Exercise 2: Identifying the Three Methods That Independently Confirm Salary's Importance — Possible Solution ==================================================================== METHOD 1: ml1-5's LOGISTIC REGRESSION COEFFICIENT ------------------------------ Per ml1-5, fitting logistic regression on the standardized features produced "salary's own coefficient... negative and substantial — higher salary genuinely associates with a lower predicted probability of leaving." This is the first, original piece of evidence: a smooth, formula-fitting algorithm ranking salary as an important predictor via its own learned coefficient size. METHOD 2: ml1-7's DECISION TREE FIRST SPLIT ------------------------------ Per ml1-7, "the tree's own very first, most impurity-reducing split plausibly lands on salary — the exact same feature ml1-5's own logistic regression coefficients already flagged as the strongest predictor." This is a second, mechanically unrelated method (greedy entropy- reduction search, rather than coefficient fitting) independently arriving at the same feature. METHOD 3: THIS CHAPTER'S OWN RANDOM FOREST feature_importances_ ------------------------------ Per this chapter's own finding-box, "the random forest's own feature_importances_ plausibly ranks salary highest, echoing ml1-5's own coefficient and ml1-7's own worked split — the third independent method now agreeing on the same answer." This is a third method — an ensemble of many trees, each trained on random data/feature subsets per ml1-7's own bagging description — likewise ranking salary as the most important feature. WHY THIS COUNTS AS "A SECOND, INDEPENDENT CONFIRMATION" ------------------------------ Per ml1-7's own earlier reasoning about the tree/logistic-regression agreement, "it's much less likely that both a formula-fitting process and a rule-splitting process would agree by pure coincidence" if there were no real, genuine pattern in the data. This chapter's random forest result extends that same logic one step further: now THREE mechanically distinct approaches (smooth coefficient fitting, a single greedy tree, and an ensemble of many randomly-varied trees) have all independently converged on salary as the standout predictor. Each additional independent method reaching the same conclusion further reduces the plausibility that the agreement is coincidental, since coincidental agreement across three genuinely different mechanisms is substantially less likely than across just two. WHY "INDEPENDENT" IS THE OPERATIVE WORD ------------------------------ None of the three methods shares its own underlying fitting mechanism with either of the other two — logistic regression fits a smooth global formula, a single decision tree performs a greedy local search, and a random forest averages many such trees each trained on a randomized subset of the data. Because they don't share assumptions, calculation methods, or even fundamentally similar mathematical approaches, their agreement carries genuine evidential weight rather than simply reflecting the same calculation performed three different ways. WHY THIS WORKS AS AN ANSWER ------------------------------ It identifies all three specific methods (ml1-5's coefficients, ml1-7's tree split, this chapter's own forest importances) by name and by chapter, and explains — using ml1-7's own "unlikely to be coincidence" reasoning extended to a third method — why three independently converging results constitute meaningfully stronger evidence than two.