Efficient Bayesian inference for multimodal problems in cosmology
Efficient Bayesian inference for multimodal problems in cosmology
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宇宙学中多模态问题的有效贝叶斯推理
DOI：
10.1111/j.1365-2966.2007.11871.x
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发表时间：
2007
影响因子：
4.8
通讯作者：
M. Hobson
中科院分区：
文献类型：
--
作者：
J. Shaw;M. Bridges;M. Hobson
贝叶斯模型选择为宇宙学家提供了一种严格的工具，可以通过贝叶斯证据来区分纯粹基于数据的竞争模型。以前计算该数量的方法要么缺乏普遍适用性，要么计算量要求较高。然而，嵌套采样（Skilling 2004）最近被 Muhkerjee 等人成功应用于宇宙学。 2006年，克服了这两个障碍。它们的实现限制了采样的参数空间，从而提高了效率，在 n 维参数空间中使用缩小的椭球界，包含高于递减似然值的参数样本。然而，如果似然函数包含在参数空间的很大一部分上分离的任何多模态，则可以防止椭圆通过小于似然峰值之间的距离来约束采样区域。在本文中，我们介绍了一种聚类嵌套采样方法，通过这种方法，可以在任何识别的峰上形成椭球体簇，从而提高效率，该系数等于小聚类椭球体集所包围的体积与不进行聚类时必然需要的大单个椭圆体所包围的体积之比。此外，我们还实现了一种确定最终证据值的期望和方差的方法，而无需使用算法重复的采样误差；这进一步减少了至少一个数量级的计算负载。我们已将我们的算法应用于一对玩具模型和一个宇宙学示例，其中我们证明所需的似然评估数量是使用先前算法所需的似然评估数量的 4%。我们已经生成了一个 FORTRAN 库，其中包含可以从任何示例代码中调用的例程，此外，为了方便起见，我们已将其作为 CosmoCLUST 合并到流行的 COSMOMC 代码中。两者均可从 www.mrao.cam.ac.uk/software/cosmoclust 下载。
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 shrinking ellipsoidal bound in the n-dimensional parameter space encompassing parameter samples above a decreasing likelihood value. 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 the expectation and variance of the final evidence value without the need to use sampling error from repetitions of the algorithm ; this further reduces the computational load by at least an order of magnitude. We have applied our algorithm to a pair of toy models and one cosmological example where we demonstrate that the number of likelihood evaluations required is 4% of that necessary for using previous algorithms. We have produced a FORTRAN library containing our routines which can be called from any sampling code, in addition for convenience we have incorporated it into the popular COSMOMC code as COSMOCLUST. Both are available for download at www.mrao.cam.ac.uk/software/cosmoclust.