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I want to build a neural network over a data set. My idea is to use cross-validation on a training set to select the "best" neural network (and evaluate it on a separate test set) and to use nested cross-validation to make some statistical predictions. I'd use nested CV to plot bias and variance of my grid search's hyper-parameters. This way I can estimate my method's performance.

If these assumptions are not wrong, what should I do first? Model selection or estimation?

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I think you misunderstood something about the nested cross validation: Hyperparameter tuning / model selection is not done before nor after nested cross validation, it is done in the inner loop of the nested cross validation.

Probably related to the misunderstanding: grid hyperparameters in themselves do not cause a bias in generalization error erstimation. The (optimistic) bias is caused by the selection of the (apparently) best hyperparameter set.

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  • $\begingroup$ Okay, I know that nested cross validation returns k models from the k inner loops (each one performs model selection). Have I to choose between one of these k models? $\endgroup$ – Stefano Nardo Dec 19 '16 at 17:39
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    $\begingroup$ You don't. Your outer cross validation estimates generalization error for a training function that includes tuning. Thus, your final model (trained on the whole data set is basically the inner CV loop applied to the whole data set. See e.g. stats.stackexchange.com/a/65156/4598, stats.stackexchange.com/a/233027/4598, stats.stackexchange.com/a/245169/4598 $\endgroup$ – cbeleites Dec 19 '16 at 17:49
  • $\begingroup$ I can't see where I'm wrong. You say that the best result is when surrogate models are similar, that is when the same hyper-parameters win in the inner loop. This is what you called a stable method. If my method is stable I can use the method over the whole set. What if is unstable? Do I change it and try again? However, let's suppose it is stable. As you said in other question, I run f (whole data set). f is essentially the inner loop. But f is just training+validation, without a third set for testing purpose. Isn't not including test set wrong? $\endgroup$ – Stefano Nardo Dec 19 '16 at 18:33
  • $\begingroup$ Tell me if I'm wrong: my goal is to get a stable f (obviously a low error too) at the end of the nested CV. If this requirement is satisfied I can run f on the whole data set. $\endgroup$ – Stefano Nardo Dec 19 '16 at 19:08
  • $\begingroup$ I think you still have important misconceptions about what cross validation does, and why it is used. "But f is just training+validation, without a third set for testing purpose." This is the last step after the cross validation, it is not a replacement for cross validation. With the CV, you test f as a training procedure. You then use the CV results as approximation to the (unmeasured) performance of f (whole data set). This approximation (which is an extrapolation to a slightly larger training set) does not work if already the surrogate models have widely varying performance. $\endgroup$ – cbeleites Dec 20 '16 at 10:16

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