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					                           Variational filtering and DEM
                                    EPSRC Symposium Workshop on
                                  Computational Neuroscience
                                 Monday 8 – Thursday 11, December 2008



                                              Abstract

This presentation reviews variational treatments of dynamic models that furnish time-dependent
conditional densities on the path or trajectory of a system's states and the time-independent
densities of its parameters. These obtain by maximizing a variational action with respect to
conditional densities. The action or path-integral of free-energy represents a lower-bound on the
model’s log-evidence or marginal likelihood required for model selection and averaging. This
approach rests on formulating the optimization in generalized coordinates of motion. The
resulting scheme can be used for online Bayesian inversion of nonlinear hierarchical dynamic
causal models and is shown to outperform existing approaches, such as Kalman and particle
filtering. Furthermore, it provides for multiple inference on a models states, parameters and
hyperparameters using exactly the same principles. Free-form (Variational filtering) and fixed
form (Dynamic Expectation Maximization) variants of the scheme will be demonstrated using
simulated (bird-song) and real data (from hemodynamic systems studied in neuroimaging).
                                             Overview

  Vu  D
                          Hierarchical dynamic models
                          Generalised coordinates (dynamical priors)
                          Hierarchal forms (structural priors)
                          Variational filtering and action (free-form)                x(t )
                          Laplace approximation and DEM (fixed-form)
                          Comparative evaluations
                          Examples (Hemodynamics and Bird songs)


                                                                                                             t



                     dx                          f
              x :         x : x[1]   f x :       x f    x : x, x, x, x           x
                     dt                          x
                                                                                                  x
                                                                                                       x
                Generalised coordinates
                               y  g ( x, v )  z
                               x  f ( x, v )  w
                               

    y  g ( x, v )  z                          x  f ( x, v )  w
    y   g x x  g v v  z                 x  f x x  f v v  w
   y  g x x  g v v  z               x  f x x  f v v  w
                                                                                  0 1    
                                                                                   0  
    ygz                                     Dx  f  w                        D       I
                                                                                       1
                                                                                         
   g  g ( x, v )                               f  f ( x, v )                          0
   g   g x x  g v v                       f   f x x  f v v
   g   g x x  g v v                    f   f x x  f v v



         Likelihood                              (Dynamical) prior
p ( y | x, v )  N ( g ,  z )              p( Dx | v )  N ( f ,  w )
                             Energies and generalised precisions

                 Instantaneous energy
                                                                                                S ( )   ( )
U (t ) : ln p  y, u   ln p  y | x, v  p  x | v  p  v 
                                                                                   1        0     (0)      1      0       1      
                    ln     
                      1
                      2
                                  1
                                  2
                                          T
                                                                                   0       (0)   0        0      1
                                                                                                                          
                                                                                                                               2

                                                                                                                                  0    
                                                                      S ( ) 1                                  2                
                                                                                    (0)    0     (0)      1    0       3
                                                                                                                                  2   
                                                                                                                                     
                                                                                                                2             4
                                                                                  
                                                                                                                                    
         z                  v  y  g 
                         x          
                     w     
                                Dx  f                                             General and Gaussian forms




                            Precision matrices in generalised coordinates and time
                            2                                     2

                            4                                     4

                            6                                     6

                            8                                     8

                           10                                 10

                           12                                 12


                                        S ( )
                           14                                 14

                           16                                 16      E 1T E 1
                                      5       10   15                 5       10       15
                                                                                                                      y  Ey(t )
                                   Hierarchical dynamic models

    y  g ( x, v)  z    y ~ z
                         ~  g ~                                 v (i 1)  g ( x (i ) , v (i ) )  z (i )         ~          ~
                                                                                                                    v (i 1)  g (i )  ~ (i )
                                                                                                                                        z
                            ~ ~                                                                                              ~
    x  f ( x, v)  w
                       D~  f  w
                         x                                          x (i )  f ( x (i ) , v (i ) )  w(i )
                                                                                                                   D~ (i )  f (i )  w(i )
                                                                                                                      x                 ~




