A sequence of \(N\) independent measurements is taken from a Poisson distribution \(\{x\}\) whose mean is \(m_{0}\) under \(H_{0}\), and \(m_{1}\) under \(H_{1}\). On what combination of the measurements should a Bayes test be based, and with what decision level should its outcome be compared, for given prior probabilities \((\xi, 1-\xi)\) and a given cost matrix \(\boldsymbol{C}\) ?

The Poisson distribution with mean \(m\) assigns a probability \(p(x)=\frac{m^{x} e^{-m}}{x !}\) to the positive integers \(x\) and probability 0 to all noninteger values of \(x\).