One of the most important queries in knowledge compilation is weighted model counting (WMC), which has been applied to probabilistic inference on various models, such as Bayesian networks. In practical situations on inference tasks, the model's parameters have uncertainty because they are often learned from data, and thus we want to compute the degree of uncertainty in the inference outcome. One possible approach is to regard the inference outcome as a random variable by introducing distributions for the parameters and evaluate the variance of the outcome. Unfortunately, the tractability of computing such a variance is hardly known. Motivated by this, we consider the problem of computing the variance of WMC and investigate this problem's tractability. First, we derive a polynomial time algorithm to evaluate the WMC variance when the input is given as a structured d-DNNF. Second, we prove the hardness of this problem for structured DNNFs, d-DNNFs, and FBDDs, which is intriguing because the latter two allow polynomial time WMC algorithms. Finally, we show an application that measures the uncertainty in the inference of Bayesian networks. We empirically show that our algorithm can evaluate the variance of the marginal probability on real-world Bayesian networks and analyze the impact of the variances of parameters on the variance of the marginal.

翻译：知识编译中最重要的查询之一是加权模型计数（WMC），它已应用于多种模型的概率推理，例如贝叶斯网络。在实际推理任务中，模型的参数往往具有不确定性，因为它们通常是从数据中学习得到的，因此我们希望计算推理结果的不确定性程度。一种可能的方法是通过为参数引入分布，将推理结果视为随机变量，并评估该结果的方差。遗憾的是，计算此类方差的易处理性尚不明确。受此启发，我们考虑计算WMC方差的问题，并研究该问题的易处理性。首先，我们推导了一个多项式时间算法，用于在输入为结构化d-DNNF时评估WMC方差。其次，我们证明了该问题对于结构化DNNF、d-DNNF和FBDD的困难性，这一点引人关注，因为后两者允许多项式时间的WMC算法。最后，我们展示了一个用于衡量贝叶斯网络推理中不确定性的应用。我们通过实验表明，我们的算法能够评估真实世界贝叶斯网络上边缘概率的方差，并分析参数方差对边缘概率方差的影响。