                                                                                                          ~
                                                                                                          v ( 2)                                                    ~ ~
                                                                                                                     p (v ( 2 ) | u ( 3 ) ,  ( 3) )  N (v ( 2 ) : g ( 3 ) ,  ( 3 ) z )
                        ~        ~
                     p (v )  N (v :  v , C v )
               ~
               v
                                                                                                         ~ ( 2)                     ( 2)
                                                   ~ ~                      ( 2 )                       x
                        x ~
                    p ( ~ | v ,  ,  )  N ( D~ : f ,  w )
                                               x                                                                                    ~                                    ~ ~
                                                                                                                      p ( ~ ( 2 ) | v ( 2 ) ,  ( 2 ) )  N ( D~ ( 2 ) : f ( 2 ) ,  ( 2 ) w )
                                                                                                                          x                                    x

               ~
                x                   
                                                                                                        ~
                                                                                                        v (1)                                                    ~ ~
                                                                                                                     p (v (1) | u ( 2 ) ,  ( 2 ) )  N (v (1) : g ( 2 ) ,  ( 2 ) z )

                ~
                y                                         ~
                         y ~ x                    y ~
                     p ( ~ | v , ~,  ,  )  N ( ~ : g ,  z )
                                                                           (1)                           ~ (1)
                                                                                                          x                           (1)
                                                                                                                                                                 ~ ~
                                                                                                                                  ~
                                                                                                                      p ( ~ (1) | v (1) ,  (1) )  N ( D~ (1) : f (1) ,  (1) w )
                                                                                                                          x                              x

           {u, , }
                                                                                                              ~
                                                                                                              y
          u  {x, v}                                                                                                                               y ~
                                                                                                                                                               ~
                                                                                                                      p( ~ | u (1) ,  (1) )  N ( ~ : g (1) ,  (1) z )
                                                                                                                         y
                                       Hierarchal forms and empirical priors
      y  g ( x (1) , v (1) )  z (1)
  x (1)  f ( x (1) , v (1) )  w(1)
  
                                                        U (t )  ln p  y, x, v |  ,                                         y   g (1) 
          
                                                                                                                                v (1)     
v ( i1)  g ( x ( i ) , v ( i ) )  z (i )                        1 ln   1  T                                             (m) 
                                                                                                                            v

                                                                                                                                       g 
                                                                    2        2
  x ( i )  f ( x ( i ) , v ( i ) )  w( i )
                                                             A simple energy function of                                       (m)       
                                                                                                                                v    
                                                                  prediction error
 v ( m )    z ( m1)

                                               p  y , x, v   p  y | x, v  p  x , v 

                                                                              m 1
                                                  p  x, v   p (v   (m)
                                                                            ) p ( x (i ) | v (i ) ) p (v ( i ) | x ( i 1) , v ( i 1) )
                                                                              i 1

                                        p  x (i ) | v (i )   N ( Dx (i ) : f (i ) , (i ) w ) Dynamical priors (empirical)
                           p  v ( i ) | x ( i 1) , v ( i 1)   N (v ( i ) : g ( i 1) , ( i ) z ) Structural priors (empirical)
                                                 p  v ( m )   N (v ( m ) :  ,  v )
                                                                                                      Priors (full)
                              Overview

  Vu  D

              Hierarchical dynamic models
              Generalised coordinates (dynamical priors)
              Hierarchal forms (structural priors)
              Variational filtering and action (free-form)
              Laplace approximation and DEM (fixed-form)
              Comparative evaluations
              Examples
                           Variational learning
Aim: To optimise the path-integral (Action) of a free-energy bound on model evidence
w.r.t. a recognition density q


                            t F  F ( y, q( ))


          Free-energy:    F  ln p ( y | m)  D( q( ) || p( | y, m))
                            GH


      Expected energy:    G  ln p ( y,  )   q
                                                   U ( )   q

               Entropy:   H  ln q ( )   q




When optimised, the recognition density approximates the true conditional density and
Action becomes a bound approximation to the integrated log-evidence; these can then be
used for inference on parameters and model space respectively
                                 Mean-field approximation
                                q ( )   q ( i )  q (u (t )) q ( ) q ( )
                                           i



         Lemma : The free energy is maximised with respect to   q( i ) when  q ( i ) F  0 

                       q(u (t ))  exp(V (t ))          V (t )  U (t )   q ( ,  )
                                                                                                     Variational energy
Recognition density       q( )  exp(V  )              V    U (t )                 dt  U 
                                                                             q ( u , )              and actions
                          q( )  exp(V  )              V    U (t )                   dt  U 
                                                                             q ( u , )




