Rule-based methods for knowledge graph completion provide explainable results, but often require tens of thousands of rules to achieve competitive performance. Although individual predictions may use only a few rules, reasoning over an entire dataset requires these massive rule sets, hampering system-level understanding. We address this by learning a probability distribution over sets of rules that work together using probabilistic circuits. Our approach achieves a 70-96% reduction in the number of rules needed to reach peak baseline performance. Using an equivalent minimal number of rules, we outperform the baseline by up to 31$\times$. When comparing our minimal rule sets against baseline's full rule sets, we preserve 91% of peak baseline performance. Empirical validation on 8 benchmark datasets shows that our reduced rule sets exhibit higher utilization---fewer rules are wasted, and each prediction requires fewer rules. We show that our framework is grounded in well-known semantics of Nilsson's probabilistic logic and does not require independence assumptions. We provide exact probabilistic inference as well as an efficient lower bound and evaluate both.

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