Fig. 1. (Graphical model) The shaded areas represent the observations y_i,t, the open ellipses represent the latent states. The top-level states (upper ellipses) are connected to all of the lower-level states (lower ellipses). Covariates x_i,t are left out for clarity.

Fig. 2. Graphical structure of the approximate models used in the variational approximation (left) and the factorial approximation (right). Ellipses represent latent states M_t for the top-level DLM and θ_i,t for the lower-level time DLMs. Observations are left out for clarity. The dashed lines in the right graph indicate that although the approximate model is fully factorized, the connections between states are incorporated in each iteration step. (a) Graphical structure of the approximate model used in the variational approximation, and (b) graphical structure of the approximate model used in the factorial approximation.

Fig. 3. Means for the first elements of the latent states over time for four outlets from the newspaper data set, and the corresponding top-level DLM. Dotted lines correspond to lower-level states, solid lines represent top-level states. The left panel plots the exact means for the hierarchical model presented in this article, means inferred from the standard hierarchical model are plotted in the right panel.

Fig. 4. Top: average variance-dependent parts of the KL-divergences between the approximated marginals and the exact marginals. The left panel plots the average divergences between marginals of the top-level DLM, the right panel plots the average divergences between the marginals of the lower-level DLMs. The dash-dotted line plots the divergence between the variational approach and the exact model, the dashed line the divergence for the factorial approach. Bottom: the variance in the first dimension of the top-level state (left) and one from a set of 10 lower-level states (right) over time. Solid lines correspond to the exact model, dash-dotted lines to the variational approximation and dashed lines to the factorial approximation.

Fig. 5. Average squared error (left) and computation time (right) as a function of the number of parallel tasks for the variational approximation (dash-dotted line), the factorial approximation (dashed line) and the exact model (solid line).

Fig. 6. The average squared error on the newspaper data for the variational approximation (bar 2), the factorial approximation (bar 3) and the standard DHM (bar 4). The performance of the extended model with exact inference is represented by bar 1. Each error is an average over 16 parallel tasks; a larger number of parallel tasks did not further decrease the sum-squared error.