# Nested cross validation for model selection

How can one use nested cross validation for model selection?

From what I read online, nested CV works as follows:

• There is the inner CV loop, where we may conduct a grid search (e.g. running K-fold for every available model, e.g. combination of hyperparameters/features)
• There is the outer CV loop, where we measure the performance of the model that won in the inner fold, on a separate external fold.

At the end of this process we end up with $K$ models ($K$ being the number of folds in the outer loop). These models are the ones that won in the grid search within the inner CV, and they are likely different (e.g. SVMs with different kernels, trained with possibly different features, depending on the grid search).

How do I choose a model from this output? It looks to me that selecting the best model out of those $K$ winning models would not be a fair comparison since each model was trained and tested on different parts of the dataset.

So how can I use nested CV for model selection?

Also I have read threads discussing how nested model selection is useful for analyzing the learning procedure. What types of analysis /checks can I do with the scores that I get from the outer K folds?

How do I choose a model from this [outer cross validation] output?

Treat the inner cross validation as part of the model fitting procedure. That means that the fitting including the fitting of the hyper-parameters (this is where the inner cross validation hides) is just like any other model esitmation routine.
The outer cross validation estimates the performance of this model fitting approach. For that you use the usual assumptions

• the $k$ outer surrogate models are equivalent to the "real" model built by model.fitting.procedure with all data.
• Or, in case 1. breaks down (pessimistic bias of resampling validation), at least the $k$ outer surrogate models are equivalent to each other.
This allows you to pool (average) the test results. It also means that you do not need to choose among them as you assume that they are basically the same. The breaking down of this second, weaker assumption is model instability.

Do not pick the seemingly best of the $k$ surrogate models - that would usually be just "harvesting" testing uncertainty and leads to an optimistic bias.

So how can I use nested CV for model selection?

The inner CV does the selection.

It looks to me that selecting the best model out of those K winning models would not be a fair comparison since each model was trained and tested on different parts of the dataset.

You are right in that it is no good idea to pick one of the $k$ surrogate models. But you are wrong about the reason. Real reason: see above. The fact that they are not trained and tested on the same data does not "hurt" here.

• Not having the same testing data: as you want to claim afterwards that the test results generalize to never seen data, this cannot make a difference.
• Not having the same training data:
• if the models are stable, this doesn't make a difference: Stable here means that the model does not change (much) if the training data is "perturbed" by replacing a few cases by other cases.
• if the models are not stable, three considerations are important:
1. you can actually measure whether and to which extent this is the case, by using iterated/repeated $k$-fold cross validation. That allows you to compare cross validation results for the same case that were predicted by different models built on slightly differing training data.
2. If the models are not stable, the variance observed over the test results of the $k$-fold cross validation increases: you do not only have the variance due to the fact that only a finite number of cases is tested in total, but have additional variance due to the instability of the models (variance in the predictive abilities).
3. If instability is a real problem, you cannot extrapolate well to the performance for the "real" model.

Which brings me to your last question:

What types of analysis /checks can I do with the scores that I get from the outer K folds?

• check for stability of the predictions (use iterated/repeated cross-validation)
• check for the stability/variation of the optimized hyper-parameters.
For one thing, wildly scattering hyper-parameters may indicate that the inner optimization didn't work. For another thing, this may allow you to decide on the hyperparameters without the costly optimization step in similar situations in the future. With costly I do not refer to computational resources but to the fact that this "costs" information that may better be used for estimating the "normal" model parameters.

• check for the difference between the inner and outer estimate of the chosen model. If there is a large difference (the inner being very overoptimistic), there is a risk that the inner optimization didn't work well because of overfitting.

### update @user99889's question: What to do if outer CV finds instability?

First of all, detecting in the outer CV loop that the models do not yield stable predictions in that respect doesn't really differ from detecting that the prediciton error is too high for the application. It is one of the possible outcomes of model validation (or verification) implying that the model we have is not fit for its purpose.

In the comment answering @davips, I was thinking of tackling the instability in the inner CV - i.e. as part of the model optimization process.

But you are certainly right: if we change our model based on the findings of the outer CV, yet another round of independent testing of the changed model is necessary.
However, instability in the outer CV would also be a sign that the optimization wasn't set up well - so finding instability in the outer CV implies that the inner CV did not penalize instability in the necessary fashion - this would be my main point of critique in such a situation. In other words, why does the optimization allow/lead to heavily overfit models?

However, there is one peculiarity here that IMHO may excuse the further change of the "final" model after careful consideration of the exact circumstances: As we did detect overfitting, any proposed change (fewer d.f./more restrictive or aggregation) to the model would be in direction of less overfitting (or at least hyperparameters that are less prone to overfitting). The point of independent testing is to detect overfitting - underfitting can be detected by data that was already used in the training process.

So if we are talking, say, about further reducing the number of latent variables in a PLS model that would be comparably benign (if the proposed change would be a totally different type of model, say PLS instead of SVM, all bets would be off), and I'd be even more relaxed about it if I'd know that we are anyways in an intermediate stage of modeling - after all, if the optimized models are still unstable, there's no question that more cases are needed. Also, in many situations, you'll eventually need to perform studies that are designed to properly test various aspects of performance (e.g. generalization to data acquired in the future). Still, I'd insist that the full modeling process would need to be reported, and that the implications of these late changes would need to be carefully discussed.

