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# Find average precision of a retrieval system, information retrieval system evaluation, how to find the average precision of a search engine, information retrieval systems solved exercises

Question:

An information retrieval system give us the following result (10 documents) where the green colored are relevant ones and the red colored are non-relevant. The numbers indicate the rank of the document. Find the average precision of the given system.

 1 2 3 4 5 6 7 8 9 10

Solution:

Precision = Fraction of retrieved documents that are relevant.

Precision = No. of relevant items retrieved/No. of retrieved items

Precision@k = Fraction of retrieved documents that are relevant in the top k documents.

Precision@k = No. of relevant docs in the top 5 results / k

Precision@5 = 3/5 = 0.6

Precision values @k is calculated and shown in the table below for all documents.

 Rank 1 2 3 4 5 6 7 8 9 10 Precision 1 0.5 0.33 0.5 0.6 0.5 0.43 0.5 0.44 0.5

We need to compute Precision@k for each of the k values in the ranked document list. Then we can calculate the average precision as follows;

Average precision = average of precision@k

Here, rel(k) is 1 if the document is relevant, otherwise 0. Hence, we can add the relevant documents precision@k and calculate the average precision as follows;

Average precision = (1.0 + 0.5 + 0.6 + 0.5 + 0.5)/5 = 0.62

Average precision for the retrieved documents = 0.62

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