Perfect aggregation for a class of general reliability models with Bayesian updating

Aggregation is an important tool for handling complex problems, because it can reduce a large number of parameters in the original problem to a smaller number of aggregate parameters. Unfortunately, most aggregated problems yield results different from those of the original problems, a fact that is not widely recognized; in other words, perfect aggregation is usually not achieved. In this paper, we develop necessary and sufficient conditions for perfect aggregation in Bayesian estimation (i.e., consistency of Bayesian analyses performed at the system and component levels) for a class of frequently used reliability models. The conditions for perfect aggregation are shown to be extremely stringent. Also, some implications of these results are discussed. In particular, because perfect aggregation is very unlikely, analyzing data at the system level when component- or subsystem-level data are available can be misleading.