Variance Computation for Weighted Model Counting with Knowledge Compilation Approach
Kengo Nakamura, Masaaki Nishino, Norihito Yasuda
TL;DR
This work addresses the problem of quantifying uncertainty in probabilistic inference by computing the variance of weighted model counting (WMC) when the weights carry uncertainty. It develops a polynomial-time algorithm for variance of WMC when the Boolean function is represented as a structured d-DNNF, while proving intractability for broader representations such as structured DNNFs, d-DNNFs, and FBDDs. The paper also demonstrates a practical application to Bayesian networks with constant treewidth, enabling variance computation of marginal probabilities, and provides empirical evidence of tractability on real networks. Overall, the approach integrates knowledge compilation techniques with probabilistic reasoning to enable uncertainty-aware inference and parameter learning. The results offer a principled framework for analyzing how parameter variance propagates to inference variance, with potential impact on robust decision-making in probabilistic systems.
Abstract
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.