DEM: a variational treatment of dynamic systems.

TLDR: A variational treatment of dynamic models that furnishes time-dependent conditional densities on the path or trajectory of a system's states and the time-independent densities of its parameters using exactly the same principles is presented.

About: This article is published in NeuroImage.The article was published on 2008-07-01 and is currently open access. It has received 281 citations till now. The article focuses on the topics: Marginal likelihood & Model selection.

Figures

Fig. 15. The simulations of the previous figure were repeated four times, using d responses were deconvolved using DEM, particle and extended Kalman filtering. W and plotted the trajectories against each other in their corresponding state-space predictions from DEM are shown on the upper right. Again, one can see that desp quickly to the true trajectories. This is not the case for particle filtering or extende track the two trajectories and succumb to the systems attractor.

Fig. 5. Variational densities on the causal and hidden states of the linear convolution model of the previous figure. These show the trajectories or

Table 1 Priors on free biophysical parameters

Fig. 18. As for the previous figure but in this instance we estimated the parameters, hyperparameters and the cause by treating it as an unknown quantity. This is an example of triple estimation, where we are trying to infer on the states of the system as well as the parameters governing its causal architecture. EM cannot do this; therefore the true and conditional estimates of the parameters are provided for the DEM scheme only (lower right). It can be seen that the true value of the causal state lies within the 90% confidence interval and that we could infer with substantial confidence that the cause was non-zero, when it occurs.

Fig. 4. This is a schematic showing the linear convolution model used in the dem directed Bayesian graph, in which the nodes are connected by arrows, depicting con as a cause to perturb dynamics among two coupled hidden states. These dynamics ar on the left. This figure also summarises the architecture of the implicit inversion variational action. Critically, the prediction errors propagate their effects up the hi predictions are passed down the hierarchy. This sort of scheme can be implemente 2006 for a neurobiological treatment of this architecture).

Fig. 8. A numerical analysis of accuracy as a function of the embedding dimension o shows accuracy in terms of the sum of squared difference between the predicted and causes was fixed at, d=2. This means that increases in accuracy are mediated solely results were obtained with a roughness of γ=4 using the linear convolution model o the cause (black line) and the true values (grey line) for n=1 (left) and n=7 (righ

Frequently asked questions

Q1. What are the contributions in "Dem: a variational treatment of dynamic systems" ?

This paper presents a variational treatment of dynamic models that furnishes time-dependent conditional densities on the path or trajectory of a system 's states and the time-independent densities of its parameters. The resulting scheme can be used for online Bayesian inversion of nonlinear dynamic causal models and is shown to outperform existing approaches, such as Kalman and particle filtering. Furthermore, it provides for dual and triple inferences on a system 's states, parameters and hyperparameters using exactly the same principles. 

Q2. How can the authors approximate the variational densities?

In the absence of closed-form solutions for the variational densities, they can be approximated using ensemble dynamics that flow on a variational energy manifold, in generalised coordinates of motion. 

Q3. What is the first review of ensemble learning?

The first reviews variational approaches to ensemble learning under the Laplace approximation, starting with static models and generalising to dynamic systems. 

Q4. What is the fundamental lemma of variational calculus?

The Fundamental Lemma of variational calculus states that F(y,q) is maximised with respect to q #i when, and only whendq #ið ÞF ¼ 0fAq #ið Þ f i ¼ 0R d#if i ¼ F ð4Þdq #ið ÞF is the variation of the free-energy with respect to q(ϑi). 

Q5. What is the simplest way to construct a scheme based on ensemble dynamics?

To construct a scheme based on ensemble dynamics the authors require the equations of motion for an ensemble whose variational density is stationary in a frame of reference that moves with its mode. 

Q6. What is the simplest way to approximate q(u,t)?

This entails integrating the path of multiple particles according to the stochastic differential equations in Eq. (17) and using their sample distribution to approximate q(u,t). 

Q7. What does the term "Variational calculus" mean?

it means that the evolution of the mode follows the peak of the variational energy as it evolves over time, such that tiny perturbations to its path do not change the variational energy. 

Q8. What is the effect of particles in other ensembles?

The effect of particles in other ensembles is mediated only through their average effect on the internal energy, V #i ¼ hU #ð Þiq #{ið Þ, hence mean-field. 

Q9. Why does the ensemble density change with time?

Because this manifold evolves with time, the ensemble will deploy itself in a time-varying way that optimises free-energy and its action. 

Q10. What is the definition of the ensemble density?

The Fokker–Planck formulation affords a useful perspective on the variational results above and shows why the variational density is also referred to as the ensemble density; it is the stationary solution to a density on an ensemble of solutions. 

Q11. What is the inverse of the Lemma?

By the fundamental Lemma, action is maximised with respect to the variational marginals when, and only whendq u;tð Þ P F ¼ 0fAq u;tð 

Q12. What is the fundamental Lemma of variational calculus?

At P V uð Þ¼ V u; tð Þ ð16ÞThis is sufficient for the mode to maximise variational action (by the Fundamental Lemma of variational calculus).