Test collections are information retrieval tools that allow researchers to quickly and easily evaluate ranking algorithms. While test collections have become an integral part of IR research, the process of data creation involves significant effort in manual annotations, which often makes it very expensive and time-consuming. Thus, test collections could become too small when the budget is limited, which may lead to unstable evaluations. As a cheaper alternative, recent studies have proposed the use of large language models (LLMs) to completely replace human assessors. However, while LLMs may seem to somewhat correlate with human judgments, their predictions are not perfect and often show bias. Thus a complete replacement with LLMs is argued to be too risky and not fully reliable. Thus, in this paper, we propose LLM-Assisted Relevance Assessments (LARA), an effective method to balance manual annotations with LLM annotations, which helps to build a rich and reliable test collection even under a low budget. We use the LLM's predicted relevance probabilities to select the most profitable documents to manually annotate under a budget constraint. With theoretical reasoning, LARA effectively guides the human annotation process by actively learning to calibrate the LLM's predicted relevance probabilities. Then, using the calibration model learned from the limited manual annotations, LARA debiases the LLM predictions to annotate the remaining non-assessed data. Empirical evaluations on TREC-7 Ad Hoc, TREC-8 Ad Hoc, TREC Robust 2004, and TREC-COVID datasets show that LARA outperforms alternative solutions under almost any budget constraint.

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