约束参数空间中贝叶斯推断的最优先验
Optimal prior for Bayesian inference in a constrained parameter space
摘要：
贝叶斯参数推断取决于所讨论的参数的先验概率分布的选择。使后验分布对数据最敏感的先验称为Jeffreys先验，它完全由似然对参数变化的响应决定。在似然是高斯分布的假设下，Jeffreys先验是常数，即平坦的。然而，如果一个参数受到物理因素的约束，高斯近似失败，平坦先验不再是Jeffreys先验。本文计算了一维约束下多元正态分布的Jeffreys先验，并将其应用于中微子质量之和$Σm_ν$与张量标量比$r$ .我们发现，这两个参数的一维边缘化后验有很大的变化，68%和95%的贝叶斯上限分别增加了9%和4%的$Σm_ν$和22%和3%的$r$。在分析过程的最后一步中，可以将优先级添加到现有链中，作为微不足道的重要性抽样。
英文摘要：
Bayesian parameter inference depends on a choice of prior probability distribution for the parameters in question. The prior which makes the posterior distribution maximally sensitive to data is called the Jeffreys prior, and it is completely determined by the response of the likelihood to changes in parameters. Under the assumption that the likelihood is a Gaussian distribution, the Jeffreys prior is a constant, i.e. flat. However, if one parameter is constrained by physical considerations, the Gaussian approximation fails and the flat prior is no longer the Jeffreys prior. In this paper we compute the correct Jeffreys prior for a multivariate normal distribution constrained in one dimension, and we apply it to the sum of neutrino masses $Σm_ν$ and the tensor-to-scalar ratio $r$. We find that one-dimensional marginalised posteriors for these two parameters change considerably and that the 68% and 95% Bayesian upper limits increase by 9% and 4% respectively for $Σm_ν$ and 22% and 3% for $r$. Adding the prior to an existing chain can be done as a trivial importance sampling in the final step of the analysis proces.
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