Modeling rapidly varying interventions with discretized dynamic Bayesian networks

Determine how to model the intervention $do\left(X_1(t) = \cos\left(\frac{2\pi t}{\Delta}\right)\right)$ using a Dynamic Bayesian Network with discretization length $\Delta$.

Background

The paper contrasts DSCMs with Dynamic Bayesian Networks (DBNs) obtained by discretizing continuous-time ODEs. The accuracy and computational cost of a DBN depend on the discretization length Δ\Delta, and the intervention considered in the quoted passage oscillates at the discretization scale itself. The authors explicitly identify the difficulty of representing this intervention in a DBN using that discretization, leaving its modeling unresolved.

References

The choice of $\Delta$ should reflect the natural timescales of the interventions to be considered too; for example, it is not clear how one would model the intervention $do\left(X_1(t) = \cos\left(\frac{2\pi t}{\Delta}\right)\right)$ with a discretisation length $\Delta$.

— From Deterministic ODEs to Dynamic Structural Causal Models  (1608.08028 - Rubenstein et al., 2016) in Section 3, subsection “Relation to ODEs and Dynamic Bayesian Networks”