Estimating the structure of Bayesian networks as directed acyclic graphs (DAGs) from observational data is a fundamental challenge, particularly in causal discovery. Bayesian approaches excel by quantifying uncertainty and addressing identifiability, but key obstacles remain: (i) representing distributions over DAGs and (ii) estimating a posterior in the underlying combinatorial space. We introduce PIVID, a method that jointly infers a distribution over permutations and DAGs using variational inference and continuous relaxations of discrete distributions. Through experiments on synthetic and real-world datasets, we show that PIVID can outperform deterministic and Bayesian approaches, achieving superior accuracy-uncertainty trade-offs while scaling efficiently with the number of variables.

翻译：从观测数据中估计贝叶斯网络的有向无环图结构是一个基础性挑战，在因果发现领域尤为突出。贝叶斯方法通过量化不确定性和处理可识别性问题表现出色，但仍面临两大关键障碍：(i) 如何表示有向无环图上的概率分布；(ii) 如何在底层组合空间中估计后验分布。本文提出PIVID方法，该方法利用变分推断和离散分布的连续松弛技术，联合推断排列与有向无环图上的联合分布。通过在合成数据集和真实数据集上的实验验证，我们证明PIVID能够超越确定性方法和传统贝叶斯方法，在变量数量增加时保持高效可扩展性的同时，实现更优的准确性与不确定性的权衡。