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I have a question about how to use cross-validation to select probability threshold for logistic regression. Suppose I want to minimize the misclassification rate. Say, I use 5-fold CV, and is this procedure correct:

1.fit 5 logistic regression models using each 4-folds of the data.

2.for each probability threshold(e.g. from 0.01 to 0.99), apply the 5 models on the left 1-fold of data, get misclassification rate. Then average these 5 error rates.

3.the optimal probability threshold is the one with smallest misclassification rate.

And suppose I fit a ridge logistic regression model, to select the tuning parameter $\lambda$, is it okay to first use CV to select an optimal $\lambda$(e.g. use cv.glmnet function in R package glmnet), then apply this parameter to the procedure above to find probability threshold?

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  • $\begingroup$ whatever classifier you have, you will have some free parameters in some parameter space. The idea is to train and classify on your data set. You do this train and classify process over cross validation, the points from 1 to 3 in your question. You do this you need a search algorithm for example sequential forward floating selection (SFFS). As a result you get the best parameters which minimize your error. For your question yes, you basically do it for many different $\lambda$ and often iteratively. $\endgroup$ – Seyhmus Güngören Jun 10 '17 at 2:52
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Yes, I think you have the right idea.

Just to put it another way, I'd say:

  1. Fix the hyper-parameters that you don't want to search for (e.g. as you mention, this could be a regularization strength $\lambda$)

  2. Choose a set of hyper-parameters $\theta$ you want to "optimize" by CV and split your dataset into folds (note it is standard practice to first remove a portion of your data as a testing set, and then use the remaining part for CV).

  3. Fix a search space (e.g. $\theta_i\in[a_i,b_i]\;\forall\; i$) and then for each $\hat{\theta}$ in the space, learn the model $f(x;\hat{\theta})$ and compute the average error $\mathcal{E}(f)$ over the CV folds. Keep the $f$ with the least error.

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