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Suppose I have a set of predictors for a regression problem. I know some of them maybe useless, but I am not sure.

So, I build multiple versions of predictor set, that each version contains/not contains some of the predictors that I am not sure about. Then, for each version, I use cross-validation (CV) to tune a same learning model. I then calculate the CV error.

Can I use the predictor set with smallest CV error? Does it introduce bias or overfitting? Why?

Will it help, if a separate test set is used?

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    $\begingroup$ Let us assume you have these predictors from prior research. If that is the case, and you testing it on new data, are you looking for model selection using a Frequentist methodology such as AIC or BIC, or are you seeking to use your data to update the quality of the predictors and do model selection afterward, in which case you would need to do a Bayesian method? $\endgroup$ Commented Feb 12, 2018 at 4:52
  • $\begingroup$ @DaveHarris Thank you for your message. It is from the prior experiment, that I know some of the predictors may not be useful. The models I am using are the simple linear regression and M5 model trees; so they are not Bayesian methods. The target is to find the best predictive model for the unseen new data. I am looking for a way to confirm the prior knowledge, and to determine whether these suspicious predictors should be removed. Should a use a wrapper method for feature selection? $\endgroup$
    – uared1776
    Commented Feb 12, 2018 at 5:18
  • $\begingroup$ I am going to bow out of answering this. I have only one practical problem that I have applied tree regression on. I am hesitant to answer because while I presume that scoring rules are the same for tree regression, I don't know that as a factual matter. There may be some unexpected effect I don't know about. It isn't really my area. $\endgroup$ Commented Feb 13, 2018 at 0:30

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Say you have sets $S_1, \dots, S_n$, where each $S_i$ a set of possible features to use. Rather than searching blindly (as in a typical feature selection problem), you've chosen these sets using some kind of prior knowledge. This gives models $M_1, \dots, M_n$, where each will be trained using the corresponding set of features. The problem is to select the model with best generalization performance. This is a standard model selection problem, and can be tackled as such.

For example, use k-fold cross validation. The training set is used to train the parameters of each model, and the validation set is used to select the best model (and any hyperparameters). Keep in mind that it's possible to overfit the validation set if you compare too many models (or hyperparameter choices) relative to the amount of data on hand (see Cawley and Talbot 2010). One can easily run into problems by blindly searching for features this way. But, if you're just comparing several models that you've carefully chosen using prior knowledge, you should be ok.

Because the validation set has been used for model selection, validation set error should not be used to estimate generalization performance of the final model, as it would be optimistically biased (see here). Instead, estimate generalization performance of the final model using an independent test set or, alternatively, use an outer cross validation loop (i.e. nested cross validation).

Cawley and Talbot (2010). On Over-fitting in Model Selection and Subsequent Selection Bias in Performance Evaluation.

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