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I would like to understand what is the difference between Permutation Importance (as outlined by Breiman in his original paper on Random Forests) and Drop Column Importance. From a cursory look at both methods, they seem to be doing almost the same thing:

Calculate a baseline score by training a model and obtaining some kind of metric (in this case R2), do something to one of the features and then calculate the score again and record the difference between the baseline and updated scores, thus generating a ranking of how much each feature influences the goodness of the model by whatever metric we are using.

The do something part is where the two methods diverge, where for permutation importance we permute one of the features considering independence with respect to the target variable and other predictor features, while instead for drop column we remove the feature entirely and retrain the model before computing the score again.

To me these two operations should give almost the same result, with the disadvantage in drop column of having to train the model P times (no. of features). Of course this is not the case, as per this blog post (emphasis mine):

Permutation importance does not require the retraining of the underlying model [...], this is a big performance win. The risk is a potential bias towards correlated predictive variables.

If we ignore the computation cost of retraining the model, we can get the most accurate feature importance using a brute force drop-column importance mechanism.

However I don't understand why the act of permutating (permuting?) one of the features, with the explicit consideration of independence between both the target variable and the other predictors, would give a bias towards correlated features.

I am aware of Strobl's work on a conditional permutation scheme which tackles specifically this issue, however it does so by modifying the null hypothesis under which the permutation is performed and considering indepence only between the feature and the target, which to me, as you can guess, is counterintuitive.

What I also don't understand is if, bias towards correlated feature notwithstanding, the two algorithms are equivalent, so to speak.

P.S.: On a side note, I would also like to know if there's a source in literature for the drop column approach, as I am including it in a text I'm currently writing and don't know how to reference it since I couldn't find an original source.

EDIT:

I now better understand why Strobl et al.'s proposed conditional permutation scheme is less biased towards correlated variables, this passage from their paper (Conditional variable importance for random forests published in BCM Bioinformatics) was quite clear:

What is crucial when we want to understand why correlated predictor variables are preferred by the original random forest permutation importance is that a positive value of the importance corresponds to a deviation from this null hypothesis – that can be caused by a violation of either part: the independence of $X_j$ and $Y$, or the independence of $X_j$ and $Z$ [other predictor variables, ed.]. However, from these two aspects only one is of interest when we want to assess the impact of $X_j$ to help predict $Y$, namely the question if $X_j$ and $Y$ are independent. This aim, to measure only the impact of $X_j$ on $Y$, would be better reflected if we could create a measure of deviation from the null hypothesis that $X_j$ and $Y$ are independent under a given correlation structure between $X_j$ and the other predictor variables, that is determined by our data set. To meet this aim we suggest a conditional permutation scheme, where $X_j$ is permuted only within groups of observations with $Z$ = $z$, to preserve the correlation structure between $X_j$ and the other predictor variables [...]

My question still stands as to what differs between the original permutation scheme and the drop column technique.

Moreover, as highlighted in the comments, in the extreme example of a duplicate column in the data the drop column algorithm will assign a score of 0 to both (due to how importance is computed in this case), and while this is certainly not biased towards correlated features it removes the importance of a potentially very influential column from the results. What's more, since the importance metric is computed from the underlying random forest (which as we have established has inherent biases), when we have two strongly correlated columns and drop one of the two the modell will still be able to obtain some of that information from the correlated feature still present in the data set, and thus there (potentially) won't be a large difference between the baseline score and the score computed when dropping either feature, meaning that both correlated features will have a lower overall importance score at the end.

This to me is in stark contrast to the "most accurate feature importace" that we are supposed to get from this very computationally expensive algorithm, and thus I'm even more confused as to what are the benefits of drop column over the original permutation scheme, not to mention the conditional variation by Strobl et al.

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    $\begingroup$ A couple of quick observations. First, Breiman developed RFs on his laptop with thousands of observations and two to three thousand predictors (or variables) and a few thousand iterations creating mini-trees or mini-models with each iteration. In his discussion of RFs spoke of it wrt two distinct kinds of permutations. For each iteration: observations are bootstrapped (with replacement) and subsets of randomly sampled predictors. In this way a distribution of how each predictor performed was constructed. Like you, I couldn't find lit wrt Drop Column but it sounds like only 50% of an RF. $\endgroup$
    – user78229
    Aug 11, 2020 at 17:13
  • $\begingroup$ Is your question not answered in the section entitled "The effect of collinear features on importance"? Two strongly correlated features will be used for roughly equal numbers of splits, & permuting either in the out-of-bag validation samples will have about the same effect on predictive performance, hence roughly equal importances. But drop either one & re-fit, & the other takes its place, resulting in a tiny decrease in performance & hence negligible importance. (Drop both together though, & the decrease in performance might be large.) $\endgroup$ Aug 11, 2020 at 21:59
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    $\begingroup$ @Scortchi-ReinstateMonica, on the contrary, that section leaves me even more confused. Drop Column is supposed to be the most accurate, but if you dupe a column both will have importance 0 (which to me is wrong), while permutation importance handles the situation a bit more gracefully and shares the importance over the two features. And when looking at "Dealing with collinear features" you see that there's even code to handle collinear/duped features so that they are treated as one and their importance is boosted as a group. $\endgroup$ Aug 11, 2020 at 22:20
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    $\begingroup$ I think what's confusing is that the authors seem to have a prior notion of what feature importance ought to be - according to which some metrics are praised & others found wanting - but they don't share it with their readers. $\endgroup$ Aug 11, 2020 at 22:42
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    $\begingroup$ ... its marginal contribution isn't the same as its conditional contribution; arguments for calculating one or the other, or something in between, come down to what you're going to use "importance" for. There's some relevant discussion in Groemping (2009), Am. Stat., 63, 4, "Variable Importance Assessment in Regression: Linear Regression versus Random Forest". $\endgroup$ Aug 17, 2020 at 8:44

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Your question and discussion with Scortchiare are very thought-provoking. I'd like to share my perspective on the difference between these two methods.

Permutation Importance and Drop Column Importance are associated with two types of data collection problems. Permutation importance provides an estimation for the performance loss when the data is collected wrongly or manipulated, like a column being shifted one row upward/downward. Drop column importance gives us an insight into what will happen if we don't collect or use this feature.

The choice depends, among other factors, on the prospective data issues.

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