VIGOR+: Iterative Confounder Generation and Validation via LLM-CEVAE Feedback Loop

Hidden confounding remains a fundamental challenge in causal inference from observational data. Recent advances leverage Large Language Models (LLMs) to generate plausible hidden confounders based on domain knowledge, yet a critical gap exists: LLM-generated confounders often exhibit semantic plausibility without statistical utility. We propose VIGOR+ (Variational Information Gain for iterative cOnfounder Refinement), a novel framework that closes the loop between LLM-based confounder generation and CEVAE-based statistical validation. Unlike prior approaches that treat generation and validation as separate stages, VIGOR+ establishes an iterative feedback mechanism: validation signals from CEVAE (including information gain, latent consistency metrics, and diagnostic messages) are transformed into natural language feedback that guides subsequent LLM generation rounds. This iterative refinement continues until convergence criteria are met. We formalize the feedback mechanism, prove convergence properties under mild assumptions, and provide a complete algorithmic framework.