Validation of Bayesian posterior distributions using a multidimensional Kolmogorov-Smirnov test
Validation of Bayesian posterior distributions using a multidimensional Kolmogorov-Smirnov test
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DOI：
10.1093/mnras/stv1110
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发表时间：
2014-04
影响因子：
4.8
通讯作者：
D. Harrison;D. Sutton;P. Carvalho;M. Hobson
中科院分区：
文献类型：
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作者：
D. Harrison;D. Sutton;P. Carvalho;M. Hobson
我们将柯尔莫哥洛夫-斯米尔诺夫（K-S）检验推广到多个维度，提出了一个R n ！[0;1]基于所考虑的参考分布的最高概率密度区域的概率内容进行映射；这种映射将问题还原为可以应用标准K-S检验的一维情况。这种映射的通用性也允许我们引入一种简单而通用的方法来验证任何维度的贝叶斯后验分布。这种新方法超越了验证软件实现；它提供了一个敏感的测试，所有的假设，明确的或隐含的，是推论的基础。特别是，该方法评估推断的后验分布是否真实地表示了模型参数的实际约束。我们通过将多维K-S检验应用于一个简单的二维高斯玩具问题来说明我们的多维K-S检验方法，并演示了我们的方法在实际天体物理应用中的后验验证，即根据微波背景数据中的Sunyaev - Zel 'dovich效应估计星系团参数的物理参数。在后一个例子中，我们展示了该方法可以在不同的对象群体中验证整个贝叶斯推理过程，其中每种情况下派生的后验是不同的。
We extend the Kolmogorov‐Smirnov (K-S) test to multiple dimensions by suggesting a R n ! [0;1] mapping based on the probability content of the highest probability density region of the reference distribution under consideration; this mapping reduces the problem back to the one-dimensional case to which the standard K-S test may be applied. The universal character of this mapping also allows us to introduce a simple, yet general, method for the validation of Bayesian posterior distributions of any dimensionality. This new approach goes beyond validating software implementations; it provides a sensitive test for all assumptions, explicit or implicit, that underlie the inference. In particular, the method assesses whether the inferred posterior distribution is a truthful representation of the actual constraints on the model parameters. We illustrate our multidimensional K-S test by applying it to a simple twodimensional Gaussian toy problem, and demonstrate our method for posterior validation in the real-world astrophysical application of estimating the physical parameters of galaxy clusters parameters from their Sunyaev‐Zel’dovich effect in microwave background data. In the latter example, we show that the method can validate the entire Bayesian inference process across a varied population of objects for which the derived posteriors are different in each case.