English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the nonparametric multivariate isotonic regression problem, wherethe regression function is assumed to be nondecreasing with respect to eachpredictor. Our goal is to construct a Bayesian credible interval for thefunction value at a given interior point with assured limiting frequentistcoverage. We put a prior on unrestricted step-functions, but make inferenceusing the induced posterior measure by an "immersion map" from the space ofunrestricted functions to that of multivariate monotone functions. This allowsmaintaining the natural conjugacy for posterior sampling. A natural immersionmap to use is a projection via a distance, but in the present context, a blockisotonization map is found to be more useful. The approach of using the induced"immersion posterior" measure instead of the original posterior to makeinference provides a useful extension of the Bayesian paradigm, particularlyhelpful when the model space is restricted by some complex relations. Weestablish a key weak convergence result for the posterior distribution of thefunction at a point in terms of some functional of a multi-indexed Gaussianprocess that leads to an expression for the limiting coverage of the Bayesiancredible interval. Analogous to a recent result for univariate monotonefunctions, we find that the limiting coverage is slightly higher than thecredibility, the opposite of a phenomenon observed in smoothing problems.Interestingly, the relation between credibility and limiting coverage does notinvolve any unknown parameter. Hence by a recalibration procedure, we can get apredetermined asymptotic coverage by choosing a suitable credibility levelsmaller than the targeted coverage, and thus also shorten the credibleintervals.

Coverage of Credible Intervals in Bayesian Multivariate Isotonic Regression