                      Where U   ln p( ) and U   ln p( ) are the prior energies

                      and the instantaneous energy is specified by a generative model

                                     U (t )  ln p( y, u |  ,  )


                         We now seek recognition densities that maximise action
                                 Ensemble learning
Lemma: q(u (t ))  exp(V (t )) is the stationary solution, in a moving frame of reference,
for an ensemble of particles, whose equations of motion and ensemble dynamics are


                u  V (t )v     
               u  V (t )v       u  V (t )u  D  
               u                                                           Variational filtering


                      q  u  [u q  quV (t )]  u  qD

 Proof: Substituting the recognition density q  exp(V (t )) gives

                       q  u  qD

 This describes a stationary density under a moving frame of reference, with velocity D 
 as seen using the co-ordinate transform

                           D 
                      q( )  q(u )  u  q  0
                                          A toy example

  Vu  D                                             V (u, t )  4(u   ) 2  (u   ) 2  1 (u   ) 2
                                                                                                 4

                                                            (t )   (t )
                       5
                                                           (t )   (t )
                       4                                   (t )  1  (t )
                                                                     4

                       3


              u (t )   2
                                                                                                   5



                                                                                                       u 
                       1
                                                                                                   0

                       0


                                                                                                            u
                                                                                                -5
                       -1                                                                        2


                       -2
                                                                                          t bins
                                                                                                        0
                                                                                                              -2
                                                                                                                    u
                            0     20      40      60      80        100         120




                                 u  V (t )u     (t )        
                                u  V (t )u     (t )        
                                u                             
                              Overview
  Vu  D

              Hierarchical dynamic models
              Generalised coordinates (dynamical priors)
              Hierarchal forms (structural priors)
              Variational filtering and action (free-form)
              Laplace approximation and DEM (fixed-form)
              Comparative evaluations
              Examples (hemodynamics and bird songs)
           Optimizing free-energy under the Laplace approximation


                     Mean-field approximation: q( )   q( )  q(u (t ))q( )q( )
                                                            i

                                                            i

                       Laplace approximation: q( )  N (  ,  )
                                                 i         i   i




The Laplace approximation enables us the specify the sufficient statistics of the recognition density very simply


                                Conditional modes           Conditional precisions

                              (t )  arg max V (t )       u  U (t )uu
                                         u

                                 arg max V                U (t ) dt  U
                                                                                  

                                         

                                 arg max V                 U (t ) dt  U 
                                                                                    

                                         



             Under these approximations, all we need to do is optimise the conditional modes
                  Approximating the mode … a gradient ascent in moving coordinates


Taking the expectation of the ensemble dynamics, we get:


                  u  V (t )u  D   
                    V (t )u  D    D  V (t )u

        
       ~ ~
Here,   D can be regarded as a gradient ascent in a frame of reference that
moves along the trajectory encoded in generalised coordinates. The stationary
solution, in this moving frame of reference, maximises variational action.

                        D  0 
                        uV (t )  0   uV u  0

by the Fundamental lemma; c.f., Hamilton's principle of stationary action.

                                                                                          D

                                                                                   D
                       Dynamic expectation maximization


  Vu  D                              V (t )uu  D
                  D-Step               (exp()  I )1 (V (t )u  D  )
               inference
                                      u  U (t )uu
                q(u (t ))
                                                       local linearisation (Ozaki 1992)




                  E-Step                  
                                   V 1V
                 learning
                  q( )
                                        
                                   U



                 M-Step                   
                                   V 1V
              uncertainty
                  q ( )
                                          
                                    U                                         q( )  q(u (t ))q( )q( )
                                                                       A dynamic recognition system that minimises
                                                                                                   prediction error
                              Overview
  Vu  D

              Hierarchical dynamic models
              Generalised coordinates (dynamical priors)
              Hierarchal forms (structural priors)
              Variational filtering and action (free-form)
              Laplace approximation and DEM (fixed-form)
              Comparative evaluations
              Examples (hemodynamics and bird songs)
     y  g ( x (1) , v (1) , (1) )  z (1)              A linear convolution model

x   (1)
           f ( x , v , )
                 (1)   (1)    (1)


v (1)  
                                                                                  Prediction error



                               y1              y2         y3    y4          y1     y2         y3           y4             (1) v



