State-of-the-art neurosymbolic learning systems use probabilistic reasoning
to guide neural networks towards predictions that conform to logical
constraints over symbols. Many such systems assume that the probabilities of
the considered symbols are conditionally independent given the input to
simplify learning and reasoning. We study and criticise this assumption,
highlighting how it can hinder optimisation and prevent uncertainty
quantification. We prove that loss functions bias conditionally independent
neural networks to become overconfident in their predictions. As a result, they
are unable to represent uncertainty over multiple valid options. Furthermore,
we prove that these loss functions are difficult to optimise: they are
non-convex, and their minima are usually highly disconnected. Our theoretical
analysis gives the foundation for replacing the conditional independence
assumption and designing more expressive neurosymbolic probabilistic models.

神经符号学习系统应用概率推理指导神经网络，以满足符号上的逻辑约束。然而，多数系统基于条件独立性假设以简化学习和推理，我们研究并批评了这一假设，指出它可能阻碍优化和使不确定性无法量化，因此我们证明了损失函数会导致条件独立神经网络过于自信，无法表示多个有效选项的不确定性。此外，我们证明了这些损失函数难以优化：它们非凸且最小值通常高度分离。我们的理论分析奠定了替代条件独立性假设并设计更具表现力的神经符号学习概率模型的基础。