Variance Computation for Weighted Model Counting with Knowledge Compilation Approach

2026-01-07

arXiv.org

AI Summary

This work addresses the challenge of efficiently quantifying uncertainty in probabilistic reasoning under parameter uncertainty. It investigates the computability of the variance of weighted model counting (WMC) results and proposes, for the first time, a polynomial-time algorithm to compute this variance over structured d-DNNF representations. Theoretical analysis reveals that variance computation remains intractable even when leveraging knowledge compilation languages—such as d-DNNF and FBDD—that support efficient WMC, thereby filling a critical theoretical gap in uncertainty quantification within knowledge compilation. Empirical evaluation on real-world Bayesian networks demonstrates the successful computation of marginal probability variances, offering concrete evidence of how parameter uncertainty propagates into inference outcomes.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Weighted Model Counting
• Variance Computation
• Knowledge Compilation
• Probabilistic Inference
• Parameter Uncertainty

Innovation

• Weighted Model Counting
• Variance Computation
• Knowledge Compilation
• d-DNNF
• Uncertainty Quantification