                                              x1           x2                    1x           2x              (1) x



                                                    v1                                  1v           ( 2)v

                                      Generation                                          Inversion
                                                1.5
                                                                hidden states
                                                                                          Variational filtering on
                                       v (t )                                                states and causes
                                                  1



  Vu  D                                     0.5



                                                  0



                                                -0.5                                           cause

 u(t )  V (t )u  D (t )  (t )               -1
                                                       5   10       15     20   25   30

 u(t )  [v, v, v,   x, x, x,    ]T

                                                                   cause
                                                1.2

                                                  1

                                                0.8
         x                                                                                      time
                                                0.6

                                                0.4
              x
                        x                     0.2

                                                  0

                                                -0.2
                                      x(t )
                                                -0.4
                                                       5   10       15     20   25   30


                                                                 time {bins}
                                   u(t )  V (t )u  D (t )  (t )                                                                                     (t )  V (t )u  D (t )
                           predicted response and error                           hidden states                                         predicted response and error                           hidden states
                0.3                                                 1                                                                                                            1
                                                                                                                            0.2
                0.2
                                                                  0.5                                                                                                          0.5
                                                                                                                            0.1
states (a.u.)




                                                                                                            states (a.u.)
                0.1
                                                                    0                                                                                                            0
                                                                                                                              0
                  0

                                                                  -0.5                                                      -0.1                                               -0.5
                -0.1


                -0.2                                               -1                                                       -0.2                                                -1
                       5      10       15     20        25   30          5   10       15     20   25   30                           5      10       15     20        25   30          5   10       15     20   25   30
                                    time {bins}                                    time {bins}                                                   time {bins}                                    time {bins}




                                Linear deconvolution with variational                                                              Linear deconvolution with Dynamic expectation
                                     filtering (SDE) – free form                                                                          maximisation (ODE) – fixed form

                              causal states - level 2                                                                                      causal states - level 2
                  1                                                                                                           1

                0.8                                                                                                         0.8

                0.6                                                                                                         0.6
states (a.u.)




                                                                                                            states (a.u.)

                0.4                                                                                                         0.4

                0.2                                                                                                         0.2

                  0                                                                                                           0

                -0.2                                                                                                        -0.2

                -0.4                                                                                                        -0.4
                       5      10       15     20        25   30                                                                     5      10       15     20        25   30
                                    time {bins}                                                                                                  time {bins}
                                                                                             The order of generalised motion
 2                         2

 4                         4

 6                         6

 8                         8

10                        10

12

14
         S ( )         12

                          14

16                        16

         5    10   15          5                                           10       15
                                                                                                                                  2
                                                                                                                                                         n 1
     Precision in generalised coordinates
                                                                                                                      v (t )   1.5

                                                                                                                                  1


                                                                       6                                                       0.5

                                                                                                                                  0
                                   sum squared error (causal states)


                                                                       5                                                       -0.5

                                                                                                                                 -1
                                                                       4
                                                                                                 Accuracy and                  -1.5
                                                                                                                                      0       10         20         30        40

                                                                       3
                                                                                                 embedding (n)                                           time

                                                                       2
                                                                                                                                 2
                                                                                                                                                    n7
                                                                       1                                              v (t )   1.5

                                                                                                                                 1

                                                                       0                                                       0.5
                                                                                1        3   5     7   9   11    13
                                                                                                                                 0

                                                                                                  n                            -0.5

                                                                                                                                 -1

                                                                                                                               -1.5
                                                                                                                                      0   5    10   15        20   25    30   35

                                                                                                                                                         time
                                          hidden states
DEM and extended      1



 Kalman filtering   0.5




                      0
                                                                                          With convergence when
                    -0.5
                                                                  DEM(0)
                                                                                                  n 1
                                                                  DEM(4)

                                                                  EKF
                      -1

                                                                  true                           hidden states
                                                                                     1
                    -1.5
                           0   5     10     15          20   25          30   35
                                                 time                              0.5



                           sum of squared error (hidden states)                      0

                    0.9
                                                                                   -0.5                          DEM(0)
                    0.8                                                                                          EKF

                    0.7                                                              -1
                                                                                          0     10      20   30           40
                                                                                                      time
                    0.6


                    0.5


                    0.4


                    0.3


                    0.2


                    0.1

                                   EKF      DEM(0)           DEM(4)
                                                                  A nonlinear convolution model
                             predicted response and error                       hidden states
                       20                                         10


                       15                                          8


                       10                                          6
      states (a.u.)




