Joint Distributional Evolution

Background: The distribution-to-distribution (D2D) framework models the evolution of marginal predictive distributions for each state dimension, rather than the full joint distribution over the complete state vector.

Question / Future Work: Extending the framework to full joint distribution-to-distribution forecasting requires scalable parameterisations of high-dimensional dependence structures and covariance representations, which is noted as a substantial challenge currently beyond the scope of the work.

Why It Matters: Modeling the evolution of the full joint probability distribution is crucial for accurately capturing inter-variable dependencies and higher-order uncertainty structures in complex dynamical systems.

Evidence: Extending the framework to full joint distribution-to-distribution forecasting would require scalable parameterisations of high-dimensional dependence structures and covariance representations, which remains a substantial challenge and is beyond the scope of the present work.