← All parts of this equation

Equation 1 · Part 19 · What Actually Happens Between a Query and an Answer in RAG

Denominator: f(q_i, D) + k_1(1 - b + bdfrac|D|avgdl)

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)}
f(qi,D)+k1(1−b+b ∣D∣avgdl)f(q_i, D) + k_1\left(1 - b + b\,\dfrac{|D|}{\mathrm{avgdl}}\right)

What this part means

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

Its job in the formula

f(qiq_i, D) + k1(1−b+bdfrac∣D∣avgdl)k_1(1 - b + bdfrac|D|avgdl) occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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…

Read this part in the article →

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.

Open the illustrated fractions: division written vertically guide →

Sources cited in the surrounding passage

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