Towards Explainability of Approximate Lifted Model Construction: A Geometric Perspective

Speller, Jan; Luttermann, Malte; Gehrke, Marcel; Braun, Tanya

Abstract

Advanced colour passing (ACP) is the state-of-the-art algorithm for lifting a propositional probabilistic model to a first-order level by combining exchangeable factors, enabling the use of lifted inference algorithms to allow for tractable probabilistic inference with respect to domain sizes. More recently, an approximate version of ACP, called ε-ACP, ensures the practical applicability of ACP by accounting for inaccurate estimates of underlying distributions. ε-ACP permits underlying distributions, encoded as potential-based factorisations, to slightly deviate depending on a hyperparameter ε while maintaining a bounded approximation error. To navigate through different levels of compression versus accuracy, a hierarchical version of ε-ACP has emerged that builds a hierarchy of ε values. In a drive towards interpretability of results, this paper looks at geometric properties of ε-equivalence, a central notion employed by ε-ACP and its hierarchical version to quantify the maximum allowed deviation between potentials. Specifically, we present a unified view on the results for ε-ACP and its hierarchical version and provide a geometric interpretation of ε-equivalence in L^p, thereby making results more interpretable.

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