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Using K-Fold, I chose to use Logistic Regression for a project of mine.

I made it learn on my X_train (80% of data), and tested it on my X_test, with good results.

My question is : now that I need the model to be used, can I rebuilt the model with the same parameters but on all my data ? Will there be overfitting, or maybe as I saw that those parameters doesn't, will it be safe ?

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Yes, you can(Atleast in most cases). Infact before going to production, you must do this, it will give you better accuracy provided you keep the cross validated parameters same.

When you participate in competitions this is a well followed strategy to increase the score.

However, sometime domain constraints prevent you to use the test/validation data.

Eg: When you have split your data based on time periods, you might not want to include the latest data just because those observations in real time might not have matured enough to be considered into a model.

A good use case is fraud identification model in Insurance. It takes time for a claim to be identified as fraud. Hence without accounting for this, if you put all your data into model, the most recent claims will have been marked as no fraud, but in near future they might actually turn out to be identified as fraud. Your model will have learnt from wrong samples.

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