Comment on Article by Berger, Bernardo, and Sun

It is our distinct pleasure to comment on a very thought provoking paper, and we first congratulate the Authors for this new masterly contribution in the field of objective priors. The main goal of the paper is to find a multi-purpose objective prior for a model that should be used by different researchers with varying goals, with the consequence that no single parameter or parametric function can be identified as a parameter of interest. In this situation, the most popular approaches either fail or, as in the case of the reference prior algorithm, they cannot be used. Three general methods are discussed by the Authors. The first one is limited to a number of particular situations where the reference prior is the same for all quantities of interest: this case is not of much concern since a natural solution exists. The second method is based on the reference prior approach: one looks for the prior which produces the marginal posteriors for the quantities of interest which are closer – in some sense to the marginal reference posteriors. Whereas this method is perfectly reasonable, the final result will depend on the particular set of the quantities of interest considered and it cannot be considered as the “overall” objective prior. The third method is based on a hierarchical representation of the model, when it is available. It shifts the problem of determining an objective prior to an upper level of the hierarchy, where the impact of the prior might be less serious.