Towards the full information chain theory: answer depth and source models

A problem of optimal information acquisition for its use in general decision making problems is considered. This motivates the need for developing quantitative measures of information sources’ capabilities for supplying accurate information depending on the particular content of the latter. A companion article developed the notion of a question difficulty functional for questions concerning input data for a decision making problem. Here, answers which an information source may provide in response to such questions are considered. In particular, a real valued answer depth functional measuring the degree of accuracy of such answers is introduced and its overall form is derived under the assumption of isotropic knowledge structure of the information source. Additionally, information source models that relate answer depth to question difficulty are discussed. It turns out to be possible to introduce a notion of an information source capacity as the highest value of the answer depth the source is capable of providing.

💡 Research Summary

The paper addresses the problem of optimal information acquisition for use in general decision‑making contexts, emphasizing that the value of information depends not only on its quantity but also on its relevance to the specific question being asked. Building on a companion article that introduced a “question difficulty” functional—a real‑valued measure of the uncertainty embedded in a query—the authors develop a complementary concept called “answer depth.” Answer depth quantifies the accuracy of the response an information source can provide to a given question.

Under the assumption that the source’s knowledge structure is isotropic (i.e., the source’s expertise is uniformly distributed over the space of possible questions), the authors derive the general form of the answer‑depth functional. They show that, in this setting, answer depth must also be a Laplacian‑type function of the question’s parameters, mirroring the form of the question‑difficulty functional. The two functionals are linked by a single scalar coefficient η, termed the source’s efficiency. Mathematically,
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