                        5                                          4


                        0                                          2


                       -5                                          0
                             5    10       15      20   25   30        5   10       15      20   25       30
                                         time {bins}                              time {bins}




                                                                                   level                                                         v
                                                                                                      g ( x, v )     f ( x, v )   z   w


                                       causal states
                                                                                  m 1                1
                                                                                                      5   x2       e v  x ln 2   e4   e16
                      2.5

                        2                                                         m2                                             2          1
                                                                                                                                             2    sin( 16 t )
                                                                                                                                                         1

                      1.5
states (a.u.)




                        1

                      0.5
                                                                                  This system has a slow sinusoidal input or cause that excites
                        0
                                                                                  increases in a single hidden state. The response is a quadratic
                      -0.5
                                                                                  function of the hidden states (c.f., Arulampalam et al 2002).
                       -1
                             5    10       15      20   25   30
                                         time {bins}
          Conditional expectation (hidden states)

                                                                                 DEM and particle filtering
14


12
                                              true
                                              DEM
10                                            PF
                                              EKF
 8


 6
                                                                                      Comparative performance
 4


 2


 0


-2




                                                          Sum of squared error
      0   5     10      15          20   25     30   35
                             time




          Conditional covariance (hidden states)
0.7


0.6              DEM
                 PF
0.5              EKF



0.4


0.3


0.2


0.1


 0
      0   5     10      15          20   25     30   35
                             time
                                     predicted response and error                             hidden states
                                                                            1
                              0.2

                                                                          0.5


  Vu  D
                              0.1




              states (a.u.)
                                                                            0
                                0


                                                                          -0.5
                                                                                                                                Inference on states
                              -0.1


                              -0.2                                         -1
                                     5    10       15      20   25   30          5       10       15      20   25          30
                                                 time {bins}                                    time {bins}




                                               Triple estimation (DEM)

                                               causal states                                  parameters
                              1.2                                         0.2

                                1
                                                                            0
                              0.8
              states (a.u.)




                              0.6

                              0.4
                                                                          -0.2
                                                                                                                    true        Learning parameters
                                                                                                                    DEM
                              0.2
                                                                          -0.4
                                0

                              -0.2                                        -0.6
                                     5    10       15      20   25   30              1                         2
                                                 time {bins}
                              Overview

  Vu  D

              Hierarchical dynamic models
              Generalised coordinates (dynamical priors)
              Hierarchal forms (structural priors)
              Variational filtering and action (free-form)
              Laplace approximation and DEM (fixed-form)
              Comparative evaluations
              Hemodynamics and Bird songs
                                      An fMRI study of attention

Stimuli 250 radially moving dots at 4.7 degrees/s

Pre-Scanning
5 x 30s trials with 5 speed changes (reducing to 1%)
Task: detect change in radial velocity

Scanning (no speed changes)
4 x 100 scan sessions; each comprising 10 scans of 4 different conditions

F A F N F A F N S .................



A – dots, motion and attention (detect changes)
N – dots and motion
S – dots
                             V5 (motion sensitive area)
F – fixation



                                                                            Buchel et al 1999
                                   A hemodynamic model

                                                                      Visual input

                                                                      Motion
                                u (t )
                                                                      Attention
                                                                                                convolution kernel
                                signal
                       s   u   s  γ( f  1)
                                                   state equations
                                                   x  f x,u, 
                                flow
                                                   
                                f  s


              volume                               dHb

           τv  f  v1 /α
                                      τq  f E ( f )   v1 /α q/v
                                                   q




                                                                                     output equation
      Output: a mixture of intra- and extravascular signal
                                                                                     y  g ( x, )
y(t )  g (v, q)  V0 (k1 (1  q)  k 2 (1  q/v )  k 3 (1  v))
                             predicted response and error                                 hidden states
                        6                                              0.6

                        4                                              0.4

                                                                       0.2
                        2




       states (a.u.)
                                                                        0
                        0
                                                                      -0.2
                        -2
                                                                      -0.4                                            Inference on states
                        -4                                            -0.6

                        -6                                            -0.8
                             50   100     150 200 250    300   350           50     100    150 200 250    300   350
                                           time {bins}                                      time {bins}




                         Hemodynamic deconvolution (V5)

                                        causal states                                     parameters
                                                                     0.025


                        1                                             0.02


                                                                     0.015
states (a.u.)




