Quantitative studies in many fields involve the analysis of multivariate data of diverse types, including measurements that we may consider binary, ordinal and continuous. One approach to the analysis of such mixed data is to use a copula model, in which the associations among the variables are parameterized separately from their univariate marginal distributions. The purpose of this article is to provide a method of semiparametric inference for copula models via the construction of what we call a marginal set likelihood function for the association parameters. The proposed method of inference can be viewed as a generalization of marginal likelihood estimation, in which inference for a parameter of interest is based on a summary statistic whose sampling distribution is not a function of any nuisance parameters. In the context of copula estimation, the marginal set likelihood is a function of the association parameters only and its applicability does not depend on any assumptions about the marginal distributions of the data, thus making it appropriate for the analysis of mixed continuous and discrete data with arbitrary margins. Estimation and inference for parameters of the Gaussian copula are available via a straightforward Markov chain Monte Carlo algorithm based on Gibbs sampling.