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
) < P
_e
(E
₂
) < P
_e
(E
₃
), but P
_e
(E
₁
,E
₃
) < P
_e
(E
₁
,E
₂
). Toussaint [2] exhibits binary valued experiments satisfying P
_e
(E
₁
) < P
_e
(E
₂
) < P
_e
(E
₃
), such that P
_e
(E
₂
,E
₃
) < P
_e
(E
₁
,E
₃
) < P
_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
) < P
_e
(E
₂
), but P
_e
(E
₂
,E
₂
') < P,
_e
(E
₁
,E
₂
) < P,
_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.