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We have a very imbalanced dataset (2% of class 1). To the best of our knowledge, there is no baseline in the literature to the problem we want to solve - so we thought of comparing our performance to a random classifier. We evaluate our model as a combination of precision and recall - we vary the threshold at which data points are classified as 1 and compute the rolling threshold and recall. We could use F1-score as well.

What would be an acceptable way to define a random predictor that we can compare to our model such that the comparison is as fair as possible?

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Since you are interested in different decision thresholds, your random model should produce scores. In that case, a reasonable base-line model assigns a score uniformly at random in $[0,1]$. Such a model will, at threshold $t$, have

$$\begin{align*} \operatorname{precision} &= \frac{2\%\cdot N\cdot (1-t)}{N(1-t)} = 0.02,\\[1em] \operatorname{recall} &= \frac{2\%\cdot N\cdot (1-t)}{2\%\cdot N} = 1-t. \end{align*} $$

(Perhaps a very simple model will serve as a better baseline.)

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