We propose a novel method ($floZ$), based on normalizing flows, to estimate the Bayesian evidence (and its numerical uncertainty) from a pre-existing set of samples drawn from the unnormalized posterior distribution. We validate it on distributions whose evidence is known analytically, up to 15 parameter space dimensions, and compare with two state-of-the-art techniques for estimating the evidence: nested sampling (which computes the evidence as its main target) and a $k$-nearest-neighbors technique that produces evidence estimates from posterior samples. Provided representative samples from the target posterior are available, our method is more robust to posterior distributions with sharp features, especially in higher dimensions. For a simple multivariate Gaussian, we demonstrate its accuracy for up to 200 dimensions with $10^5$ posterior samples. $floZ$ has wide applicability, e.g., to estimate evidence from variational inference, Markov Chain Monte Carlo samples, or any other method that delivers samples and their likelihood from the unnormalized posterior density. As a physical application, we use $floZ$ to compute the Bayes factor for the presence of the first overtone in the ringdown signal of the gravitational wave data of GW150914, finding good agreement with nested sampling.

翻译：我们提出了一种基于归一化流的新方法（$floZ$），用于从一组预先存在的、从未归一化后验分布中抽取的样本中估计贝叶斯证据（及其数值不确定性）。我们在证据可解析计算的分布上验证了该方法，参数空间维度最高达15维，并与两种最先进的证据估计技术进行了比较：嵌套采样（以计算证据为主要目标）和一种基于$k$近邻技术、从后验样本生成证据估计的方法。只要能够获得目标后验的代表性样本，我们的方法对具有尖锐特征的后验分布（尤其是在高维情况下）具有更强的鲁棒性。对于一个简单的多元高斯分布，我们证明了其在高达200维、使用$10^5$个后验样本时的准确性。$floZ$具有广泛的适用性，例如可用于估计来自变分推断、马尔可夫链蒙特卡洛样本或任何其他能够从未归一化后验密度中提供样本及其似然值的方法的证据。作为一个物理应用，我们使用$floZ$计算了GW150914引力波数据中铃荡信号是否存在第一泛音的贝叶斯因子，结果与嵌套采样具有良好的一致性。