arXiv:2508.06230v1 Announce Type: new Abstract: Unifying probabilistic and logical learning is a key challenge in AI. We introduce a Bayesian inductive logic programming approach that learns minimum message length programs from noisy data. Our approach balances hypothesis complexity and data fit through priors, which explicitly favour more general programs, and a likelihood that favours accurate programs. Our experiments on several domains, including game playing and drug design, show that our method significantly outperforms previous methods, notably those that learn minimum description length programs. Our results also show that our approach is data-efficient and insensitive to example balance, including the ability to learn from exclusively positive examples.