We show how to find bounds on posterior expectations of arbitrary functions of the parameters when the prior marginals are specified but when the complete joint prior is unspecified. We also give a theorem that is useful for finding posterior bounds in a wide range of Bayesian robustness problems. We apply these techniques to two examples. The first example involves a recent clinical trial for extracorporeal membrane oxygenation (ECMO). Our analysis may be regarded as a follow-up to a detailed Bayesian analysis given by Kass and Greenhouse who concluded that the posterior probability that the treatment is superior to the control is about .95. Their analysis, however, assumed a priori independence of the parameters. We consider other prior distributions with the same marginals as Kass and Greenhouse, but in which the parameters are not independent and conclude that, as long as a priori independence is at least approximately tenable, then ECMO seems superior to the control. The second example is the product of means problem, which has been studied in the Bayesian context by Berger and Bernardo. Here the goal is to find the posterior expectation of αβ, where α and β are the means of conditionally independent random variables X and Y. Berger and Bernardo recommended a joint prior π0 proportional to (α2 + β2)1/2. We find that among all priors with the same marginals as π0, the posterior expectation of αβ can be made arbitrarily large or arbitrarily close to 0. Furthermore, the parameterization is important: with a different parameterization the upper bound is strictly finite. © 1991 Taylor & Francis Group, LLC.