The Best Two Independent Measurements Are Not the Two Best

Consider an item that belongs to one of two classes, θ = 0 or θ = 1, with equal probability. Suppose also that there are two measurement experiments E 1 and E 2 that can be performed, and suppose that the outcomes are independent (given θ). Let E í denote an independent performance of experiment E I . Let P e (E) denote the probability of error resulting from the performance of experiment E. Elashoff 1 gives an example of three experiments E 1 ,E 2 ,E 3 such that P e (E 1 ) e (E 2 ) e (E 3 ), but P e (E 1 ,E 3 ) e (E 1 ,E 2 ). Toussaint 2 exhibits binary valued experiments satisfying P e (E 1 ) e (E 2 ) e (E 3 ), such that P e (E 2 ,E 3 ) e (E 1 ,E 3 ) e (E 1 ,E 2 ). We shall give an example of binary valued experiments E 1 and E 2 such that P e (E 1 ) e (E 2 ), but P e (E 2 ,E 2 ') e (E 1 ,E 2 ) e (E 1 ,E 1 '). Thus if one observation is allowed, E 1 is the best experiment. If two observations are allowed, then two independent copies of the ''worst'' experiment E 2 are preferred. This is true despite the conditional independence of the observations.