Computer vision is the subfield of computer science devoted to developing algorithms that can “understand” images. For example, some security systems use facial recognition software to decide whether...

Computer vision is the subfield of computer science devoted to developing algorithms that can “understand” images. For example, some security systems use facial recognition software to decide whether to grant access to a particular person. We desire to maximize the probability that the vision algorithm we choose gets the answer right—that is, grants access to the person if and only if that person is authorized to enter

Suppose that we have two algorithms, A and B, that we have employed on two different cameras in a test run. Suppose that algorithm A is deployed on Camera I. It makes the correct decision on 75% of the CS majors at Camera I and 60% of philosophy majors at Camera I. (That is, when a CS major arrives at Camera I, algorithm A correctly decides whether to grant her access 75% of the time.) Algorithm B, deployed at Camera II, makes the correct decision on 70% of CS majors and 50% of philosophy majors. The following claim seems obvious, because Algorithm A performed better for both philosophy majors and CS majors:

Claim: Algorithm A is right a higher fraction of the time (overall, combining both majors) than Algorithm B. But the claim is false, as you’ll show!

The falsehood of this claim (for example, in the scenario illustrated by the next exercise) is called Simpson’s Paradox because the behavior is so counterintuitive. State precisely where the following argument goes wrong: