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I want to train a binary classifier but I have an expected unbalanced dataset (90%/10%). Should I train an unbalanced dataset or to downsample the first class? This ratio is natural and will be always present in my data for prediction.

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Try and see which works better.

Given that you have a training set, a validation set and a metric that you want to optimize, you can answer all sorts of questions like this by simply trying.

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