We discuss inference for a human phage display experiment with three stages. The data are tripeptide counts by tissue and stage. The primary aim of the experiment is to identify ligands that bind with high affinity to a given tissue. We formalize the research question as inference about the monotonicity of mean counts over stages. The inference goal is then to identify a list of peptide–tissue pairs with significant increase over stages. We use a semiparametric Dirichlet process mixture of Poisson model. The posterior distribution under this model allows the desired inference about the monotonicity of mean counts. However, the desired inference summary as a list of peptide–tissue pairs with significant increase involves a massive multiplicity problem. We consider two alternative approaches to address this multiplicity issue. First we propose an approach based on the control of the posterior expected false discovery rate. We notice that the implied solution ignores the relative size of the increase. This motivates a second approach based on a utility function that includes explicit weights for the size of the increase.