[论文解读] Clustered nested sampling: efficient Bayesian inference for cosmology

Ruth G. Shaw, M. Bridges+1 more|arXiv (Cornell University)|Jan 30, 2007

Bayesian Methods and Mixture Models被引用 6

一句话总结

本文提出聚类嵌套采样（clustered nested sampling），通过用多个以不同似然峰值为中心的较小椭球体替代单一的大椭球约束，提升了宇宙学中贝叶斯证据计算的效率。该方法通过适应多峰似然函数，使采样效率提升的倍数等于小聚类总体积与单个包围椭球体积之比。

ABSTRACT

Bayesian model selection provides the cosmologist with an exacting tool to distinguish between competing models based purely on the data, via the Bayesian evidence. Previous methods to calculate this quantity either lacked general applicability or were computationally demanding. However, nested sampling (Skilling 2004), which was recently applied successfully to cosmology by Muhkerjee et al. 2006, overcomes both of these impediments. Their implementation restricts the parameter space sampled, and thus improves the efficiency, using a decreasing ellipsoidal bound in the n-dimensional parameter space centred on the maximum likelihood point. However, if the likelihood function contains any multi-modality, separated over a significant portion of the parameter space then the ellipse is prevented from constraining the sampling region by less than the distance between the likelihood peaks. In this paper we introduce a method of clustered nested sampling whereby ellipsoidal clusters can form on any peaks identified –thus improving the efficiency by a factor which is equal to the ratio of the volumes enclosed by the set of small clustered ellipsoids and the large single ellipse that would necessarily be required without clustering. In addition we have implemented a method for determining

研究动机与目标

• 解决标准嵌套采样在宇宙学模型选择中常见的多峰参数空间中效率低下的问题。
• 克服单一椭球约束无法探索分离的似然峰值的局限性。
• 开发一种能够动态识别并围绕多个似然峰值进行采样的方法，以提升计算效率。
• 实现对复杂宇宙学模型更准确且可扩展的贝叶斯证据评估。

提出的方法

• 在嵌套采样中用多个以识别出的似然峰值为中心的较小椭球体，替代标准的单个椭球约束。
• 使用聚类算法检测并分组参数空间中对应于似然函数不同局部最大值的区域。
• 在每个聚类内部独立应用嵌套采样，以更有效地探索高似然区域。
• 根据活点的分布和似然梯度动态更新聚类边界。
• 将所有聚类的证据贡献合并，以计算总贝叶斯证据。
• 集成一种在无需先验了解似然结构的情况下确定聚类边界和峰值识别的方法。

实验结果

研究问题

• RQ1 在宇宙学参数推断中存在多峰似然函数时，如何使嵌套采样更加高效？
• RQ2 与单个包围椭球相比，使用多个椭球聚类在嵌套采样中具有何种计算优势？
• RQ3 聚类能否提升具有分离似然峰值的模型中贝叶斯证据估计的准确性和收敛性？
• RQ4 在高维参数空间中，如何在不依赖先验知识的情况下可靠地检测并隔离似然峰值？

主要发现

• 聚类嵌套采样使采样效率提升的倍数等于小聚类总体积与单个包围椭球体积之比。
• 该方法能有效采样多峰似然函数，而标准嵌套采样因被束缚在椭球约束中而失效。
• 该方法使具有复杂、分离似然结构的宇宙学模型中贝叶斯证据计算更加准确。
• 实现中包含一种无需依赖参数空间拓扑先验知识即可识别似然峰值并形成聚类的机制。