Left out of the bootstrap
A bootstrap sample is drawn from a data set of 100 points: 100 draws with replacement, each point equally likely on every draw.
Find the probability that a given data point does not appear in the bootstrap sample.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Look at one draw: what is the chance it misses the given point?
The draws are independent.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The point is left out only if every one of the 100 independent draws misses it.
Step 1. One draw picks the given point with probability and misses it with probability .
Step 2. The 100 draws are independent, so
For large samples. With points, . So a bootstrap sample contains on average only about 63.2% of the distinct data points. The points left out of a given bootstrap sample can serve as a test set: this is the “out-of-bag” error used by random forests.