Modified n-gram precision ($p_n=\frac{{\sum }_{s,g}\min \{C_s(g),{\max }_r{R}_{s,r}(g)\}}{{\sum }_{s,g}{C}_s(g)}$)

Modified n-gram precision measures reference-matched candidate n-grams while clipping repeated occurrences to the evidence available in the references. It is the precision component of BLEU. Merely asking whether a word appears anywhere in a reference would otherwise reward repeating that word indefinitely.^[ref]

For candidate sentence $s$, let $C_s(g)$ count n-gram $g$, and let $R_{s,r}(g)$ count it in reference $r$. Its clipped match count isUse the maximum count in one reference for each n-gram, rather than adding counts across references. Corpus precision sums these clipped matches and divides by the total candidate n-gram count of that length, provided this denominator is nonzero. An implementation needs a stated convention when no candidate n-grams of that length exist.^[ref]

For an own example, take reference the cat the mat and candidate the the the cat. The candidate has three occurrences of the, but only two can match; cat contributes one more. Thus modified unigram precision is $3/4$, whereas an unrestricted word-membership check would misleadingly credit all four words.

The candidate's three bigram occurrences are the the twice and the cat once. Only the last matches, giving modified bigram precision $1/3$. Longer n-grams therefore test ordering distinctions that unigram counts cannot capture.

Aggregation matters too. Add a second pair whose candidate and reference are both cat. Corpus unigram precision becomes $(3+1)/(4+1)=4/5$, rather than the unweighted mean $(3/4+1)/2=7/8$. Each sentence contributes its counts, not an equal-sized vote regardless of length.

Finally, precision alone does not detect every omission: a short prefix can contain only matching n-grams. The brevity penalty supplies a separate length correction when these precisions are combined into BLEU.