayesian model comparison SPM applications

3 Overview of the talk An introduction to probabilistic modelling Bayesian model comparison SPM applications

4 Probability theory: basics Degree of plausibility desiderata: - should be represented using real numbers (D1) - should conform with intuition (D2) - should be consistent (D3) normalization: a=2 marginalization: a=2 b=5 conditioning : (Bayes rule)

5 Deriving the likelihood function - Model of data with unknown parameters: y f e.g., GLM: f X - But data is noisy: y f - Assume noise/residuals is small : f exp p P Distribution of data, given fixed parameters: p y y f exp 2

6 Forward and inverse problems forward problem p y, m likelihood model posterior distribution p y, m data inverse problem

7 Likelihood, priors and the model evidence Likelihood: Prior: generative model m Bayes rule:

8 Bayesian model comparison Principle of parsimony : «plurality should not be assumed without necessity» Model evidence: y=f(x) y = f(x) x model evidence p(y m) Occam s razor : space of all data sets

9 Hierarchical models inference causality

10 Directed acyclic graphs (DAGs)

11 Variational approximations (VB, EM, ReML) ln p y m ln p y, m S q KL p y, m ; q q Free energy F q VB : maximize the free energy F(q) w.r.t. the approximate posterior q(θ) under some (e.g., mean field, Laplace) simplifying constraint 2 1 p p, y, m or 2, q 1 or 2 y m

12 Overview of the talk An introduction to probabilistic modelling Bayesian model comparison SPM applications

13 Frequentist versus Bayesian inference define the null, e.g.: pth 0 H 0 : 0 Ptt* H 0 define two alternative models, e.g.: m : p m 0 0 m : p m N 0, if 0 0 otherwise pym 0 pym 1 if 0 t * Ptt* H t t Y estimate parameters (obtain test stat.) apply decision rule, i.e.: then reject H0 classical (null) hypothesis testing apply decision rule, e.g.: y Bayesian Model Comparison Y space of all datasets P m0 y if then accept m 0 P m y 1

14 Family-level inference P(m 1 y) = 0.04 P(m 2 y) = 0.25 A B A B model selection error risk: 1 1max P e y P m y 0.3 m P(m 2 y) = 0.01 P(m 2 y) = 0.7 A B A B u u

15 Family-level inference P(m 1 y) = 0.04 P(m 2 y) = 0.25 A B A B model selection error risk: 1 1max P e y P m y 0.3 m P(m 2 y) = 0.01 P(m 2 y) = 0.7 family inference (pool statistical evidence) A B A B Pm y P f y m f u P(f 1 y) = 0.05 u P(f 2 y) = max P e y P f y 0.05 f

16 Sampling subjects as marbles in an urn m m r i i 1 0 i th marble is blue i th marble is purple = proportion of blue marbles in the urn r (binomial) probability of drawing a set of n marbles: m1 m2 mn n m 1 1 i p m r r r i1 m i Thus, our belief about the proportion of blue marbles is: i1 1 n pr n m 1m 1 i i pr m prr 1r Er m m n i1 i

17 Group-level model comparison At least, we can measure how likely is the i th subject s data under each model! p y1 m1 p y2 m2 p yi mi p yn mn r n, i i i p r m y p r p y m p m r i1 m1 m2 y 1 y 2 mn y n Our belief about the proportion of models is: pr, m y p r y m Exceedance probability: k Prk rk' k y

18 Overview of the talk An introduction to probabilistic modelling Bayesian model comparison SPM applications

19 segmentation and normalisation posterior probability maps (PPMs) dynamic causal modelling multivariate decoding realignment smoothing general linear model normalisation statistical inference p <0.05 Gaussian field theory template

20 class variances amri segmentation mixture of Gaussians (MoG) model 1 2 k 1 2 i th voxel label y i c i i th voxel value class frequencies k class means grey matter white matter CSF

21 fixation cross + Decoding of brain images recognizing brain states from fmri pace >> response log-evidence of X-Y sparse mappings: effect of lateralization log-evidence of X-Y bilateral mappings: effect of spatial deployment

22 fmri time series analysis spatial priors and model comparison short-term memory design matrix (X) PPM: regions best explained by short-term memory model prior variance of data noise GLM coeff prior variance of GLM coeff AR coeff (correlated noise) long-term memory design matrix (X) PPM: regions best explained by long-term memory model fmri time series

23 Dynamic Causal Modelling network structure identification m 1 m 2 m 3 m 4 attention attention attention attention PPC PPC PPC PPC stim V1 V5 stim V1 V5 stim V1 V5 stim V1 V models marginal likelihood ln p y m attention 0.10 PPC estimated effective synaptic strengths for best model (m 4 ) 5 stim 0.26 V V5 0 m 1 m 2 m 3 m 4

24 SPM: frequentist vs Bayesian RFX analysis 0? parameter estimates p 0.05 subjects p p p 0.05

25 I thank you for your attention.

26 A note on statistical significance lessons from the Neyman-Pearson lemma Neyman-Pearson lemma: the likelihood ratio (or Bayes factor) test p y H p y H is the most powerful test of size 1 0 u p u H 0 to test the null. what is the threshold u, above which the Bayes factor test yields a error I rate of 5%? ROC analysis 1 - error II rate MVB (Bayes factor) u=1.09, power=56% CCA (F-statistics) F=2.20, power=20% error I rate