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 efforts in manual annotations, which often makes it very expensive and time-consuming. Thus, the test collections could become small when the budget is limited, which may lead to unstable evaluations. As an alternative, recent studies have proposed the use of large language models (LLMs) to completely replace human assessors. However, while LLMs seem to somewhat correlate with human judgments, they are not perfect and often show bias. Moreover, even if a well-performing LLM or prompt is found on one dataset, there is no guarantee that it will perform similarly in practice, due to difference in tasks and data. Thus a complete replacement with LLMs is argued to be too risky and not fully trustable. Thus, in this paper, we propose \textbf{L}LM-\textbf{A}ssisted \textbf{R}elevance \textbf{A}ssessments (\textbf{LARA}), an effective method to balance manual annotations with LLM annotations, which helps to make a rich and reliable test collection. We use the LLM's predicted relevance probabilities in order to select the most profitable documents to manually annotate under a budget constraint. While solely relying on LLM's predicted probabilities to manually annotate performs fairly well, with theoretical reasoning, LARA guides the human annotation process even more effectively via online calibration learning. 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-COVID and TREC-8 Ad Hoc datasets show that LARA outperforms the alternative solutions under almost any budget constraint.

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