CombMNZ

The algorithm is defined as:

CombMNZ(d) = (Number of Non-Zero Scores) × Σ NormalizedScore_i(d)

• CombSUM Component: The sum of all normalized relevance scores for document d across n retrieval systems.
• MNZ Multiplier: The count of retrieval systems that actually returned document d in their result set.
• Zero Handling: If a system does not retrieve the document, its score is 0 and it does not increment the MNZ multiplier.
• Effect: A document ranked 3rd by three different systems will outrank a document ranked 1st by only one system, assuming comparable score magnitudes.

CombMNZ is highly sensitive to the scale of input scores. Raw scores from BM25 and vector similarity are not directly comparable.

• Min-Max Normalization: Rescales all scores to a [0, 1] range before fusion.
• Z-Score Normalization: Standardizes scores based on mean and standard deviation, though less common for this use case.
• Scale Mismatch Risk: Without normalization, a system producing scores in the 100s will dominate a system producing scores between 0 and 1, rendering the MNZ multiplier ineffective.
• Curve Calibration: Some implementations apply a sigmoid function to dense retrieval scores before normalization to better align their distribution with sparse scores.

CombMNZ implements a consensus-based relevance principle. If multiple independent retrieval strategies agree a document is relevant, it is likely truly relevant.

• Signal Diversity: Combines lexical (BM25) and semantic (dense vector) signals.
• Spam Resistance: A document optimized solely for keyword density will score high on BM25 but low on semantic search, reducing its MNZ multiplier.
• Example: For a query 'how to fix a flat tire', a document retrieved by both a keyword index and a semantic vector index gets a multiplier of 2, boosting it above a keyword-stuffed page only found by BM25.

CombMNZ is an extension of CombSUM, which simply sums normalized scores without the MNZ multiplier.

• CombSUM: Score = Σ NormalizedScore_i(d)
• CombMNZ: Score = MNZ × Σ NormalizedScore_i(d)
• Key Difference: CombSUM treats all documents equally based on score magnitude. CombMNZ adds a penalty for documents found by fewer systems.
• Performance: Studies on TREC datasets show CombMNZ often outperforms CombSUM, particularly when the retrieval systems being fused are diverse and produce complementary result sets.

Implementing CombMNZ in a hybrid search pipeline involves a specific sequence of steps:

• Step 1: Execute parallel queries against sparse (BM25) and dense (vector) indexes.
• Step 2: Collect top-K results from each system into a candidate pool.
• Step 3: Apply Min-Max normalization to each system's scores independently.
• Step 4: For each unique document, sum its normalized scores and count how many systems retrieved it.
• Step 5: Multiply the sum by the count to produce the final CombMNZ score.
• Step 6: Sort documents by the final score in descending order.

CombMNZ is not universally optimal and has known failure modes:

• System Homogeneity: If all fused systems are highly correlated (e.g., two similar dense models), the MNZ multiplier provides little benefit and can amplify noise.
• Outlier Dominance: A document with an extremely high score from one system can still outrank a document with moderate scores from all systems, undermining the consensus mechanism.
• Recall Boundary: If a relevant document is only found by one system, it is penalized even if that single system is highly confident.
• Alternative: Reciprocal Rank Fusion (RRF) is often preferred in modern systems because it is hyperparameter-free and less sensitive to score distribution anomalies.