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I've been looking for papers that discuss the pros and cons of positive unlabeled learning but I haven't been able to find anything.

I'm looking to compare the general differences between creating a positive-unlabeled based problem vs a regression classification. I have a biological dataset where it's hard to definitively define a sample as negative but I can make rules that would find something as close as possible to negative - with my idea being that I can assign scores to samples (e.g. like 0.1 instead of 0 to imply the lack of certainty but the closeness of the sample to being negative). However, I am trying to get an understanding of if I should consider positive unlabelled learning (in theory I could label my positive samples and ignore everything else even if other samples are capable of having a close to negative label/score) but I'm struggling to find information on pros and cons of trying positive-unlabelled learning.

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I don't think it's possible to know for sure if PU learning would work in your setting or not. It's certainly relevant to cases like the one you describe, so it would be worth trying. But there are other valid options, and even within PU learning there are different approaches to choose from (you might be interested in this question).

In my opinion the alternative you propose with regression makes some sense and it might work, but it's not very "clean" in terms of design: first the choice of 0.1 is arbitrary (why not 0.2 or 0.05 or ...?). Second, it means that you're telling the regression algorithm that "this instance should have probability 0.1" for many negative instances and also for a few negative instances: this is different than saying "I don't know the target value for this instance".

Note that you could also consider one class classification in this kind of setting, (as part of PU learning or not).

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