The construction of proper Bayes tests which are invariant under non-compact groups has been treated in Kiefer and Schwartz (1965) and Schwartz (1966, 1969). These papers are limited to testing problems concerning exponential families' $\operatorname{pdf}$'s. The present note shows, by means of two simple examples, that the same methods can sometimes be applied more generally. One example considers estimation of the ratio of the mean to the variance for the normal distribution. The second example considers testing problems in which the pdf's have an exponential factor as well as a non-exponential factor.