The main idea of nested sampling is to substitute the high-dimensional likelihood integral over the parameter space $\Omega$ by an integral over the unit line $[0,1]$ by employing a push-forward with respect to a suitable transformation. For this substitution, it is often implicitly or explicitly assumed that samples from the prior are uniformly distributed along this unit line after having been mapped by this transformation. We show that this assumption is wrong, especially in the case of a likelihood function with plateaus. Nevertheless, we show that the substitution enacted by nested sampling works because of more interesting reasons which we lay out. Although this means that analytically, nested sampling can deal with plateaus in the likelihood function, the actual performance of the algorithm suffers under such a setting and the method fails to approximate the evidence, mean and variance appropriately. We suggest a robust implementation of nested sampling by a simple decomposition idea which demonstrably overcomes this issue.
翻译:嵌套取样的主要想法是用单位线$[0,1,1]美元的一个整体部分来取代参数空间的高度可能性($\Omega$),代之以单位线$[0,1,1]美元的一个整体部分。对于这一替代,人们往往隐含或明确假定,在经过这一变换后绘制了图后,原原体样品在单位线上均匀分布。我们表明,这一假设是错误的,特别是在高原的概率功能方面。然而,我们表明,由于我们提出的更有趣的理由,嵌套取样工作所实行的替代方法。尽管这意味着,从分析角度来说,嵌套式取样可以处理概率功能中的高原,但算法的实际性能在这种变换法下受到影响,而且该方法未能适当地接近证据、平均值和差异。我们建议,通过简单的分解法来强有力地实施嵌套式取样,这可以明显地克服这一问题。