Score Fusion & Normalization

QMD combines scores from multiple search backends using Reciprocal Rank Fusion (RRF) and position-aware blending.

Score Normalization

Different search backends produce scores in different ranges. QMD normalizes all scores to [0, 1] for consistency.

BM25 (FTS5) Normalization

FTS5 BM25 scores are negative (lower = better match):

// Raw BM25 score: -10 (strong), -2 (weak), 0 (no match)
const bm25_score = row.bm25_score;  // e.g., -10

// Normalize to [0, 1] where higher = better
const score = Math.abs(bm25_score) / (1 + Math.abs(bm25_score));

Raw BM25	Normalized	Interpretation
-10	0.91	Strong match
-5	0.83	Good match
-2	0.67	Medium match
-0.5	0.33	Weak match
0	0.00	No match

This transformation is:

Monotonic: Preserves ranking order
Query-independent: No per-query normalization needed
Stable: Same raw score → same normalized score

Vector (Cosine) Normalization

Vector search returns cosine distance (0 = identical, 1 = orthogonal):

const distance = vecResult.distance;  // e.g., 0.3

// Convert distance to similarity [0, 1] where higher = better
const score = 1 - distance;

Cosine Distance	Normalized	Interpretation
0.0	1.00	Identical
0.1	0.90	Very similar
0.3	0.70	Similar
0.5	0.50	Somewhat similar
0.7	0.30	Different
1.0	0.00	Orthogonal

Reranker Normalization

Qwen3-Reranker returns scores already in [0, 1] range:

const rerankScore = result.score;  // Already 0.0 to 1.0

No normalization needed.

Reciprocal Rank Fusion (RRF)

Formula

For each document appearing in multiple ranked lists:

RRF_score = Σ (weight_i / (k + rank_i + 1))

Where:

weight_i = weight for list i (default 1.0, original query gets 2.0)
k = constant (60, from RRF literature)
rank_i = 0-based rank in list i

Implementation

export function reciprocalRankFusion(
  resultLists: RankedResult[][],
  weights: number[] = [],
  k: number = 60
): RankedResult[] {
  const scores = new Map<string, { result: RankedResult; rrfScore: number; topRank: number }>();

  for (let listIdx = 0; listIdx < resultLists.length; listIdx++) {
    const list = resultLists[listIdx];
    const weight = weights[listIdx] ?? 1.0;

    for (let rank = 0; rank < list.length; rank++) {
      const result = list[rank];
      const rrfContribution = weight / (k + rank + 1);
      const existing = scores.get(result.file);

      if (existing) {
        existing.rrfScore += rrfContribution;
        existing.topRank = Math.min(existing.topRank, rank);
      } else {
        scores.set(result.file, {
          result,
          rrfScore: rrfContribution,
          topRank: rank,
        });
      }
    }
  }

  // Top-rank bonus
  for (const entry of scores.values()) {
    if (entry.topRank === 0) {
      entry.rrfScore += 0.05;  // #1 rank bonus
    } else if (entry.topRank <= 2) {
      entry.rrfScore += 0.02;  // #2-3 rank bonus
    }
  }

  return Array.from(scores.values())
    .sort((a, b) => b.rrfScore - a.rrfScore)
    .map(e => ({ ...e.result, score: e.rrfScore }));
}

Top-Rank Bonus

Documents ranking #1 in any list get a +0.05 bonus, #2-3 get +0.02:

if (entry.topRank === 0) {
  entry.rrfScore += 0.05;  // Rank #1 bonus
} else if (entry.topRank <= 2) {
  entry.rrfScore += 0.02;  // Rank #2-3 bonus
}

This preserves exact matches that might get diluted by expanded queries.

Weight Configuration

Original query results get 2× weight:

// First 2 lists (original FTS + original vector) get 2x weight
const weights = rankedLists.map((_, i) => i < 2 ? 2.0 : 1.0);

List	Weight	Rationale
Original query FTS	2.0	Preserve exact keyword matches
Original query vector	2.0	Preserve semantic intent
Expanded query 1	1.0	Supporting evidence
Expanded query 2	1.0	Supporting evidence

Example Calculation

Document appears in 3 lists:

List 0 (original FTS, weight=2.0): rank 0
List 1 (original vec, weight=2.0): rank 5
List 2 (expanded lex, weight=1.0): rank 2

