I have a classifier which derives from RandomForestClassifier, in
order to implement a custom "score" method. This obviously affects
scoring results obtained with cross-validation, but I observed that it
seems to also affect the actual predictions. In other words, the same
RF classifier with two different scoring functions will produce
different predictions. Perhaps I'm confused, but this doesn't seem
obvious to me, as I assumed that the scoring mechanism was somewhat
independent from the training one. So my question is: does a
classifier's "score" method play any role in the working of its "fit"
method?

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