                       0.5
                                                                                                                      Learning parameters
                                                                      0.01
                        0
                                                                     0.005

                  -0.5
                                                                        0
                             50   100     150 200 250    300   350            vis              mot        att
                                           time {bins}
                         hidden states
  1.6

  1.4


  1.2

   1


  0.8

  0.6
                                                        visual stimulation
  0.4                                                   signal
                                                        flow
  0.2                                                   volume
                                                        dHb
   0

        0   20    40     60                 80   100   120             140
                              time (bins)


            … and a closer look at the states
                        neuronal causes
0.05


0.04


0.03


0.02


0.01


   0


-0.01


-0.02
        0   20    40     60                 80   100   120             140
                              time (bins)
                              Overview

  Vu  D

              Hierarchical dynamic models
              Generalised coordinates (dynamical priors)
              Hierarchal forms (structural priors)
              Variational filtering and action (free-form)
              Laplace approximation and DEM (fixed-form)
              Comparative evaluations
              Hemodynamics and Bird songs
                                 Synthetic song-birds




                                    syrinx                                            simulus
                                                                         5000                                             5000

                                                                         4500                                             4500

                                                                         4000                                             4000




                                                        Frequency (Hz)




                                                                                                         Frequency (Hz)
                                                                         3500                                             3500

                                                                         3000                                             3000

                                                                         2500                                             2500

                                                                         2000                                             2000
                                                                                0.5       1        1.5                           0.5
                                                                                      Time (sec)
hierarchy of Lorenz attractors
       prediction and error                         hidden states
50                                       50

40                                       40

30                                       30

20                                       20

10                                       10

 0                                        0

-10                                      -10

-20                                      -20
      20    40   60     80   100   120         20   40   60     80   100   120
                 time                                    time                                                  simulus                                          percept
                                                                                                  5000                                             5000

                                                                                                  4500                                             4500


           Song recognition with DEM                                                              4000                                             4000




                                                                                 Frequency (Hz)




                                                                                                                                  Frequency (Hz)
                                                                                                  3500                                             3500

                                                                                                  3000                                             3000

                                                                                                  2500                                             2500

           causes - level 2                         hidden states                                 2000
                                                                                                         0.5       1        1.5
                                                                                                                                                   2000
                                                                                                                                                          0.5       1        1.5
50                                       50                                                                    Time (sec)                                       Time (sec)


40                                       40

                                         30
30
                                         20
20
                                         10
10
                                          0

 0                                       -10

-10                                      -20
      20    40   60     80   100   120         20   40   60     80   100   120
                 time                                    time
                                  … and broken birds
                                percept                                                   LFP
                 5000                                                  60

                 4500                                                  40

                 4000




                                                   LFP (micro-volts)
Frequency (Hz)
                                                                       20
                 3500
                                                                        0
                 3000

                 2500                                                  -20

                 2000                                                  -40
                          0.5       1        1.5                             0   500        1000        1500   2000
                                Time (sec)                                         peristimulus time (ms)


                        no structural priors                                              LFP
                 5000                                                  60

                 4500

                 4000
                                                                       40

                                                                       20
                                                                                                                      m 1


                                                   LFP (micro-volts)
Frequency (Hz)




                 3500                                                   0

                 3000                                                  -20

                 2500                                                  -40

                 2000                                                  -60
                          0.5       1        1.5                             0   500        1000        1500   2000
                                Time (sec)                                         peristimulus time (ms)


                        no dynamical priors                                               LFP
                 5000                                                  60

                 4500                                                  40

                 4000                                                  20
                                                   LFP (micro-volts)
Frequency (Hz)




                 3500

                 3000
                                                                        0

                                                                       -20
                                                                                                                      n 1
                 2500                                                  -40

                 2000                                                  -60
                          0.5       1        1.5                             0   500        1000        1500   2000
                                Time (sec)                                         peristimulus time (ms)
                            Summary

  Vu  D

              Hierarchical dynamic models
              Generalised coordinates (dynamical priors)
              Hierarchal forms (structural priors)
              Variational filtering and action (free-form)
              Laplace approximation and DEM (fixed-form)
              Comparative evaluations
              Hemodynamics and Bird songs

				
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