RAG Retrieval Metrics: Recall@k, MRR, nDCG

Before tuning prompts, check that your retriever finds the right documents. Three standard metrics, for one query with a ranked list retrieved and a set of relevant document ids:

• Recall@k: the fraction of relevant docs that appear in the top k.
• Reciprocal rank: $1/\text{rank}$ of the first relevant doc (ranks start at 1), or 0 if none appears.
• nDCG@k with binary relevance: $\text{DCG} = \sum_{i=1}^{k} \frac{rel_i}{\log_2(i + 1)}$, divided by the best possible DCG, where all relevant docs come first.

Implement recall_at_k, reciprocal_rank and ndcg_at_k, each taking (retrieved, relevant) (plus k where named). Then implement evaluate(runs, k), where runs is a list of (retrieved, relevant) pairs, returning a dict with the mean "recall@k", "mrr" and "ndcg@k" across queries.

If relevant is empty, recall and nDCG are 0.