Probabilistic Circuits (PCs) are prominent tractable probabilistic models, allowing for a wide range of exact inferences. This paper focuses on a main algorithm for training PCs, LearnSPN, arguably a gold standard due to its efficiency, performance, and ease of use, in particular for tabular data. We show that LearnSPN is a greedy likelihood maximizer under mild assumptions. While inferences in PCs may use the entire circuit structure for processing queries, LearnSPN applies a hard method for learning PCs, propagating at each sum node a data point through one and only one of the children/edges as in a hard clustering process. We propose a new learning procedure named SoftLearn, that induces a PC using a soft clustering process. We investigate the effect of this learning-inference compatibility in PCs. Our experiments show that SoftLearn outperforms LearnSPN in many situations, yielding better likelihoods and arguably better samples. We also analyze comparable tractable models to highlight the differences between soft/hard learning and model querying.

Deep neural networks are in the limelight of machine learning with their excellent performance in many data-driven applications. However, they can lead to inaccurate predictions when queried in out-of-distribution data points, which can have detrimental effects especially in sensitive domains, such as healthcare and transportation, where erroneous predictions can be very costly and/or dangerous. Subsequently, quantifying the uncertainty of the output of a neural network is often leveraged to evaluate the confidence of its predictions, and ensemble models have proved to be effective in measuring the uncertainty by utilizing the variance of predictions over a pool of models. In this paper, we propose a novel approach for uncertainty quantification via ensembles, called Random Activation Functions (RAFs) Ensemble, that aims at improving the ensemble diversity toward a more robust estimation, by accommodating each neural network with a different (random) activation function. Extensive empirical study demonstrates that RAFs Ensemble outperforms state-of-the-art ensemble uncertainty quantification methods on both synthetic and real-world datasets in a series of regression tasks.