In inverse problems, one attempts to infer spatially variable functions from indirect measurements of a system. To practitioners of inverse problems, the concept of "information" is familiar when discussing key questions such as which parts of the function can be inferred accurately and which cannot. For example, it is generally understood that we can identify system parameters accurately only close to detectors, or along ray paths between sources and detectors, because we have "the most information" for these places. Although referenced in many publications, the "information" that is invoked in such contexts is not a well understood and clearly defined quantity. Herein, we present a definition of information density that is based on the variance of coefficients as derived from a Bayesian reformulation of the inverse problem. We then discuss three areas in which this information density can be useful in practical algorithms for the solution of inverse problems, and illustrate the usefulness in one of these areas -- how to choose the discretization mesh for the function to be reconstructed -- using numerical experiments.

翻译：反问题旨在从系统的间接测量中推断空间变化函数。对反问题研究者而言，"信息"这一概念在探讨诸如函数的哪些部分可精确推断、哪些部分不可精确推断等关键问题时为人熟知。例如，通常认为我们只能精确识别靠近探测器或沿源与探测器之间射线路径的系统参数，因为我们对这些位置拥有"最多信息"。尽管众多文献中提及，但此类语境下引用的"信息"并非一个被充分理解且明确定义的量。本文提出一种基于贝叶斯反问题重新表述所推导系数方差的信息密度定义，进而讨论该信息密度可在反问题求解实用算法中发挥作用的三个领域，并通过数值实验说明其中一个领域（即如何为待重构函数选择离散化网格）的应用价值。