Reference priors with partial information

In this paper, reference priors are derived for three cases where partial information is available. If a subjective conditional prior is given, two reasonable methods are proposed for finding the marginal reference prior. If, instead, a subjective marginal prior is available, a method for defining the conditional reference prior is proposed. A sufficient condition is then given under which this conditional reference prior agrees with the conditional reference prior derived in the first stage of the reference prior algorithm of Berger & Bernardo (1989, 1992). Finally, under the assumption of independence, a method for finding marginal reference priors is also proposed. Various examples are given to illustrate the methods.