Compression versus Accuracy: A Hierarchy of Lifted Models

Probabilistic graphical models that encode indistinguishable objects and relations among them use first-order logic constructs to compress a propositional factorised model for more efficient (lifted) inference. To obtain a lifted representation, the state-of-the-art algorithm Advanced Colour Passing (ACP) groups factors that represent matching distributions. In an approximate version using $\varepsilon$ as a hyperparameter, factors are grouped that differ by a factor of at most $(1\pm \varepsilon)$. However, finding a suitable $\varepsilon$ is not obvious and may need a lot of exploration, possibly requiring many ACP runs with different $\varepsilon$ values. Additionally, varying $\varepsilon$ can yield wildly different models, leading to decreased interpretability. Therefore, this paper presents a hierarchical approach to lifted model construction that is hyperparameter-free. It efficiently computes a hierarchy of $\varepsilon$ values that ensures a hierarchy of models, meaning that once factors are grouped together given some $\varepsilon$, these factors will be grouped together for larger $\varepsilon$ as well. The hierarchy of $\varepsilon$ values also leads to a hierarchy of error bounds. This allows for explicitly weighing compression versus accuracy when choosing specific $\varepsilon$ values to run ACP with and enables interpretability between the different models.