Title: The differential structure shared by probability and moment matching priors on non-regular statistical models via the Lie derivative Show Chinese title

Title: 非正则统计模型上概率和矩匹配先验共享的微分结构通过李导数

Abstract: In Bayesian statistics, the selection of noninformative priors is a crucial issue. There have been various discussions on theoretical justification, problems with the Jeffreys prior, and alternative objective priors. Among them, we focus on two types of matching priors consistent with frequentist theory: the probability matching priors and the moment matching priors. In particular, no clear relationship has been established between these two types of priors on non-regular statistical models, even though they share similar objectives. Considering information geometry on a one-sided truncated exponential family, a typical example of non-regular statistical models, we find that the Lie derivative along a particular vector field provides the conditions for both the probability and moment matching priors. Notably, this Lie derivative does not appear in regular models. These conditions require the invariance of a generalized volume element with respect to differentiation along the non-regular parameter. This invariance leads to a suitable decomposition of the one-sided truncated exponential family into one-dimensional submodels. This result promotes a unified understanding of probability and moment matching priors on non-regular models. Show Chinese abstract

Abstract: 在贝叶斯统计中，非信息先验的选择是一个关键问题。关于理论依据、Jeffreys先验的问题以及替代的客观先验，已经有许多讨论。其中，我们关注两种与频率学派理论一致的匹配先验：概率匹配先验和矩匹配先验。特别是，在非正则统计模型上，这两种先验之间没有明确的关系，尽管它们具有相似的目标。考虑到单边截断指数族的信息几何，这是非正则统计模型的一个典型例子，我们发现沿着特定向量场的Lie导数提供了概率匹配先验和矩匹配先验的条件。值得注意的是，这种Lie导数在正则模型中不会出现。这些条件要求在非正则参数的方向上的微分中广义体积元素的不变性。这种不变性导致单边截断指数族被分解为一维子模型。这一结果促进了对非正则模型上概率匹配先验和矩匹配先验的统一理解。