RRF_score = (2.0 / (60 + 0 + 1)) + (2.0 / (60 + 5 + 1)) + (1.0 / (60 + 2 + 1))
          = (2.0 / 61) + (2.0 / 66) + (1.0 / 63)
          = 0.0328 + 0.0303 + 0.0159
          = 0.0790

# Top-rank bonus (rank 0 in list 0)
RRF_score += 0.05
Final RRF_score = 0.1290

Position-Aware Blending

Motivation

Pure reranker scores can contradict high-confidence retrieval results. For example:

BM25 finds an exact keyword match (rank 1)
Reranker gives it a low score (0.3) because it lacks semantic context
Result gets buried despite being the user’s intent

Position-aware blending trusts retrieval for top ranks and gradually increases reranker influence for lower ranks.

Algorithm

const blended = reranked.map(r => {
  const rrfRank = rrfRankMap.get(r.file) || candidateLimit;
  
  // Position-aware weights
  let rrfWeight: number;
  if (rrfRank <= 3) {
    rrfWeight = 0.75;      // Trust retrieval
  } else if (rrfRank <= 10) {
    rrfWeight = 0.60;      // Balanced
  } else {
    rrfWeight = 0.40;      // Trust reranker
  }
  
  // Convert rank to score (inverse rank)
  const rrfScore = 1 / rrfRank;
  
  // Blend retrieval and reranker scores
  const blendedScore = rrfWeight * rrfScore + (1 - rrfWeight) * r.score;
  
  return { ...r, score: blendedScore };
}).sort((a, b) => b.score - a.score);

Weight Table

RRF Rank	RRF Weight	Reranker Weight	Rationale
1-3	75%	25%	Preserve exact matches
4-10	60%	40%	Balanced trust
11+	40%	60%	Trust reranker for semantic matches

Example 1: Top Rank Preserved

Document at RRF rank 2:

rrfScore = 1 / 2 = 0.50
rerankScore = 0.30  // Reranker disagrees
rrfWeight = 0.75    // Rank 2 → trust retrieval

blendedScore = 0.75 × 0.50 + 0.25 × 0.30
             = 0.375 + 0.075
             = 0.45  ✓ Preserved despite reranker disagreement

Example 2: Reranker Promotes

Document at RRF rank 15:

rrfScore = 1 / 15 = 0.067
rerankScore = 0.85  // Reranker strongly agrees
rrfWeight = 0.40    // Rank 15 → trust reranker

blendedScore = 0.40 × 0.067 + 0.60 × 0.85
             = 0.027 + 0.51
             = 0.537  ✓ Reranker elevated this result

Example 3: Middle Ground

Document at RRF rank 7:

rrfScore = 1 / 7 = 0.143
rerankScore = 0.65
rrfWeight = 0.60    // Rank 7 → balanced

blendedScore = 0.60 × 0.143 + 0.40 × 0.65
             = 0.086 + 0.260
             = 0.346  ✓ Balanced contribution

Full Pipeline Example

User query: "machine learning algorithms"

Step 1: Query Expansion

Original: "machine learning algorithms"
Expanded:
  lex: "ML classification regression"
  vec: "artificial intelligence models"
  hyde: "A guide explaining neural networks and deep learning"

Step 2: Multi-Backend Search

List 0 (original FTS, weight=2.0):
  [doc1: -8.5, doc2: -3.2, doc3: -1.5]
  → [doc1: 0.89, doc2: 0.76, doc3: 0.60]

List 1 (original vec, weight=2.0):
  [doc2: 0.15, doc4: 0.25, doc1: 0.30]
  → [doc2: 0.85, doc4: 0.75, doc1: 0.70]

List 2 (lex expanded, weight=1.0):
  [doc1: -5.0, doc3: -2.0]
  → [doc1: 0.83, doc3: 0.67]

List 3 (vec expanded, weight=1.0):
  [doc4: 0.20, doc5: 0.35]
  → [doc4: 0.80, doc5: 0.65]

