ARD is also a bad measure for relevance as it measures mostly non-linearity [1510.04813] Projection predictive model selection for Gaussian processes (published in Projection predictive model selection for Gaussian processes | IEEE Conference Publication | IEEE Xplore)

With that ratio, no need for MCMC. Use something like GPy, GPflow or in R CRAN - Package gplite

Sorry, I don’t understand what you are saying.

If you want to model all interactions, have binomial observation model, 5000 observations and 14 variables, use something else than Stan.

If you are happy with an additive model, have binomial observation model, 5000 observations and 14 variables, you can try the basis function approximated GP also in Stan (probably good to use glm compound functions for speedup).