  Probabilistic Circuits (PCs) are prominent tractable probabilistic models,
allowing for a range of exact inferences. This paper focuses on the main
algorithm for training PCs, LearnSPN, 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 them, 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.