Step 3: RRF Fusion

doc1:
  List 0 rank 0: 2.0 / (60 + 0 + 1) = 0.0328
  List 1 rank 2: 2.0 / (60 + 2 + 1) = 0.0317
  List 2 rank 0: 1.0 / (60 + 0 + 1) = 0.0164
  RRF = 0.0809 + 0.05 (rank #1 bonus) = 0.1309

doc2:
  List 0 rank 1: 2.0 / (60 + 1 + 1) = 0.0323
  List 1 rank 0: 2.0 / (60 + 0 + 1) = 0.0328
  RRF = 0.0651 + 0.05 (rank #1 bonus) = 0.1151

doc3:
  List 0 rank 2: 2.0 / (60 + 2 + 1) = 0.0317
  List 2 rank 1: 1.0 / (60 + 1 + 1) = 0.0161
  RRF = 0.0478 + 0.02 (rank #2 bonus) = 0.0678

... (doc4, doc5)

RRF ranking: [doc1, doc2, doc3, doc4, doc5]

Step 4: Reranking

Reranker scores (on best chunks):
  doc1: 0.45
  doc2: 0.85
  doc3: 0.30
  doc4: 0.75
  doc5: 0.60

Step 5: Position-Aware Blending

doc1 (RRF rank 1):
  rrfScore = 1/1 = 1.0
  rerankScore = 0.45
  rrfWeight = 0.75
  blended = 0.75 × 1.0 + 0.25 × 0.45 = 0.8625  ← Preserved

doc2 (RRF rank 2):
  rrfScore = 1/2 = 0.5
  rerankScore = 0.85
  rrfWeight = 0.75
  blended = 0.75 × 0.5 + 0.25 × 0.85 = 0.5875

doc3 (RRF rank 3):
  rrfScore = 1/3 = 0.333
  rerankScore = 0.30
  rrfWeight = 0.75
  blended = 0.75 × 0.333 + 0.25 × 0.30 = 0.325

doc4 (RRF rank 4):
  rrfScore = 1/4 = 0.25
  rerankScore = 0.75
  rrfWeight = 0.60
  blended = 0.60 × 0.25 + 0.40 × 0.75 = 0.450

Final ranking: [doc1: 0.863, doc2: 0.588, doc4: 0.450, doc3: 0.325, doc5: ...]

Result

doc1 stays #1 (strong keyword + semantic match, preserved by position-aware blending)
doc2 stays #2 (strong semantic match)
doc4 promoted to #3 (reranker elevated it)
doc3 drops to #4 (weak reranker score)

The position-aware blending prevented doc1 from being demoted despite a mediocre reranker score, while still allowing doc4 to be promoted based on semantic relevance.

Get Started

Core Concepts

Usage Guides

Architecture

Score Normalization

BM25 (FTS5) Normalization

Vector (Cosine) Normalization

Reranker Normalization

Reciprocal Rank Fusion (RRF)

Formula

Implementation

Top-Rank Bonus

Weight Configuration

Example Calculation

Position-Aware Blending

Motivation

Algorithm

Weight Table

Example 1: Top Rank Preserved

Example 2: Reranker Promotes

Example 3: Middle Ground

Full Pipeline Example

Step 1: Query Expansion

Step 2: Multi-Backend Search

Step 3: RRF Fusion

Step 4: Reranking

Step 5: Position-Aware Blending

Result

Build docs developers (and LLMs) love

Get Started

Core Concepts

Usage Guides

Architecture

​Score Normalization

​BM25 (FTS5) Normalization

​Vector (Cosine) Normalization

​Reranker Normalization

​Reciprocal Rank Fusion (RRF)

​Formula

​Implementation

​Top-Rank Bonus

​Weight Configuration

​Example Calculation

​Position-Aware Blending

​Motivation

​Algorithm

​Weight Table

​Example 1: Top Rank Preserved

​Example 2: Reranker Promotes

​Example 3: Middle Ground

​Full Pipeline Example

​Step 1: Query Expansion

​Step 2: Multi-Backend Search

​Step 3: RRF Fusion

​Step 4: Reranking

​Step 5: Position-Aware Blending

​Result

Build docs developers (and LLMs) love

Score Normalization

BM25 (FTS5) Normalization

Vector (Cosine) Normalization

Reranker Normalization

Reciprocal Rank Fusion (RRF)

Formula

Implementation

Top-Rank Bonus

Weight Configuration

Example Calculation

Position-Aware Blending

Motivation

Algorithm

Weight Table

Example 1: Top Rank Preserved

Example 2: Reranker Promotes

Example 3: Middle Ground

Full Pipeline Example

Step 1: Query Expansion

Step 2: Multi-Backend Search

Step 3: RRF Fusion

Step 4: Reranking

Step 5: Position-Aware Blending

Result