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I am reading the chapter on random forests by Leo Breiman (found here: https://www.stat.berkeley.edu/~breiman/randomforest2001.pdf).

In section 3.1 Using out-of-bag estimates to monitor error, strength, and correlation (page 11), it says:

In each bootstrap training set, about one-third of the instances are left out. Therefore, the out-of-bag estimates are based on combining only about one-third as many classifiers as in the ongoing main combination.

I am not sure I understand how the first sentence (that about one-third of cases are left out of each bootstrap sample) implies the second (that each case is OOB in about one-third of the trees)?

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In a bootstrap sample we replace. The probability of a subject x being sampled with replacement is at about 2/3. If we build for example 1000 trees with different bootstrap samples, we therefore expect that x will be in (2/3)*1000 of these samples. One third of the trees were therefore not build with subject x. The OOB only calculates the error for the trees that have not been build with x, which is (1/3)*1000 in that case.

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