Trust but Verify: Prover-Verifier Deliberation for Selective LLM Prediction

Reliably knowing when a language model is correct is almost as important as being correct. We introduce prover-verifier deliberation (PVD), an inference-time protocol grounded in interactive proof theory, as a mechanism for selective prediction: the protocol produces both an answer and a structured confidence verdict, allowing a system to report high-confidence answers while abstaining on uncertain cases. In each dialogue, a prover defends a candidate answer through checkable sub-claims while a verifier issues targeted challenges and returns \textsc{Accept}, \textsc{Challenge}, or \textsc{Reject}. Because frozen language models are imperfect provers and verifiers operating over a noisy channel, formal soundness and completeness guarantees do not transfer; instead, we characterize the protocol empirically through its coverage-precision behavior. Our main experiment uses Claude Sonnet 4.6 as prover and Claude Haiku 4.5 as verifier on GPQA Diamond. Questions accepted with no answer revision, which we call Accept + No Change (ANC), are reported as the high-confidence subset; we evaluate this subset by its precision and coverage. ANC separates reliable from unreliable answers, yielding a $\sim$30pp HC-Prec gap over the non-ANC complement. Robustness experiments with GPT and Gemini pairings show that high HC-Prec can transfer across model families, while verifier strictness and domain competence largely determine the size of the selection gap. On Humanity's Last Exam, weaker prover-verifier pairings can collapse or invert the ANC signal, illustrating a practical failure mode when the verifier operates outside its effective region. Comparisons with self-consistency, universal self-consistency, multi-agent debate, and Reflexion suggest that prover-verifier deliberation supplies a distinct argument-defensibility signal for selective prediction.

A graph-based multimodal framework to predict gentrification

Gentrification--the transformation of a low-income urban area caused by the influx of affluent residents--has many revitalizing benefits. However, it also poses extremely concerning challenges to low-income residents. To help policymakers take targeted and early action in protecting low-income residents, researchers have recently proposed several machine learning models to predict gentrification using socioeconomic and image features. Building upon previous studies, we propose a novel graph-based multimodal deep learning framework to predict gentrification based on urban networks of tracts and essential facilities (e.g., schools, hospitals, and subway stations). We train and test the proposed framework using data from Chicago, New York City, and Los Angeles. The model successfully predicts census-tract level gentrification with 0.9 precision on average. Moreover, the framework discovers a previously unexamined strong relationship between schools and gentrification, which provides a basis for further exploration of social factors affecting gentrification.