Why are precision, recall and F1 score equal when using micro averaging in a multi-class problem?

In a recent project I was wondering why I get the exact same value for precision, recall and the F1 score when using scikit-learn’s metrics. The project is about a simple classification problem where the input is mapped to exactly \(1\) of \(n\) classes. I was using micro averaging for the metric functions, which means the following according to sklearn’s documentation:

Calculate metrics globally by counting the total true positives, false negatives and false positives.

According to the documentation this behaviour is correct:

Note that for “micro”-averaging in a multiclass setting with all labels included will produce equal precision, recall and F, while “weighted” averaging may produce an F-score that is not between precision and recall.

After thinking about it a bit I figured out why this is the case. In this article, I will explain the reasons.

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NLP: Approaches for Sentence Embeddings (Overview)

In 2013, Mikolov et. al published ‘Distributed Representations of Words and Phrases and their Compositionality‘, a paper about a new approach to represent words by dense vectors. This was an improvement over the alternative, representing words as one-hot vectors, as these dense vector embeddings encode some meaning of the words they represent. In other terms, words with similar meaning are be close to each other in the vector space of the embedding. For example, “blue” would be close to “red” but far from “cat”. A commonly used name for their approach is word2vec.

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