An increasingly common setting in machine learning involves multiple parties, each with their own data, who want to jointly make predictions on future test points. Agents wish to benefit from the collective expertise of the full set of agents to make better predictions than they would individually, but may not be willing to release labeled data or model parameters.

In this section, we analyze the vanilla gradient-based explainers and GNNExplainer [24] under the explanation model framework. The proof that this explanation method falls into the class ofadditive feature attribution methods is quite straight-forward. TheconditionG S indicates thattherealization of G must be consistent with the realization of subgraphS. Thus, GNNExplainer would fail to explain predictions of thosemodels. In Figure 1, we provide an example illustrating the impact of theno-child constraint (3) onto the PGMexplanation. However, the constraint changes the edges in the Bayesian network.

Graph Neural Networks (GNNs) have been emerging as powerful solutions to many real-world applications in various domains where the datasets are in form of graphs such as social networks, citationnetworks,knowledgegraphs,andbiologicalnetworks [1,2,3].

Hierarchical Bayesian methods enable information sharing across regression problems on multiple groups of data.

Below, we proceed to describe our contributions in greater detail. Our framework builds upon the seminal work of Jordan et al. [1998], which introduced the celebrated