Using uncertainty functions: comparison of experiments

Suppose that two populations, X and Y are available to an experimenter. Population X has probability density function (p.d.f.) f and population Y has p.d.f. h. It is assumed that of the populations has been replaced by another population with p.d.f. g. Let 𝐻1 denote the hypothesis that population Y has been replaced and let 𝐻2 denote the hypothesis that population X has been replaced. It is assumed that the experimenter can express a prior distribution which hypothesis is true. The experimenter can take 𝑛 observations (𝑛 ≥ 1), and he must allocate these observations such that the amount of information about which hypothesis is true, using a given uncertainty function, is maximised. Sequential and non-sequential allocation of observations is considered for various uncertainty functions and different p.d.f.'s. A generalisation of the problem is given for k+1 populations (𝑘 ≥ 1). In particular, suppose that k+1 coins are available to the experimenter. It is known that one of these coins been replaced by a two-headed coin. Each toss of a coin has a certain cost. The experimenter can take observations until the two-headed coin is found. An optimal strategy that minimises expected cost of finding the two-headed coin is given.