Signal Processing - May 2016 - 26

Bayesian Model
Inversion

Stochastic dynamic causal models
Stochastic DCM entails inverting a model of the form given
by (10) in the time domain, which includes state noise. This
requires estimation of not only the model parameters (and any
hyperparameters that parameterize the precision of generalized

Endogenous
Fluctuations

50

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.
x(t ) = f (x(t ), u, θ)

Posterior Density

In p(Σ|m )
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Model

p(Σ|m )
1
Probability

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This is referred to as stochastic DCM [116]. The second
approach makes predictions in the frequency domain and
is based on fitting second-order data features, such as cross
spectra. This is referred to as spectral DCM [114], [121].
We briefly review both schemes and illustrate their clinical
applications. For a schematic illustration of DCM of intrinsic
dynamics, see Figure 4. Figure 5 presents a comparison of the
two schemes.

log-Probability

w(t )

generalized inverse, then the Lyapunov exponents N (m) of this
linear dynamical system will always be negative. In general,
the Jacobian is not symmetrical (causal effects are asymmet-
ric); the modes and eigenvalues take complex values. See [119]
for a detailed treatment of the special case of symmetrical con-
nectivity, in which the eigenmodes of functional and effective
connectivity become the same. It is worth noting that these
eigenmodes are also closely related to (group) independent
component analysis (ICA) except with a rotation based on
higher-order statistics (for details, see [120]).
There are currently two schemes to invert models of
the form (9). They differ in what data features they use for
the parameter estimation. The first inverts data in the time
domain, and the model is used to predict the time series per se.

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Model

p(θ|Σ, m ) ≈ q(θ|µ)
In p(Σ|m ) = F(Σ, µ)

y(t )

Log Model Evidence

Bayesian Model
Comparison

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Σ(τ )

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p(θ|Σ) = ∑ m p(θ|Σ, m ) p(m |Σ)
Bayesian Model Averaging

FIGure 4. This schematic shows a DCM that embodies the best effective connectivity-identified using Bayesian model inversion (top left panel)-

among hidden neuronal states that explains the observed functional connectivity, / (t), among hemodynamic responses. This explanation is possible
because the cross spectra contain all the information about (second-order) statistical dependencies among regional dynamics. Bayesian model inversion
furnishes posterior estimates for the parameters of each model and provides the associated log model evidence in terms of a variational free-energy
bound. Because the mapping from functional connectivity to effective connectivity is not objective (there may be many combinations of effective connectivity parameters that induce the same functional connectivity), one can use a Bayesian model comparison (top right panel) to score competing models.
The model with the highest model evidence can then be selected. Alternatively, one can use Bayesian model averaging to average all possible models
(bottom right panel).

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IEEE Signal Processing Magazine

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May 2016

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Table of Contents for the Digital Edition of Signal Processing - May 2016
