基于归一化流的 Savage-Dickey 密度比估计用于贝叶斯模型比较
Savage-Dickey density ratio estimation with normalizing flows for Bayesian model comparison
AI总结：
本文提出用归一化流实现 Savage-Dickey 密度比估计，以解决高维嵌套模型贝叶斯因子计算难题，并在宇宙学场级推断中验证了 SBI 与 BHM 方法的一致性。
摘要：
科学的一个核心动机是评估哪种科学模型能最好地解释观测数据。贝叶斯模型比较为比较科学模型提供了一种具有原则性的统计方法，并已在宇宙学和天体物理学中得到广泛应用。计算贝叶斯证据在计算上具有挑战性，尤其是当我们持续探索越来越复杂的模型时。Savage-Dickey 密度比（SDDR）提供了一种仅使用超模型的后验样本来计算两个嵌套模型之间贝叶斯因子（证据比）的方法。SDDR 需要计算超模型额外参数上的归一化边缘分布，这通常使用经典密度估计器（如直方图）来完成。然而，经典密度估计器难以扩展到高维场景。我们引入了一种使用归一化流的神经 SDDR 方法，该方法可以扩展到超模型包含大量额外参数的场景。我们展示了这种神经 SDDR 方法在玩具示例和真实宇宙学示例上的有效性。在场级推断场景中，我们表明针对贝叶斯层次模型（BHM）和基于模拟的推断（SBI）方法计算的贝叶斯因子是一致的，这进一步验证了 SBI 从场中提取的宇宙学信息与 BHM 方法一样多。基于归一化流的 SDDR 估计器已在开源 harmonic Python 包中实现。
英文摘要：
A core motivation of science is to evaluate which scientific model best explains observed data. Bayesian model comparison provides a principled statistical approach to comparing scientific models and has found widespread application within cosmology and astrophysics. Calculating the Bayesian evidence is computationally challenging, especially as we continue to explore increasingly more complex models. The Savage-Dickey density ratio (SDDR) provides a method to calculate the Bayes factor (evidence ratio) between two nested models using only posterior samples from the super model. The SDDR requires the calculation of a normalised marginal distribution over the extra parameters of the super model, which has typically been performed using classical density estimators, such as histograms. Classical density estimators, however, can struggle to scale to high-dimensional settings. We introduce a neural SDDR approach using normalizing flows that can scale to settings where the super model contains a large number of extra parameters. We demonstrate the effectiveness of this neural SDDR methodology applied to both toy and realistic cosmological examples. For a field-level inference setting, we show that Bayes factors computed for a Bayesian hierarchical model (BHM) and simulation-based inference (SBI) approach are consistent, providing further validation that SBI extracts as much cosmological information from the field as the BHM approach. The SDDR estimator with normalizing flows is implemented in the open-source harmonic Python package.
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