Exploiting the indistinguishability of objects in a probabilistic graphical model such as a factor graph is key to lifted probabilistic inference algorithms and allows for tractable probabilistic inference problems with respect to domain sizes. A central building block for the exploitation of indistinguishable objects in factor graphs is the identification of commutative factors, i.e., factors whose output values are invariant under permutations of input values assigned to a subset of their arguments. In this paper, we revisit the theoretical foundations underlying the state-of-the-art algorithm to detect commutative factors. Specifically, we show that in its current form, the state-of-the-art algorithm relies on a central theorem that is mistakenly regarded as a sufficient condition to identify commutative factors, while it actually only implies necessary condition. Consequently, the state of the art might, as we show in this paper, deliver incorrect results. To fix the flaws currently present in the state of the art, we prove a slightly modified version of the aforementioned theorem, which serves as a necessary condition to identify commutative factors. Moreover, we present a corrected version of the state-of-the-art algorithm, which keeps its efficiency while ensuring correctness and introduce a complementary algorithm with tighter worst-case bounds.

翻译：在因子图等概率图模型中利用对象的不可区分性是提升概率推理算法的关键，使得可处理概率推理问题得以摆脱领域规模限制。因子图中利用不可区分对象的核心基础在于识别可交换因子，即那些输入变量子集值排列不改变输出值的因子。本文重新审视了现有最优算法检测可交换因子的理论基础。具体而言，我们发现当前最优算法所依赖的核心定理被误认为识别可交换因子的充分条件，而实际上该定理仅蕴含必要条件。正如本文所示，这一错误可能导致当前最优算法输出不正确结果。为修正现有缺陷，我们证明了一个经微调后的定理版本，该版本可作为识别可交换因子的必要条件。此外，我们提出了当前最优算法的修正版本，在保持效率的同时确保正确性，并引入了一种具有更紧最坏情况界的新补充算法。