BM25 ranking

Problem

BM25 is the standard ranking function for keyword search. It gives a high score to a document that contains rare query terms many times, and it corrects for the length of the document. Hybrid retrieval pipelines run BM25 next to a dense vector search.

Task

Write three functions. The documents and the query are lists of tokens that are already tokenized.

1. idf(corpus, term) returns the inverse document frequency of one term.
2. bm25_scores(corpus, query, k1=1.5, b=0.75) returns the BM25 score of each document for the query.
3. rank(corpus, query, top_k, k1=1.5, b=0.75) returns the indices of the top_k documents with the highest scores.

Let N be the number of documents. Let n(t) be the number of documents that contain the term t at least once. Use this idf:

Let f(t, d) be the number of times t occurs in document d. Let |d| be the number of tokens in d, and \mathrm{avgdl} the mean of |d| over all documents. The score of document d is

The sum runs over the query list. A term that occurs twice in the query adds its weight twice. A term with f(t, d) = 0 adds nothing.

Tie rule: rank sorts the scores from high to low. When two documents have the same score, the document with the lower index comes first.

Input and output

• idf returns a Python float.
• bm25_scores returns a float array of shape (N,). Entry i is the score of document i.
• rank returns a list of top_k int document indices, best first.

Constraints

• 1 <= N <= 200 and each document has at most 200 tokens.
• A document can be empty, but at least one document has a token.
• 0 <= b <= 1, k1 > 0 and 1 <= top_k <= N.
• Do not change the corpus or the query.
• Use Python and numpy only.

Examples

pythonCopycorpus = [
    ["the", "cat", "sat"],
    ["the", "dog", "sat", "on", "the", "mat"],
    ["cat", "cat", "food"],
    ["dogs", "and", "cats"],
]
idf(corpus, "cat"), idf(corpus, "zebra")

textCopy(0.6931471805599453, 2.302585092994046)

pythonCopybm25_scores(corpus, ["cat", "sat"])

textCopy[1.5234004  0.54578518 1.05823997 0.        ]

pythonCopyrank(corpus, ["cat", "sat"], 3)

textCopy[0, 2, 1]

Document 3 contains cats, not cat. Tokens must match exactly, so its score is 0.