Explaining the Inferences on Large Graph Flexibly via Subgraph Search

Probabilistic inferences distill knowledge from graphs to aid human make important decisions. Due to the inherent uncertainty in the model and the complexity of the knowledge, it is desirable to help the end-users understand the inference outcomes. Different from deep or high-dimensional parametric models, the lack of interpretability in graphical models is due to the cyclic and long-range dependencies and the byzantine inference procedures. Prior works did not tackle cycles and make the inferences interpretable. To close the gap, we formulate the problem of explaining probabilistic inferences as a constrained cross-entropy minimization problem to find simple subgraphs that faithfully approximate the inferences to be explained. We prove that the optimization is NP-hard, while the objective is not monotonic and submodular to guarantee efficient greedy approximation. We propose a general beam search algorithm to find simple trees to enhance the interpretability and diversity in the explanations, with parallelization and a pruning strategy to allow efficient search on large and dense graphs without hurting faithfulness. We demonstrate superior performance on 10 networks from 4 distinct applications, comparing favorably to other explanation methods. Regarding the usability of the explanation, we visualize the explanation in an interface that allows the end-users to explore the diverse search results and find more personalized and sensible explanations. Introduction Distilling knowledge from graphs is an important task found ubiquitously in applications, such as fraud detection (1), user interest modeling in social networks (2; 3; 4), and bioinformatics (5; 6; 7). The knowledge helps humans make high-stake decisions, such as whether to investigate a business or account for fraud detection on Yelp or to conduct expensive experiments on a promising protein in drug discovery. The state-of-the-art approaches model the graphs as directed or undirected graphical models, such as Bayesian networks, Markov Random Fields (MRF) (8), and Conditional Random Fields (CRF) (9), and distill knowledge from the graphs using inferences based on optimization (10; 11) and message passing (12). Different from predictive models on i.i. d. vectors, graphical models capture the dependencies among random variables and carry more insights for decision making. However, the inferences involve iterative and recursive computations, making the inference outcomes cognitively difficult to understand, verify, and ratify, locking away more applications of graphical models (EU law requires algorithmic transparency (13)) and more accurate models through debugging (14). We focus on MRFs and belief propagation (BP) inferences (15) that compute marginal distributions, aiming to make the inference outcomes more interpretable and thus cognitively easier for humans. Fig. 1 depicts the problem definition and the proposed solution. Several challenges are due. First, simple but faithful explanations are desired (16) but have not been defined for inferences on MRFs. Prior work (17; 18) approximates a high-dimensional Gaussian through a sparse covariance matrix, which does not explain belief propagation. To explain the marginal distribution on MRFs, using sensitivity analysis, the authors in (19; 20) proposed to find influential parameters inherent in the model but not in any particular inference algorithms and computations. Explanation of parametric linear models and deep networks using surrogate models, differentiation, and feature selection (21; 22; 23; 24; 25) cannot be applied to graphical model inferences, although our proposal can explain inferences on deep graphical models (26). Explainable RNNs (27) handles linear chains but not general MRFs. In terms of usability, previous works have studied how to visualize explanations of other models and utilize the explanations in the end tasks such as model debugging (28). It is less know what’s the best graph explanation complexity and faithfulness trade-off for the end-users, how to effectively communicate the probabilistic explanations, and how to utilize the explanations. Second, algorithmically, an MRF can be large, densely connected, and cyclic, while simple and faithful explanations need to be found efficiently. BP computes each message using other messages iteratively and recursively until convergence (Eq.