Variance Computation for Weighted Model Counting with Knowledge Compilation Approach

2026-04-16 · Source: cs.AI updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This paper introduces a novel query for computing the variance of Weighted Model Counting (WMC), a critical operation in knowledge compilation applied to probabilistic inference, particularly in Bayesian networks. The authors present a polynomial-time algorithm for evaluating WMC variance when the input is a structured d-DNNF (st-d-DNNF). This is significant because st-d-DNNFs subsume Sentential Decision Diagrams (SDDs), which are widely used. Conversely, the study proves the intractability of this problem for broader representations like structured DNNFs, d-DNNFs, and FBDDs, even though WMC itself is polynomial-time for the latter two. An application is demonstrated for measuring uncertainty in Bayesian network inference, showing that the algorithm can evaluate the variance of marginal probability on real-world Bayesian networks and identify parameters whose variances most impact the marginal's uncertainty. Experiments on 70 binary Bayesian networks from bnRep confirm the practical tractability, with maximum variance computation taking 0.025 seconds for an SDD size of 3,888.

Key takeaway

Research scientists working with probabilistic inference in Bayesian networks should prioritize compiling their models into structured d-DNNFs (like SDDs) to enable efficient, polynomial-time computation of WMC variance. This capability allows you to quantify the uncertainty in inference outcomes and identify specific model parameters whose variances most significantly impact overall uncertainty, guiding targeted data collection efforts to improve model reliability and decision-making.

Key insights

Computing WMC variance is tractable for structured d-DNNFs but intractable for broader DNNF variants.

Principles

• Uncertainty in model parameters necessitates variance computation for inference outcomes.
• Knowledge compilation tractability varies significantly across NNF representations.
• Parameter independence assumption is crucial for WMC variance calculations.

Method

The proposed algorithm recursively decomposes covariance calculations using vtrees and caches results, adjusting variable sets with pre-computed expectations and variances for efficient WMC variance computation.

In practice

• Quantify inference uncertainty in Bayesian networks using WMC variance.
• Identify critical parameters for targeted data collection to reduce inference uncertainty.
• Utilize SDD compilers for efficient st-d-DNNF generation in practical applications.

Topics

• Weighted Model Counting Variance
• Knowledge Compilation Tractability
• Structured d-DNNF Algorithm
• Bayesian Network Inference
• Probabilistic Parameter Uncertainty

Code references

• nttcslab/variance-wmc

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

Related on AIssential

• Dynestyx: A Probabilistic Programming Library for Dynamical Systems — Dynestyx unifies state-space model specification and Bayesian inference, making advanced methods accessible in probabilistic programming.
• Pinning the group-level variance parameters to speed computation for hierarchical models — Pinning group-level variance parameters can significantly speed up hierarchical model computation.
• Finite Neural Networks as Mixtures of Gaussian Processes: From Provable Error Bounds to Prior Selection — Finite neural networks can be approximated by Gaussian process mixtures with provable error bounds.
• Structured Nonparametric Variational Inference for Dependent Latent Modeling — SN-VI models complex latent variable dependencies using multivariate splines for accurate posterior approximation.
• Variational Consensus Monte Carlo for Bayesian Mixture — VCMC extends federated Bayesian mixture models to infer all parameters and cluster counts without conjugacy.
• Quantifying Consistency in LLM Logical Reasoning via Structural Uncertainty — Structural uncertainty quantifies LLM logical reasoning consistency by measuring the stability of self-preference-induced rankings over sampled solutions.

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by cs.AI updates on arXiv.org.