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Published equation contexts

DCG@k=∑i=1k2 reli−1log⁡2(i+1),nDCG@k=DCG@kIDCG@k\mathrm{DCG@}k = \sum_{i=1}^{k} \frac{2^{\,\mathrm{rel}_i}-1}{\log_2(i+1)}, \qquad \mathrm{nDCG@}k = \frac{\mathrm{DCG@}k}{\mathrm{IDCG@}k}

Why this formula appears here

Normalised discounted cumulative gain generalises this to graded relevance and to the whole list rather than only the first hit: DCG@k=∑i=1k2 reli−1log⁡2(i+1),nDCG@k=DCG@kIDCG@k\mathrm{DCG@}k = \sum_{i=1}^{k} \frac{2^{\,\mathrm{rel}_i}-1}{\log_2(i+1)}, \qquad \mathrm{nDCG@}k = \frac{\mathrm{DCG@}k}{\mathrm{IDCG@}k}. where reli\mathrm{rel}_i is the graded relevance of the result at position i and IDCG@\mathrm{IDCG@}k is the DCG of the ideal ordering, so the ratio is bounded near one regardless of how many relevant documents exist for a given query. The logarithmic discount encodes a specific judgement about attention: a relevant document at position one is worth far more than the same document at position ten, and nDCG is the standard way the information-retrieval literature makes that judgement quantitative. Heterogeneous retrieval benchmarks built to compare…

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kk

Symbol k

the DCG of the ideal ordering, so the ratio is bounded near one regardless of how many relevant documents exist for a given query.

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i=1i=1

Starting index or lower bound: i=1

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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kk

Ending index or upper bound: k

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

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log⁡2(i+1)\log_2(i+1)

Denominator: log_2(i+1)

The complete quantity below the fraction bar; it must be nonzero for this division.

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IDCG@k\mathrm{IDCG@}k

Denominator: IDCG@k

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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DCG@k=∑i=1k2 reli−1log⁡2(i+1),nDCG@k=DCG@kIDCG@k\mathrm{DCG@}k = \sum_{i=1}^{k} \frac{2^{\,\mathrm{rel}_i}-1}{\log_2(i+1)}, \qquad \mathrm{nDCG@}k = \frac{\mathrm{DCG@}k}{\mathrm{IDCG@}k}

Equation 11 · AI Agents & Systems

Measuring What a RAG System Retrieves, Not Just What It Answers

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Normalised discounted cumulative gain generalises this to graded relevance and to the whole list rather than only the first hit: DCG@k=∑i=1k2 reli−1log⁡2(i+1),nDCG@k=DCG@kIDCG@k\mathrm{DCG@}k = \sum_{i=1}^{k} \frac{2^{\,\mathrm{rel}_i}-1}{\log_2(i+1)}, \qquad \mathrm{nDCG@}k = \frac{\mathrm{DCG@}k}{\mathrm{IDCG@}k}. where reli\mathrm{rel}_i is the graded relevance of the result at position i and IDCG@\mathrm{IDCG@}k is the DCG of the ideal ordering, so the ratio is bounded near one regardless of how many relevant documents exist for a given query. The logarithmic discount encodes a specific judgement about attention: a relevant document at position one is worth far more than the same document at position ten, and nDCG is the standard way the information-retrieval literature makes that judgement quantitative. Heterogeneous retrieval benchmarks built to compare…

Meanings in this article

  • kk: the DCG of the ideal ordering, so the ratio is bounded near one regardless of how many relevant documents exist for a given query.
  • ii: a position index.
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