Slide 1 Wellcome Trust Centre for Neuroimaging UCL (PowerPoint)

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Slide 1 Wellcome Trust Centre for Neuroimaging UCL (PowerPoint) Powered By Docstoc
					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



                                                          statistical       Gaussian
                                                          inference        field theory
        normalisation
                                                                 p <0.05



                        template
Bayesian segmentation         Smoothness
  and normalisation            modelling




 realignment      smoothing        general linear model



                                                          statistical       Gaussian
                                                          inference        field theory
        normalisation
                                                                 p <0.05



                        template
Bayesian segmentation         Smoothness         Posterior probability
  and normalisation            estimation          maps (PPMs)




 realignment      smoothing         general linear model



                                                                statistical         Gaussian
                                                                inference          field theory
        normalisation
                                                                         p <0.05



                        template
Bayesian segmentation         Smoothness         Posterior probability        Dynamic Causal
  and normalisation            estimation          maps (PPMs)                  Modelling




 realignment      smoothing         general linear model



                                                                statistical         Gaussian
                                                                inference          field theory
        normalisation
                                                                         p <0.05



                        template
                         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
                         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
     General Linear Model

             Model:         y  X  e


X
          Prior

           Model:           y  X  e


2         Prior:   p(  2 )  N k (0,  2 1 I k )
                                          


                               exp( 2 T  / 2)



     1
                   Prior

                    Model:             y  X  e


     2
                    Prior:     p(  2 )  N k (0,  2 1 I k )
                                                     


                                          exp( 2 T  / 2)



              1

                             Sample curves from prior
                             (before observing any data)
Z
                             Mean curve



          x
                   Priors and likelihood

                            Model:                 y  X  e


     2
                            Prior:         p(  2 )  N k (0,  2 1 I k )
                                                                 


                                                      exp( 2 T  / 2)
                            Likelihood:
                                                       N
                                     p( y  , 1 )   p( yi |  , 11 )
              1                                       i 1

                                     p( yi  , 1 )  N ( X i , 11 )
                                                    exp(1 ( yi  X i ) 2 / 2)

X



          x
                   Priors and likelihood

                            Model:                 y  X  e


     2
                            Prior:         p(  2 )  N k (0,  2 1 I k )
                                                                 


                                                      exp( 2 T  / 2)
                            Likelihood:
                                                       N
                                     p( y  , 1 )   p( yi |  , 11 )
              1                                       i 1

                                     p( yi  , 1 )  N ( X i , 11 )
                                                    exp(1 ( yi  X i ) 2 / 2)

X



          x
          Posterior after one observation

                        Model:                      y  X  e


     2
                        Prior:           p(  2 )  N k (0,  2 1 I k )
                                                               


                                                      exp( 2 T  / 2)
                        Likelihood:
                                                       N
                                     p ( y  ,  1 )   p ( yi |  ,  1 )
              1                                      i 1


                        Bayes Rule:
                                     p( y, )  p( y |  , ) p( |  )

X
                        Posterior:       p | y,               N  , C 
                                                
                                         C  1 X T X   2 I k              1


                                           1CX T y
          x
          Posterior after two observations

                        Model:                      y  X  e


     2
                        Prior:           p(  2 )  N k (0,  2 1 I k )
                                                               


                                                      exp( 2 T  / 2)
                        Likelihood:
                                                       N
                                     p ( y  ,  1 )   p ( yi |  ,  1 )
              1                                      i 1


                        Bayes Rule:
                                     p( y, )  p( y |  , ) p( |  )

X
                        Posterior:       p | y,               N  , C 
                                                
                                         C  1 X T X   2 I k              1


                                           1CX T y
          x
          Posterior after eight observations

                         Model:                      y  X  e


     2
                         Prior:           p(  2 )  N k (0,  2 1 I k )
                                                                


                                                       exp( 2 T  / 2)
                         Likelihood:
                                                        N
                                      p ( y  ,  1 )   p ( yi |  ,  1 )
               1                                      i 1


                         Bayes Rule:
                                      p( y, )  p( y |  , ) p( |  )

X
                         Posterior:       p | y,               N  , C 
                                                 
                                          C  1 X T X   2 I k              1


                                            1CX T y
           x
                         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
SPM Interface
               Posterior Probability Maps

            Y  X                                    
                                            p   N 0,  2 1 L1
                                                           
                                                                        



                                                  prior precision   prior precision
     aMRI          Smooth Y (RFT)                  of GLM coeff      of AR coeff

                                                                       
                                    Observation
                                       noise

                                                     
                                                           GLM         A          AR coeff
                                                                              (correlated noise)



ML                 Bayesian                            Y    observations
              ROC curve

Sensitivity




                  1-Specificity
                         Posterior Probability Maps
                                                                             Display only voxels
                                                                            that exceed e.g. 95%
                                     activation
                                                                                   p  pth
                                     threshold
                                        sth

                                                    Probability mass p



 Mean (Cbeta_*.img)




                                        Posterior density                    PPM (spmP_*.img)

                         probability of getting an effect, given the data

                                     q( n )  N (  n ,  n )     mean: size of effect
Std dev (SDbeta_*.img)
                                                                   covariance: uncertainty
                         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
                Dynamic Causal Models

                     SPC    Posterior Density

    V1


                V5




Priors
Are
Physiological



                                V5->SPC
                         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
                             Model Evidence

