The method of minimum necessary samples formulated by Novick and Hall from a suggestion by Lindley is refined and extended to cover multiparameter models. A number of such models are studied and Bayesian indifference specifications are obtained for each within extended natural conjugate families. In all cases the one-parameter conditional specifications derived from the multiparameter specifications are consistent with specifications derivable from corresponding one-parameter models. Models studied include the multinomial models, one- and two-parameter uniform models, an exponential model with a range parameter, and the two-parameter normal model. For these models the specifications seem entirely straightforward. Indifference specifications are also obtained for the simple regression model and the bivariate normal model. It is noted that these specifications do not agree but the relationship between these two specifications is described and in part justified. It is pointed out that these differences could, in fact, mirror important differences in the structural assumptions of these models. A random effects model of value in mental test theory is studied briefly. This model provides a framework for a discussion of the multiple comparisons problem. Finally some summary comments on the methods of this paper are offered.