\[\hat{s}= \sum_{k \in \mathcal{D}} k\,p(k).\]This produces a smooth score such as (5.4), rather than forcing the model to commit to a single sampled integer. In practice, this is substantially more stable than naive score sampling and better reflects the model’s uncertainty. It also handles cases where the judge distribution is broad or multimodal. For example, two candidates may both have mean score (5.4), while one has most of its mass tightly concentrated around (5) and (6), and the other splits mass between much lower and much higher ratings. The mean alone is the same, but the underlying judgement is very different.
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