Why can two highly correlated features both look unimportant under permutation importance?

## The mechanism Permutation importance holds every other column fixed while it scrambles one. That design is exactly what makes it a statement about *this* model rather than about the data — and it is also what breaks when two columns carry the same information. Take a manufacturing-yield model whose training table happens to include inlet temperature twice, once in Celsius and once in Fahrenheit. The two columns are a deterministic transform of each other, so any pattern the model could learn from one it can equally learn from the other. During training, the fit distributes its reliance between them in some arbitrary way. Now permute the Celsius column: the Fahrenheit column is untouched, still carries the full signal, and the model's predictions barely move. Score drop: near zero. Permute Fahrenheit instead: same story. Report both numbers side by side and temperature — the dominant physical driver of yield — appears to be irrelevant. This is not a bug in the arithmetic. The measurement is literally correct: **removing either column alone costs this model almost nothing**, because the other one is a perfect substitute. The mistake is reading 'unimportant column' as 'unimportant quantity'. ## It is a matter of degree Exact duplicates are the clean teaching case, but the effect is continuous. Two features correlated at 0.95 will each absorb most of the other's shuffle, and both drops shrink toward zero. At 0.6 the masking is partial and both features still show some importance, but less than either would show alone. Any correlated cluster — three sensor channels on the same shaft, several ratios built from the same two raw columns — dilutes the same way, spreading a real dependency across several small numbers that individually look like noise. ## The second problem: impossible rows Shuffling one column of a correlated pair does not just leave a substitute in place; it also *creates rows that could never occur*. After the shuffle, a row may show 5 degrees Celsius beside 200 degrees Fahrenheit, or a low-power reading beside a high-vibration reading that never co-occur in the plant. The model is being evaluated on inputs far from anything it was trained on, and its behaviour there is unconstrained. Whatever drop you measure is then a mixture of 'the model lost information' and 'the model extrapolated badly'. Both effects are real, but they are different findings, and the single number does not separate them. ## What to do instead - **Group and permute jointly.** Cluster the features by correlation — a hierarchical clustering on the correlation matrix with a cut at a chosen threshold is the usual route — then permute every column in a group together and record one drop for the group. In the temperature example, shuffling both columns at once produces the large drop that reflects the real dependency. - **Report the ranking at the group level.** Present 'temperature block: 0.09 AUC' rather than two near-zero rows that invite the wrong conclusion. - **Confirm with a refit.** If a group's zero importance is surprising, drop the whole group and retrain. That answers the different but often more useful question of whether the information is available anywhere else in the table. - **De-duplicate before you explain.** If two columns are the same quantity in different units, the honest fix is upstream: keep one. - **Condition instead of marginalise.** Conditional variants permute a feature only within groups of rows that are similar on the correlated columns, which keeps the rows plausible. They cost more and change the question slightly — they measure additional information beyond the correlated peers rather than total reliance — so state which one you ran. ## What to say in an interview Name the mechanism (the intact substitute), name the second-order effect (off-distribution rows), and land on the practical rule: **never conclude 'this feature does not matter' from a low permutation importance without first checking what it is correlated with.** A candidate who volunteers group permutation as the fix, and who notes that group importance and individual importance answer different questions, is showing the judgment the question is testing.