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In ensemble learning with bagging, why is it important to do bootstrap resampling (sampling with replacement) instead of just sub-sampling (sampling without replacement)?

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It's not. Sampling with replacement is not required. Two issues come up when you use subsampling without replacement instead of the usual bootstrap samples: 1. You must determine what sub-sample size to use, and 2. Out of bag observations are no longer created in the same way.

See Buja and Stuetzel, 2006, section 7 (entitled "Equivalence of Resampling With and Without Replacement in Bagging") here http://www.stat.washington.edu/wxs/Learning-papers/paper-bag.pdf for a reference.

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