Token-Level Attribution of Multi-Mode LLM Watermarks: Optimal Rule and Belief Weighting

Abstract

A watermark verifier may hold many modes, i.e., secret keys, and a document may mix spans of different modes with unwatermarked text. Directly reporting the mode with the largest pivotal statistic (pivot) inflates the Type I error, and restoring the nominal level reduces power as candidate modes are added. We formulate token-level multi-mode attribution for the Gumbel-max watermark as maximizing correct-label power under a Type I error constraint given a mode prior. We show that the optimal rule reports the mode with the largest prior-weighted alternative density and rejects when this density exceeds one common threshold. Based on this rule, we propose the Belief-weighted Likelihood Attribution with Density Estimation (BLADE) detector. It estimates the mode prior from neighboring pivots and the alternative density from reference token probabilities. Under a conditional product model of the pivots, it keeps the exact Type I error. We further show that BLADE approaches the optimal rule as its prior and density estimates improve, with a sharp coefficient on the density error. On OPT-1.3B and other open-weight large language models, BLADE attains higher correct-label power than the compared detectors across candidate counts, token-substitution rates, generation temperatures and generator models.