Bayesian Inference. Thomas Nichols. With thanks Lee Harrison

Transcription

1 Bayesian Inference Thomas Nichols With thanks Lee Harrison

2

3 Attention to Motion Paradigm Results Attention No attention Büchel & Friston 1997, Cereb. Cortex Büchel et al. 1998, Brain - fixation only - observe static dots + photic V1 - observe moving dots + motion V5 - task on moving dots + attention V5 + parietal cortex

4 Attention to Motion Paradigm Dynamic Causal Models Model 1 (forward): attentional modulation of V1 V5: forward Model 2 (backward): attentional modulation of SPC V5: backward - fixation only - observe static dots - observe moving dots - task on moving dots

5 Responses to Uncertainty Long term memory Short term memory

6 Responses to Uncertainty Paradigm Stimuli sequence of randomly sampled discrete events Model simple computational model of an observers response to uncertainty based on the number of past events (extent of memory) Question which regions are best explained by short / long term memory model? trials??

7 Overview Introductory remarks Some probability densities/distributions Probabilistic (generative) models Bayesian inference A simple example Bayesian linear regression SPM applications Segmentation Dynamic causal modeling Spatial models of fmri time series

8 Probability distributions and densities k=2

9 Probability distributions and densities k=2

10 Probability distributions and densities k=2

11 Probability distributions and densities k=2

12 Probability distributions and densities k=2

13 Probability distributions and densities k=2

14 Probability distributions and densities k=2

15 Generative models estimation space space generation time

16 Bayesian statistics

17 Bayes rule θ θ

18 Principles of Bayesian inference likelihood p(y θ) prior distribution p(θ) y

19 Univariate Gaussian

20 Bayesian GLM: univariate case

21 Bayesian GLM: multivariate case β 2 β 1

22 Approximate inference: optimization mean-field approximation True posterior iteratively improve Approximate posterior free energy Objective function Value of parameter

23 Simple example linear regression Data Ordinary least squares

24 Simple example linear regression Data and model fit Ordinary least squares Bases (explanatory variables) Sum of squared errors

25 Simple example linear regression Data and model fit Ordinary least squares Bases (explanatory variables) Sum of squared errors

26 Simple example linear regression Data and model fit Ordinary least squares Bases (explanatory variables) Sum of squared errors

27 Simple example linear regression Data and model fit Ordinary least squares Over-fitting: model fits noise Inadequate cost function: blind to overly complex models Solution: include uncertainty in model parameters Bases (explanatory variables) Sum of squared errors

28 Bayesian linear regression: priors and likelihood Model:

29 Bayesian linear regression: priors and likelihood Model: Prior:

30 Bayesian linear regression: priors and likelihood Model: Prior: Sample curves from prior (before observing any data) Mean curve

31 Bayesian linear regression: priors and likelihood Model: Prior: Likelihood:

32 Bayesian linear regression: priors and likelihood Model: Prior: Likelihood:

33 Bayesian linear regression: priors and likelihood Model: Prior: Likelihood:

34 Bayesian linear regression: posterior Model: Prior: Likelihood: Bayes Rule:

35 Bayesian linear regression: posterior Model: Prior: Likelihood: Bayes Rule: Posterior:

36 Bayesian linear regression: posterior Model: Prior: Likelihood: Bayes Rule: Posterior:

37 Bayesian linear regression: posterior Model: Prior: Likelihood: Bayes Rule: Posterior:

38 Posterior Probability Maps (PPMs) s th y s th p th

39 Bayesian linear regression: model selection Bayes Rule: Model evidence: normalizing constant

40 amri segmentation PPM of belonging to grey matter white matter CSF

41 Dynamic Causal Modelling: generative model for fmri and ERPs

42 Bayesian Model Selection for fmri [Stephan et al., Neuroimage, 2008]

43 fmri time series analysis with spatial priors degree of smoothness Spatial precision matrix amri Smooth Y (RFT) prior precision of GLM coeff prior precision of AR coeff prior precision of data noise GLM coeff AR coeff (correlated noise) ML estimate of β VB estimate of β observations Penny et al 2005

44 fmri time series analysis with spatial priors: posterior probability maps activation threshold Display only voxels that exceed e.g. 95% Probability mass p n Posterior density q(β n ) probability of getting an effect, given the data mean: size of effect covariance: uncertainty

45 fmri time series analysis with spatial priors: Bayesian model selection Log-evidence maps model 1 subject 1 subject N model K Compute log-evidence for each model/subject

46 fmri time series analysis with spatial priors: Bayesian model selection Log-evidence maps BMS maps model 1 subject 1 subject N PPM model K EPM Compute log-evidence for each model/subject Probability that model k generated data model k Joao et al, 2009

47 Reminder Long term memory Short term memory

48 Compare two models Short-term memory model long-term memory model onsets IT indices: H,h,I,i Missed trials IT indices are smoother H=entropy; h=surprise; I=mutual information; i=mutual surprise

49 Group data: Bayesian Model Selection maps Regions best explained by shortterm memory model Regions best explained by long-term memory model primary visual cortex frontal cortex (executive control)

50 Thank-you