Also, aggregation including and out-of-bag analogue CV estimate of performance would be possible from the already available results - which is the other type of "post-processing" of the model that I'd be willing to consider benign here. Yet again, it then would have been better if the study were designed from the beginning to check that aggregation provides no advantage over individual predcitions (which is another way of saying that the individual models are stable).

• W.r.t. model selection, if the classifier is unstable, should we choose the one with the median performance amongst the best ones? This choice would be analogous to your suggestion to compare inner performance with outer performance. – viyps Mar 31 '14 at 14:08
• @davips: If the models are unstable, the optimization will not work (instability causes additional variance). Choosing the one model with median (or average) performance will not help, though. Instead, if the models are unstable I'd recommend either to go for more restrictive models (e.g. stronger regularization) or to build a model ensemble (which is fundametally different from selecting one model). – cbeleites Mar 31 '14 at 16:11
• the ensemble approach is almost a bagging. It seems reasonable to not discard the models. Therefore the outer accuracies (aka out-of-bag error) are useless for model selection. – viyps Mar 31 '14 at 20:31
• @user99889: please see updated answer. – cbeleites May 24 '17 at 16:10
• @cbeleites: a related hypothetical. Suppose I decide to search a parameter space c:[1,2,3]. I perform nested CV on my whole dataset and find the performance not so great. I therefore expand my search space to c:[0.5,1,1.5,2,2.5,3,3.5,4]. Have I done something very bad? It seems that I have essentially changed my parameter space (which is a part of the modeling process) based on knowledge gotten from the test data, and therefore need to evaluate on a dataset external to my current dataset? Happy to make this a separate question if you think it's best. – user0 May 31 '17 at 17:23

In addition to cebeleites excellent answer (+1), the basic idea is that cross-validation is used to assess the performance of a method for fitting a model, not of the model itself. If you need to perform model selection, then you need to perform that independently in each fold of the cross-validation procedure, as it is an integral part of the model fitting procedure. If you use a cross-validation based model selection procedure, this means you end up with nested cross-validation. It is helpful to consider the purpose of each cross-validation - one is for model selection, the other for performance estimation.

I would make my final model by fitting the model (including model selection) to the whole dataset, after using nested cross-validation to get an idea of the performance I could reasonably expect to get from that model.

• Why do you need to get an idea of the performance? – viyps Mar 31 '14 at 14:13
• @davips Generally if a statistical method is going to be used for some practical purpose, then the users will often want to have some idea of how well it works (e.g. medical screening test). Also if you are developing a machine learning algorithm then it is useful to have an unbiased estimate of how well it performs compared with competing methods. It is also a useful means of validating whether the method actually works (which is invalidated if the cross-validation is used both to select parameters and estimate performance). – Dikran Marsupial Mar 31 '14 at 16:35
• So to actually decide what parameter to use in the final model you would do the inner loop once? So if the inner loop was 10fold validation you would hold out 1/10 of the data train each model repeat this 10 times and then pick the parameter value with the smallest average error? Then retrain the model with that parameter value on the whole data set? – emschorsch Oct 4 '15 at 0:31
• Yes, that is correct. r – Dikran Marsupial Oct 5 '15 at 7:19
• (+1) I am thinking of re-organizing the ambiguous [nested] tag: see stats.meta.stackexchange.com/questions/4306. Do you think we could use a [nested-cross-validation] tag in addition to the existing [cross-validation], or do you think it's not really necessary and we are fine using [cross-validation] alone? (If you have something to say, maybe it's better if you comment over there on Meta; I will erase this off-topic comment after some time. Thanks.) – amoeba Apr 4 '17 at 8:54

I don't think anyone really answered the first question. By "Nested cross-validation" I think he meant combining it with GridSearch. Usually GridSearch has CV built in and takes a parameter on how many folds we wish to test. Combining those two I think its a good practice but the model from GridSearch and CrossValidation is not your final model. You should pick the best parameters and train a new model with all your data eventually, or even do a CrossValidation here too on unseen data and then if the model really is that good you train it on all your data. That is your final model.

• to clarify, in python scikit-learn, GridSearchCV(refit=True) does actually refit a model on the FULL data using the best parameters, so that extra step is not necessary. See docs – Paul Paczuski May 11 at 22:18
• You are right about the refit option. I was just stating tbe obvious !! – anselal May 13 at 11:10
• "the model from GridSearch is not your final model". But my points is that the grid search model with refit=True is the final model. Do you mean you and I are on the same page? But then I still don't see where the nesting happens in Grid search with CV. It seems like a single layer of CV to me (eg, 5-fold CV in grid search is a single layer of CV). – Paul Paczuski May 14 at 22:51
• We are on the same page about the refit. But with nested CV we mean that you create another CV loop outside your GridSearch, leaving some data out of the training and testing your final-final model to see if it generalises (makes good predictions on unknown data) – anselal May 16 at 5:27