Towards agreement: Bayesian experimental design

An experimenter wishes to design an experiment to settle an inferential question about the value of a parameter $\theta$. The data $X\sb{1}, \..., X\sb{n}$ from such an experiment will be viewed by a class $\Gamma$ of Bayesians, where each such Bayesian $\gamma$ has a prior distribution $\pi\sb\gamma (\theta)$ for $\theta$. Denote by $A\sb\theta$ the event: the collection of all samples $X\sb{1}, \..., X\sb{n}$ for which all Bayesians in $\Gamma$ agree to the correct decision concerning $\theta$. Using his own prior distribution $\pi\sb* (\theta)$, the experimenter wishes the preposterior probability $P(A\sb\theta)$ to be at least as large as a prespecified constant $\epsilon\ (0 \u3c \epsilon \u3c 1)$. In the case of hypothesis testing, this paper gives necessary conditions for the existence of a sample size $N\sb\epsilon$ achieving these goals, and also gives some sufficient conditions for $N\sb\epsilon$ to exist. Interestingly, $P(A\sb\theta)$ need not be monotone increasing in n, so that observing data additional to the experiment can cause $P(A\sb\theta)$ to decrease from above $\epsilon$ to below $\epsilon$. Consequently, to better settle the correct decision concerning $\theta$, the smallest value of $N\sb\epsilon$ such that $P(A\sb\theta) \geq \epsilon$ for all $n \geq N\sb\epsilon$ is sought. Bounds and numerical algorithms for $N\sb\epsilon$ are given. Some results extending the theory to estimation problems involving $\theta$ are also presented. Restrict the event $A\sb\theta$ so that each Bayesian, in addition to choosing the correct decision, also satisfies his own goal for a low posterior expected loss using that correct decision. This definition of $A\sb\theta$ extends the theory, reducing to the original theory through an induced set of new priors in the case of hypothesis testing