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Abstract:
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We study the asymptotic coverage of Bayesian credibility intervals for monotone functions. We consider a projection-posterior distribution induced by the projection on the space of monotone functions of a sample from an unrestricted posterior distribution. Here, the centered and scaled posterior distribution does not converge to a fixed distribution in probability, but only weakly. Nevertheless, we can evaluate the asymptotic coverage of a posterior quantile interval. For models index by smooth functions, Bayesian credible regions may have low or zero asymptotic coverage. In the present context, the limiting coverage is not equal to the credibility level, but is positive, depends only on the credibility level used, and is higher than the latter. Moreover, we can achieve any targeted coverage by starting with a lower credibility level that can be calculated from the expression for the limiting coverage. We illustrate the results for monotone regression function, monotone density function and a monotone regression quantile. Some ideas for an extension to multivariate monotone regression will also be discussed.
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