Nonparametric Full Bayesian Significance Testing for Bayesian Histograms

In this article, we present an extension of the Full Bayesian Significance Test (FBST) for nonparametric settings, termed NP-FBST, which is constructed using the limit of finite dimension histograms. The test statistics for NP-FBST are based on a plug-in estimate of the cross-entropy between the null hypothesis and a histogram. This method shares similarities with Kullback–Leibler and entropy-based goodness-of-fit tests, but it can be applied to a broader range of hypotheses and is generally less computationally intensive. We demonstrate that when the number of histogram bins increases slowly with the sample size, the NP-FBST is consistent for Lipschitz continuous data-generating densities. Additionally, we propose an algorithm to optimize the NP-FBST. Through simulations, we compare the performance of the NP-FBST to traditional methods for testing uniformity. Our results indicate that the NP-FBST is competitive in terms of power, even surpassing the most powerful likelihood-ratio-based procedures for very small sample sizes.