Using quasi-densities to summarize and present the posterior distribution of parameter contrasts in statistical models

Consider Bayesian inference on statistical models in which contrasts among parameters are of interest. Usually, the multivariate posterior distribution of contrasts is not available in closed form and approximate Bayesian inference relies on a sample from that distribution. This, however, makes it difficult to report the posterior density in practice, for example in a journal publication, and therefore to allow subsequent readers to perform inference on contrasts of their own interest. We propose an approximation to the posterior distribution, which can easily be reported in published work. The approximation is in terms of a set of univariate densities qj(x) such that the posterior of any set of contrasts can be approximated by considering the original parameters as independent random variables, with the j-th parameter having density qj . This approximation resembles the quasi-variance approximation to the covariance matrix of contrasts, so the densities qj may be called quasi-densities. In order to calculate quasi-densities, we model the logarithm of each density as a spline function. We present ways of estimating the parameters of the log-spline quasidensities and discuss their numerical implementation. Some alternative parametric forms are also considered. It is also discussed how to assess the accuracy of the quasi-density approximation by using suitable error functionals, such as the total variation distance and the Kolmogorov-Smirnov distance. Finally, the use of quasi-densities is illustrated in some real-data examples. Statistical models considered in these examples include generalized linear models, mixed-effects models, Bradley-Terry models and association models