Bayes Rule:
                  p ( y |  , m) p ( | m)
   p ( y, m) 
                           p ( y m)
                   normalizing constant




 Model evidence

p( y m)   p( y |  , m) p( | m)d
Model                                  Model                          Bayes factor:
Posterior        Evidence              Prior                     p(y | m  i)
                                                           Bij 
                                                                 p(y | m  j )
                   p ( y | m) p ( m)
       p(m | y ) 
                          p(y )



              Model, m=i                            Model, m=j

                           SPC                                  SPC


            V1                                 V1


                      V5                                   V5
Model                                  Model        Bayes factor:
Posterior       Evidence               Prior         p(y | m  i)
                                               Bij 
                                                     p(y | m  j )
                   p ( y | m) p ( m)
       p(m | y ) 
                          p(y )



  For
  Equal
  Model
  Priors
                         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
Bayes Factors versus p-values
           Two sample t-test




      Subjects




                 Conditions
                       p=0.05
Bayesian




                                BF=3




           Classical
Bayesian




                       BF=20
                       BF=3




           Classical
                       p=0.05
Bayesian




                                BF=20
                                BF=3




           Classical
Bayesian        p=0.01   p=0.05




                                  BF=20
                                  BF=3




           Classical
Model Evidence Revisited

             p( y m)   p( y |  , m) p( | m)d




            log p ( y | m) 
            accuracy(m)  complexity(m)



             accuracy(m)  1 y  Z  ...
                                             2



            complexity(m)   2   0            ...
                                             2
                         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
                          Free Energy Optimisation
                Initial
                 Point
Precisions, 




                                  Parameters, 
                         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
                           incorrect model (m2)                 correct model (m1)

                                     u2                                 u2


                                                x3                               x3



                           u1        x1              x2        u1       x1            x2



                                m2                                               m1
Simulated data sets




                      -5        -4   -3    -2         -1   0    1   2        3        4    5
                                          Log model evidence differences
                                                                                               Figure 2
                                           LD                                                      LD|LVF




                    MOG         FG              FG           MOG                     MOG     FG       FG            MOG

           LD|RVF                                                  LD|LVF           LD                                    LD


                                LG              LG                                           LG       LG



                          RVF        LD              LVF                                   RVF LD|RVF       LVF
                          stim.                      stim.                                 stim.            stim.




                                                m2                                            m1
Subjects




                                                                                             Models from
                                                                                             Klaas Stephan
           -35      -30   -25        -20        -15          -10    -5      0   5
                           Log model evidence differences
              Random Effects (RFX) Inference

                                            A         log p(yn|m)
                                                       log p(y|a)
    0.8

    0.6                            5


    0.4                 Subjects   10
r




    0.2                            15


     0                             20
          1 2 3 4 5 6                   1 2 3 4 5 6
            Models                        Models
                            Gibbs Sampling
                  Initial
                   Point


p(r | A, y )
 Frequencies, r




                                                Stochastic Method

                               Assignments, A               p( A | r , Y )
                                                          log p(yn|m)
                                                           log p(y|a)




                  u nm  exp[log p ( yn | m)  log rm ]
 m     anm
       0
       m
                            unm
              n   g nm                                         Gibbs
r  Dir( )                 unm'
                           m'
                                                                Sampling
   p(r | A, y )   an  Mult( g n )

                       p( A | r , Y )
                                           LD                                                      LD|LVF




                    MOG         FG              FG           MOG                     MOG     FG       FG            MOG

           LD|RVF                                                  LD|LVF           LD                                    LD


                                LG              LG                                           LG       LG



                          RVF        LD              LVF                                   RVF LD|RVF       LVF
                          stim.                      stim.                                 stim.            stim.




                                                m2                                            m1
Subjects




                                                                                             11/12=0.92

           -35      -30   -25        -20        -15          -10    -5      0   5
                           Log model evidence differences
                                   p(r >0.5 | y) = 0.997
                                         1
            5

           4.5

            4

           3.5

            3
p(r 1|y)




           2.5

            2

           1.5

            1

           0.5                                                     r1  0.843

            0
             0   0.1   0.2   0.3   0.4       0.5       0.6   0.7   0.8    0.9   1
                                               r
                                                   1
                         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
                              PPMs for Models
    log p( y m)  F (q)

    Log-evidence maps

               subject 1
model 1
                      subject N




model K




    Compute log-evidence
    for each model/subject
                               PPMs for Models
    log p( y m)  F (q)

    Log-evidence maps                                                         BMS maps

               subject 1
model 1
                      subject N
                                        q(rk  0.5)  0.941
                                                                     rk                PPM


                             q(rk )

model K
                                                                     k                EPM
                                                  rk
                                      Probability that model k
                                          generated data
    Compute log-evidence                                         Rosa et al Neuroimage, 2009
    for each model/subject
       Computational fMRI: Harrison et al (in prep)

                                                 Long
    Short                                        Time
 Time Scale                                      Scale




                                               Frontal cortex
   Primary
visual cortex
Non-nested versus nested comparison
                  For detecting model B:




Non-nested:

Compare model A
versus model B

Nested:

Compare model A
versus model AB




                                           Penny et al, HBM,2007
                Double Dissociations
                                         Long
    Short                                Time
 Time Scale                              Scale




                                       Frontal cortex
   Primary
visual cortex
                         Summary

• 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

				
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