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Many recent state-of-the-art results in language tasks were achieved using compound systems that perform multiple Language Model (LM) calls and aggregate their responses. However, there is little understanding of how the number of LM calls -- e.g., when asking the LM to answer each question multiple times and taking a majority vote -- affects such a compound system's performance. In this paper, we initiate the study of scaling properties of compound inference systems. We analyze, theoretically and empirically, how the number of LM calls affects the performance of Vote and Filter-Vote, two of the simplest compound system designs, which aggregate LM responses via majority voting, optionally applying LM filters. We find, surprisingly, that across multiple language tasks, the performance of both Vote and Filter-Vote can first increase but then decrease as a function of the number of LM calls. Our theoretical results suggest that this non-monotonicity is due to the diversity of query difficulties within a task: more LM calls lead to higher performance on "easy" queries, but lower performance on "hard" queries, and non-monotone behavior can emerge when a task contains both types of queries. This insight then allows us to compute, from a small number of samples, the number of LM calls that maximizes system performance, and define an analytical scaling model for both systems. Experiments show that our scaling model can accurately predict the performance of Vote and Filter-Vote systems and thus find the optimal number of LM calls to make. 

We present an approach for estimating the fraction of text in a large corpus which is likely to be substantially modified or produced by a large language model (LLM). Our maximum likelihood model leverages expert-written and AI-generated reference texts to accurately and efficiently examine real-world LLM-use at the corpus level. We apply this approach to a case study of scientific peer review in AI conferences that took place after the release of ChatGPT: ICLR 2024, NeurIPS 2023, CoRL 2023 and EMNLP 2023. Our results suggest that between 6.5% and 16.9% of text submitted as peer reviews to these conferences could have been substantially modified by LLMs, i.e. beyond spell-checking or minor writing updates. The circumstances in which generated text occurs offer insight into user behavior: the estimated fraction of LLM-generated text is higher in reviews which report lower confidence, were submitted close to the deadline, and from reviewers who are less likely to respond to author rebuttals. We also observe corpus-level trends in generated text which may be too subtle to detect at the individual level, and discuss the implications of such trends on peer review. We call for future interdisciplinary work to examine how LLM use is changing our information and knowledge practices.