We conduct a novel theoretical analysis to demonstrate CODA's

Markov Condition, are not independent of each other .

Based onLemma 5.1anditsproof, weknownthatthePMP oftheauxiliary control system, (S.2), is exactly the differential PMP equations (13). Thus below, we only look at the differential PMP equationsin(S.2). In the system identification experiment, we collect a total number of five trajectories from systems (in Table 2) with dynamics known, wherein different trajectoriesξo = {xo0:T,u0:T 1}havedifferent initial conditionsx0 andhorizonsT (T ranges from10to20),with randominputsu0:T 1 drawnfromuniformdistribution. In fact, throughout the entire learning process, PDP always guarantees that the policyconstraint isperfectly respected (as the forward pass strictly follows the policy). Please seeAppendix Fig. S4for validation.

Automated decision-making under uncertainty requires balancing exploitation and exploration. Classical methods treat these separately using heuristics, while Active Inference unifies them through Expected Free Energy (EFE) minimization. However, EFE minimization is computationally expensive, limiting scalability. We build on recent theory recasting EFE minimization as variational inference, formally unifying it with Planning-as-Inference and showing the epistemic drive as a unique entropic contribution. Our main contribution is a novel message-passing scheme for this unified objective, enabling scalable Active Inference in factored-state MDPs and overcoming high-dimensional planning intractability.

We conduct a novel theoretical analysis to demonstrate CODA's