The Copas parametric model is aimed at exploring the potential impact of publication bias via sensitivity analysis, by making assumptions regarding the probability of publication of individual studies related to the standard error of their effect sizes. Reviewers often have prior assumptions about the extent of selection in the set of studies included in a meta-analysis. However, a Bayesian implementation of the Copas model has not been studied yet. We aim to present a Bayesian selection model for publication bias and to extend it to the case of network meta-analysis where each treatment is compared either with placebo or with a reference treatment creating a star-shaped network. We take advantage of the greater flexibility offered in the Bayesian context to incorporate in the model prior information on the extent and strength of selection. To derive prior distributions, we use both external data and an elicitation process of expert opinion. Copyright © 2012 John Wiley & Sons, Ltd.