Inferences for hierarchical models with partial prior information

Suppose items can be purchased from one of k-suppliers and it is required to purchase from the one with the smaller failure rate or equivalently from the one with the larger mean-time-to-failure. It is assumed that data $\mathit{d}$, in the form of the times-to-failure for $n_1,\dots ,n_k$ items from suppliers $1,\dots ,k$, respectively is available. There are two suggested selection criteria studied in this paper and when comparing only two suppliers they reduce to$P({\theta }_1<b{\theta }_2|d)\text{and}P(Y_1>{\mathit{cY}}_2|d)\text{,}$where b and c are prespecified practical constants; ${\theta }_1$ and ${\theta }_2$ are the respective mean failure rates; $Y_1$ and $Y_2$ are the predicted times to failure for individual items purchased from each supplier.In addition partial prior information about the $k$-suppliers collectively is assumed to have been elicited. This situation is modelled using the hierarchical Bayesian approach, which easily facilitates interpreting the elicited partial prior information as constraints on the hyperpriors, i.e. hyperpriors that are known only to be contained in families with specified properties. In this paper these properties are assumed to be in the form of specifying certain quantiles arising from the elicited information. Minimum and maximum values of the above selection criteria are obtained and are used to indicate whether or not the elicited prior information is useful. Specific examples are given for comparing two suppliers but generalisation to k-suppliers follows easily.