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Equation 1 · Part 10 · What Actually Happens Between a Query and an Answer in RAG

fraction

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)}
fraction

What this part means

Divide the expression above the line by the one below it.

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

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 fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.