On the Latent Variable Interpretation in Sum-Product Networks
This work resolves foundational theoretical issues in SPNs, which are important for probabilistic graphical models in machine learning, though it is incremental as it builds on prior interpretations.
The paper addresses a conflict in the latent variable interpretation of Sum-Product Networks (SPNs), where existing methods violate completeness and fail to fully specify the model, by proposing SPN augmentation to remedy this. It establishes theoretical foundations for probabilistic interpretation, EM algorithm derivation, and correctness of MPE inference, with experimental validation on synthetic data and 103 real-world datasets.
One of the central themes in Sum-Product networks (SPNs) is the interpretation of sum nodes as marginalized latent variables (LVs). This interpretation yields an increased syntactic or semantic structure, allows the application of the EM algorithm and to efficiently perform MPE inference. In literature, the LV interpretation was justified by explicitly introducing the indicator variables corresponding to the LVs' states. However, as pointed out in this paper, this approach is in conflict with the completeness condition in SPNs and does not fully specify the probabilistic model. We propose a remedy for this problem by modifying the original approach for introducing the LVs, which we call SPN augmentation. We discuss conditional independencies in augmented SPNs, formally establish the probabilistic interpretation of the sum-weights and give an interpretation of augmented SPNs as Bayesian networks. Based on these results, we find a sound derivation of the EM algorithm for SPNs. Furthermore, the Viterbi-style algorithm for MPE proposed in literature was never proven to be correct. We show that this is indeed a correct algorithm, when applied to selective SPNs, and in particular when applied to augmented SPNs. Our theoretical results are confirmed in experiments on synthetic data and 103 real-world datasets.