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Equation 1 · Part 6 · What Actually Happens Between a Query and an Answer in RAG

Symbol f

score(D,Q)=∑i=1nIDF(qi)⋅f(qi,D) (k1+1)f(qi,D)+k1(1−b+b ∣D∣avgdl)\mathrm{score}(D,Q) = \sum_{i=1}^{n} \mathrm{IDF}(q_i) \cdot \frac{f(q_i, D)\,(k_1+1)}{f(q_i, D) + k_1\left(1 - b + b\,\dfrac{|D|}{\mathrm{avgdl}}\right)}
ff

What this part means

f is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

f is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

Lexical retrieval scores a document by how well its terms overlap the query’s terms, weighted by how rare each term is and normalised for document length. The canonical scoring function, still the default first-stage ranker across a large share of production search two decades after its formulation, is BM25: score(D,Q)=∑i=1nIDF(qi)⋅f(qi,D) (k1+1)f(qi,D)+k1(1−b+b ∣D∣avgdl)\mathrm{score}(D,Q) = \sum_{i=1}^{n} \mathrm{IDF}(q_i) \cdot \frac{f(q_i, D)\,(k_1+1)}{f(q_i, D) + k_1\left(1 - b + b\,\dfrac{|D|}{\mathrm{avgdl}}\right)}. Robertson and Zaragoza’s account of the probabilistic relevance framework behind this formula is worth reading past the equation for one design choice it exposes: the term-frequency component saturates rather than growing linearly, so a document repeating a query term fifty times is scored only marginally higher than one repeating it five times, and the…

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Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the surrounding passage

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