integral priors for binomial regression

Diego Salmerón and Juan Antonio Cano from Murcia, Spain (check the movie linked to the above photograph!), kindly included me in their recent integral prior paper, even though I mainly provided (constructive) criticism. The paper has just been arXived.

A few years ago (2008 to be precise), we wrote together an integral prior paper, published in TEST, where we exploited the implicit equation defining those priors (Pérez and Berger, 2002), to construct a Markov chain providing simulations from both integral priors. This time, we consider the case of a binomial regression model and the problem of variable selection. The integral equations are similarly defined and a Markov chain can again be used to simulate from the integral priors. However, the difficulty therein follows from the regression structure, which makes selecting training datasets more elaborate, and  whose posterior is not standard. Most fortunately, because the training dataset is exactly the right dimension, a re-parameterisation allows for a simulation of Bernoulli probabilities, provided a Jeffreys prior is used on those.  (This obviously makes the “prior” dependent on the selected training dataset, but it should not overly impact the resulting inference.)