Reciprocal Rank Fusion for Score Merging
Implement reciprocal rank fusion to merge ranked result lists from dense and sparse retrievers without needing to normalize incompatible similarity scores.
The Score Incompatibility Problem
Dense retrievers produce cosine similarity scores between -1 and 1, while BM25 produces unbounded positive scores that depend on corpus size and term frequencies. You cannot add these numbers directly — a BM25 score of 3.7 and a cosine similarity of 0.85 mean completely different things. Simple score normalization (dividing by max score) is brittle because outlier documents distort the scale. We need a method that is agnostic to absolute score values.
The Core Idea Behind RRF
Reciprocal Rank Fusion (RRF) sidesteps the score incompatibility problem by converting each retriever's results into ranked positions and fusing those ranks rather than raw scores. A document ranked first gets a high RRF contribution, a document ranked tenth gets a much lower one, and the final score is the sum of RRF contributions across all retrievers. The formula is: RRF(d) = sum(1 / (k + rank_i(d))) where k is a smoothing constant (typically 60).
# RRF formula
# For each retriever i, document d receives:
# contribution = 1 / (k + rank_i(d))
# Final RRF score = sum of contributions from all retrievers
# k = 60 is the standard constant from the original 2009 paper
# Example:
# Document A: rank 1 in BM25, rank 4 in dense
# RRF(A) = 1/(60+1) + 1/(60+4) = 0.01639 + 0.01563 = 0.03202
# Document B: rank 2 in BM25, rank 2 in dense
# RRF(B) = 1/(60+2) + 1/(60+2) = 0.01613 + 0.01613 = 0.03226
# Document B scores slightly higher due to consistent top-2 rankingAll lessons in this course
- Dense vs Sparse Retrieval: Trade-offs
- Implementing BM25 Keyword Search
- Reciprocal Rank Fusion for Score Merging
- Hybrid Search in Pinecone and pgvector