Mean reciprocal rank (MRR)

Background

Mean Reciprocal Rank evaluates systems that return a ranked list where you mostly care about the first correct answer — question answering, entity retrieval, "I'm feeling lucky" search. For each query the reciprocal rank is $1/(\text{position of the first relevant result})$; MRR averages it over all queries.

Problem statement

Implement mean_reciprocal_rank(rankings), where rankings[q] is a binary relevance list for query $q$ in ranked order (1 = relevant):

$$\text{MRR} = \frac{1}{Q}\sum_{q=1}^{Q}\frac{1}{\text{rank}_q}$$

where $\text{rank}_q$ is the 1-indexed position of the first relevant item in query $q$'s list (the term is $0$ if no relevant item appears).

Input

• rankings — list[list[int]]: each inner list holds 0/1 relevance flags in ranked order.

Output

Returns a float in $[0, 1]$.

Examples

Example 1

Input:  rankings = [[0, 1, 0], [1, 0, 0], [0, 0, 1]]
Output: 0.6111

Explanation: the first relevant items sit at ranks 2, 1, 3, giving reciprocal ranks $1/2$, $1$, $1/3$. Their mean is $(0.5 + 1 + 0.333)/3 = 0.6111$.

Constraints

• Ranks are 1-indexed (the top result is rank 1).
• A query with no relevant item contributes $0$ to the sum.
• MRR is the mean of the per-query reciprocal ranks; the result lies in $[0, 1]$.

Notes

• MRR rewards only the first hit — a perfect second result still caps a query's contribution at $1/2$.
• Contrast with NDCG/MAP, which credit all relevant results; MRR is the right choice when there is essentially one correct answer per query.