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Bayesian Inference

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kengel
56 diapositives · 713 Views · Jan 06, 2020

Bayesian Inference. Will Penny. Wellcome Centre for Neuroimaging, UCL, UK. SPM for fMRI Course, London, October 21st, 2010. What is Bayesian Inference ?. (From Daniel Wolpert). Bayesian segmentation and normalisation. realignment. smoothing. general linear model. Gaussian

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Bayesian Inference
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  1. Bayesian Inference Will Penny Wellcome Centre for Neuroimaging, UCL, UK. SPM for fMRI Course, London, October 21st, 2010

  2. What is Bayesian Inference ? (From Daniel Wolpert)

  3. Bayesian segmentation and normalisation realignment smoothing general linear model Gaussian field theory statistical inference normalisation p <0.05 template

  4. Bayesian segmentation and normalisation Smoothness modelling realignment smoothing general linear model Gaussian field theory statistical inference normalisation p <0.05 template

  5. Bayesian segmentation and normalisation Smoothness estimation Posterior probability maps (PPMs) realignment smoothing general linear model Gaussian field theory statistical inference normalisation p <0.05 template

  6. Bayesian segmentation and normalisation Smoothness estimation Posterior probability maps (PPMs) Dynamic Causal Modelling realignment smoothing general linear model Gaussian field theory statistical inference normalisation p <0.05 template

  7. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

  8. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

  9. General Linear Model Model:

  10. Prior Model: Prior:

  11. Prior Model: Prior: Sample curves from prior (before observing any data) Mean curve

  12. Priors and likelihood Model: Prior: Likelihood:

  13. Priors and likelihood Model: Prior: Likelihood:

  14. Posterior after one observation Model: Prior: Likelihood: Bayes Rule: Posterior:

  15. Posterior after two observations Model: Prior: Likelihood: Bayes Rule: Posterior:

  16. Posterior after eight observations Model: Prior: Likelihood: Bayes Rule: Posterior:

  17. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

  18. SPM Interface

  19. q Smooth Y(RFT) prior precision of GLM coeff prior precision of AR coeff aMRI Observation noise GLM AR coeff (correlated noise) ML Bayesian observations Posterior Probability Maps

  20. ROC curve Sensitivity 1-Specificity

  21. Display only voxels that exceed e.g. 95% activation threshold Probability mass p Posterior density probability of getting an effect, given the data mean: size of effectcovariance: uncertainty Posterior Probability Maps Mean (Cbeta_*.img) PPM (spmP_*.img) Std dev (SDbeta_*.img)

  22. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

  23. SPC V1 V5 Dynamic Causal Models Posterior Density Priors Are Physiological V5->SPC

  24. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

  25. Bayes Rule: normalizing constant Model Evidence Model evidence

  26. SPC V1 V5 SPC V1 Model Model Evidence Prior Posterior V5 Bayes factor: Model, m=j Model, m=i

  27. Model Model Evidence Prior Posterior Bayes factor: For Equal Model Priors

  28. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

  29. Bayes Factors versus p-values Two sample t-test Subjects Conditions

  30. p=0.05 Bayesian BF=3 Classical

  31. BF=20 Bayesian BF=3 Classical

  32. p=0.05 BF=20 Bayesian BF=3 Classical

  33. p=0.01 p=0.05 BF=20 Bayesian BF=3 Classical

  34. Model Evidence Revisited

  35. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

  36. Free Energy Optimisation Initial Point Precisions, a Parameters, q

  37. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

  38. u2 u2 x3 x3 x2 x2 x1 x1 u1 u1 incorrect model (m2) correct model (m1) m2 m1 Figure 2

  39. LD LD|LVF LD|RVF LD|LVF LD LD RVF stim. LD LVF stim. RVF stim. LD|RVF LVF stim. MOG MOG MOG MOG LG LG LG LG FG FG FG FG m2 m1 Models from Klaas Stephan

  40. Random Effects (RFX) Inference log p(yn|m)

  41. Gibbs Sampling Initial Point Frequencies, r Stochastic Method Assignments, A

  42. log p(yn|m) Gibbs Sampling

  43. LD LD|LVF LD|RVF LD|LVF LD LD RVF stim. LD LVF stim. RVF stim. LD|RVF LVF stim. MOG MOG MOG MOG LG LG LG LG FG FG FG FG m2 m1 11/12=0.92

  44. Overview • Parameter Inference • GLMs, PPMs, DCMs • Model Inference • Model Evidence, Bayes factors (cf. p-values) • Model Estimation • Variational Bayes • Groups of subjects • RFX model inference, PPM model inference

  45. Log-evidence maps subject 1 model 1 subject N model K Compute log-evidence for each model/subject PPMs for Models