论文标题
关于模式和最大后验估计的订单理论观点
An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems
论文作者
论文摘要
通常希望按照模式或地图估算器（即最大概率点）对空格$ x $进行概率度量。可以使用小拉迪乌斯极限的公制球来严格定义这样的点。但是，该理论并非完全直接：文献包含多个模式的概念和各种没有模式的病理措施的例子。由于大量的球会在$ x $的点上引起自然顺序，因此本文旨在通过采用订单理论的观点来阐明非参数地图估算中的某些问题，这似乎是反问题社区中的新问题。这种观点根据Cantor和Kuratowski交集定理打开了有吸引力的证明策略。它还揭示了许多病理是由订单的最大元素和最大元素之间的区别以及$ x $的无与伦比元素的存在而产生的，我们证明，即使在$ x $上也可以密集，即使是在$ x = \ mthbb {r} $上进行绝对连续的度量。
It is often desirable to summarise a probability measure on a space $X$ in terms of a mode, or MAP estimator, i.e.\ a point of maximum probability. Such points can be rigorously defined using masses of metric balls in the small-radius limit. However, the theory is not entirely straightforward: the literature contains multiple notions of mode and various examples of pathological measures that have no mode in any sense. Since the masses of balls induce natural orderings on the points of $X$, this article aims to shed light on some of the problems in non-parametric MAP estimation by taking an order-theoretic perspective, which appears to be a new one in the inverse problems community. This point of view opens up attractive proof strategies based upon the Cantor and Kuratowski intersection theorems; it also reveals that many of the pathologies arise from the distinction between greatest and maximal elements of an order, and from the existence of incomparable elements of $X$, which we show can be dense in $X$, even for an absolutely continuous measure on $X = \mathbb{R}$.
