The Best Two Independent Measurements Are Not the Two Best

Consider an item that belongs to one of two classes, ¿ = 0 or ¿ = 1, with equal probability. Suppose also that there are two measurement experiments E1 and E2 that can be performed, and suppose that the outcomes are independent (given ¿). Let Ei¿ denote an independent performance of experiment Ei. Let Pe(E) denote the probability of error resulting from the performance of experiment E. Elashoff [1] gives an example of three experiments E1,E2,E3 such that Pe(E1) < Pe(E2) < Pe(E3), but Pe(E1,E3) < Pe(E1,E2). Toussaint [2] exhibits binary valued experiments satisfying Pe(E1) < Pe(E2) < Pe(E3), such that Pe(E2,E3) < Pe(E1,E3) < Pe(E1,E2). We shall give an example of binary valued experiments E1 and E2 such that Pe(E1) < Pe(E2), but Pe(E2,E2¿) < Pe(E1,E2) < Pe(E1,E1¿). Thus if one observation is allowed, E1 is the best experiment. If two observations are allowed, then two independent copies of the ``worst´´ experiment E2 are preferred. This is true despite the conditional independence of the observations.