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

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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

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)


Generalised coordinates

Likelihood

(Dynamical) prior


Energies and generalised precisions

Instantaneous energy

General and Gaussian forms

Precision matrices in generalised coordinates and time


Hierarchical dynamic models


A simple energy function of prediction error

Hierarchal forms and empirical priors

Dynamical priors (empirical)

Structural priors (empirical)

Priors (full)


Overview

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 densityq

Free-energy:

Expected energy:

Entropy:

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


Lemma : The free energy is maximised with respect to when

Where and are the prior energies

and the instantaneous energy is specified by a generative model

Mean-field approximation

Variational energy and actions

Recognition density

We now seek recognition densities that maximise action


Lemma: is the stationary solution, in a moving frame of reference, for an ensemble of particles, whose equations of motion and ensemble dynamics are

Ensemble learning

Variational filtering

Proof: Substituting the recognition density gives

This describes a stationary density under a moving frame of reference, with velocity

as seen using the co-ordinate transform


5

0

-5

2

0

-2

A toy example

5

4

3

2

1

0

-1

-2

0

20

40

60

80

100

120


Overview

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:

Laplace approximation:

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

Conditional modes

Conditional precisions

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:

Here, 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.

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


Dynamic expectation maximization

D-Step

inference

local linearisation (Ozaki 1992)

E-Step

learning

M-Step

uncertainty

A dynamic recognition system that minimises prediction error


Overview

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)


A linear convolution model

Prediction error

Generation

Inversion


hidden states

1.5

1

0.5

0

-0.5

cause

-1

5

10

15

20

25

30

cause

1.2

1

0.8

0.6

time

0.4

0.2

0

-0.2

-0.4

5

10

15

20

25

30

time {bins}

Variational filtering on states and causes


Linear deconvolution with variational filtering (SDE) – free form

Linear deconvolution with Dynamic expectation maximisation (ODE) – fixed form


6

5

Accuracy and embedding (n)

4

sum squared error (causal states)

3

2

2

2

1.5

1.5

1

1

1

0

0.5

0.5

1

3

5

7

9

11

13

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

0

10

20

30

40

0

5

10

15

20

25

30

35

time

time

The order of generalised motion

Precision in generalised coordinates


1

DEM(0)

DEM(4)

0.5

EKF

0

true

DEM(0)

-0.5

EKF

-1

-1.5

0

5

10

15

20

25

30

35

time

0.9

hidden states

0.8

1

0.7

0.5

0.6

0

0.5

0.4

-0.5

0.3

-1

0

10

20

30

40

0.2

time

0.1

EKF

DEM(0)

DEM(4)

hidden states

DEM and extended Kalman filtering

With convergence when

sum of squared error (hidden states)


level

A nonlinear convolution model

This system has a slow sinusoidal input or cause that excites increases in a single hidden state. The response is a quadratic function of the hidden states (c.f., Arulampalam et al 2002).


Comparative performance

Sum of squared error

DEM and particle filtering


Inference on states

Triple estimation (DEM)

Learning parameters


Overview

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

F – fixation

V5 (motion sensitive area)

Buchel et al 1999


A hemodynamic model

Visual input

Motion

Attention

convolution kernel

state equations

output equation

Output: a mixture of intra- and extravascular signal


Inference on states

Hemodynamic deconvolution (V5)

Learning parameters


… and a closer look at the states


Overview

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

hierarchy of Lorenz attractors


Song recognition with DEM


… and broken birds


Summary

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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