Can I use a continuous version vs a bucketed version of the same field in Machine learning? I am training a classification model using tree based algorithms. I have a feature which is the price of an object. I'd like to check if the I keep the price as it is (as a continuous variable) & I also create price buckets (and then one hot encode them) and keep that has a feature too is a problem?
The results are better with both variables included in training in terms of accuracy and F1 score. Also, no sign of overfitting are observed.
 A: Yes, we can do that. Given that a model is adequately regularised this should not be a grave issue. Ideally we always want to avoid providing redundant explanatory features $x$ to our classifiers but most classifiers can handle such a situation reasonably well.
Conceptually, this "double use" of a feature will be akin to having the same signal being represented in different sampling frequencies. It can also be thought as having a noisy/unfiltered as well as a very smoothed/filtered version of a signal and trying to classify it.
The immediate issue that arises from the "double use" regards the interpretation of an individual feature's contribution when trying to do variable importance calculations and general explainability tasks; e.g what the feature $x_{\text{contin.}} = -4.35$ means for our prediction $\hat{y} = f(\dots)$ when the feature $x_{\text{discete}} = -4$ and even more so what it means if we have an obviously impossible situation where the feature $x_{\text{contin.}} = 3.2$ but due to permutation importance procedure we still assume that our $x_{\text{discete}} = -2$; in this situation our metrics are nonsensical. Similarly, exactly because the underlaying information encapsulated in  $x_{\text{contin.}}$ and  $x_{\text{discete}}$ is partially duplicated, we anticipate lower overall importance metrics for each of the features.
