 BLEU

BLEU is a reference-based metric for evaluating machine translation. It combines modified n-gram precision at several lengths using a geometric mean, then applies a brevity penalty to discourage very short outputs. Kishore Papineni and colleagues introduced it as a way to make repeated translation evaluation less costly than repeatedly collecting human judgments.^[ref]

For each candidate n-gram, count matches only up to its maximum occurrence count in a single reference. Sum these clipped matches across the corpus and divide by the total number of candidate n-grams of that length to obtain $p_n$. This clipping prevents repeated use of a common word from earning unlimited matches. Corpus aggregation uses counts, rather than simply averaging independently calculated sentence scores.^[ref]

With candidate length $c$, effective reference length $r$ and weights summing to one,The original baseline uses $N=4$ and $w_n=1/4$. Effective reference length sums the closest reference sentence lengths for the candidates.^[ref]

For a synthetic one-pair calculation, take reference the cat is on the mat and candidate the cat is on mat. The clipped precisions areHere $c=5$, $r=6$ and the precision product is $1/4$, givingThis is about $57.893$ on a scale multiplied by one hundred; it is not $57.893\%$ translation accuracy.

The unsmoothed score becomes zero when a required precision is zero. Short sentences make this particularly conspicuous, while legitimate rephrasing can reduce overlap. State tokenization, casing, reference set, smoothing and scale when comparing results. The example illustrates the arithmetic; the paper emphasizes corpus-level use rather than treating each sentence score as an independent human-quality judgment.^[ref]