Coverage of credible intervals in Bayesian multivariate isotonic regression

Abstract

We consider the nonparametric multivariate isotonic regression problem, where the regression function is assumed to be nondecreasing with respect to each predictor. Our goal is to construct a Bayesian credible interval for the function value at a given interior point with assured limiting frequentist coverage. A natural prior on the regression function is given by a random step function with a suitable prior on increasing step-heights, but the resulting posterior distribution is hard to analyze theoretically due to the complicated order restriction on the coefficients. We instead put a prior on unrestricted step-functions, but make inference using the induced posterior measure by an “immersion map” from the space of unrestricted functions to that of multivariate monotone functions. This allows for maintaining the natural conjugacy for posterior sampling. A natural immersion map to use is a projection with respect to a distance function, but in the present context, a block isotonization map is found to be more useful. The approach of using the induced “immersion posterior” measure instead of the original posterior to make inference provides a useful extension of the Bayesian paradigm, particularly helpful when the model space is restricted by some complex relations. We establish a key weak convergence result for the posterior distribution of the function at a point in terms of some functional of a multiindexed Gaussian process that leads to an expression for the limiting coverage of the Bayesian credible interval. Analogous to a recent result for univariate monotone functions, we find that the limiting coverage is slightly higher than the credibility, the opposite of a phenomenon observed in smoothing problems. Interestingly, the relation between credibility and limiting coverage does not involve any unknown parameter. Hence, by a recalibration procedure, we can get a predetermined asymptotic coverage by choosing a suitable credibility level smaller than the targeted coverage, and thus also shorten the credible intervals.