Variance Computation for Weighted Model Counting with Knowledge Compilation Approach
Summary: arXiv:2601.03523v2 Announce Type: replace
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.
Introduction
In the field of knowledge compilation, weighted model counting (WMC) plays a pivotal role, especially in probabilistic inference. This technique is extensively utilized for various probabilistic models, including Bayesian networks. However, the challenge arises when the parameters of the models carry uncertainty due to their reliance on learned data. This uncertainty necessitates the computation of variance in inference outcomes, which remains a complex issue.
Motivation and Problem Statement
The main motivation behind this research is to tackle the problem of variance computation in WMC. By treating inference outcomes as random variables, we can introduce distributions for their parameters, leading to the evaluation of variance. Despite the importance of this computation, its tractability has not been thoroughly explored. This paper investigates the conditions under which the variance of WMC can be computed effectively.
Key Contributions
- Polynomial Time Algorithm: We present a polynomial time algorithm capable of evaluating the variance of WMC when the input is provided in the form of a structured d-DNNF (Decomposable Negation Normal Form). This advancement offers a significant step towards efficient variance computation in structured environments.
- Complexity Results: Our research establishes the hardness of computing variance for structured DNNFs, d-DNNFs, and FBDDs (Functional Binary Decision Diagrams). This finding is particularly noteworthy as it highlights the complexity of variance computation in frameworks that otherwise permit polynomial time algorithms for WMC.
- Application to Bayesian Networks: We demonstrate a practical application by measuring uncertainty in Bayesian network inference. Our empirical studies reveal that the proposed algorithm can effectively evaluate the variance of marginal probabilities in real-world Bayesian networks, further illustrating the impact of parameter variances on the overall inference variance.
Conclusion
The research presented in this paper contributes significantly to the understanding of variance computation in weighted model counting. By introducing a polynomial time algorithm and establishing the complexity of the problem, we pave the way for future studies that could enhance probabilistic inference in various applications. The ability to quantify uncertainty in inference outcomes is crucial, especially in fields where decision-making relies heavily on probabilistic models.
Future Work
Further research is needed to explore additional structures and frameworks that may facilitate variance computation in WMC. Investigating alternative approaches to handling uncertainty in parameters and improving algorithm efficiency will also be vital for advancing knowledge compilation techniques.