A Unified Framework for Bayesian and Non-Bayesian Decision Making and Inference
本文通过一系列例子展示，用不同的近似概念替换贝叶斯范式中的近似，就能得到其他范式，并提出了一个统一框架，将统计模型映射到希尔伯特空间上的算子代数，通过选择范数或散度函数来产生不同的决策与推断理论。
After showing, by means of a series of examples, that paradigms alternative to the Bayesian one obtain by simply replacing the notion of approximation associated with the latter, the paper presents a unified framework for theories of decision making and inference. Given a statistical model, the algebra of bounded random variables on the sample space is mapped homomorphically into an algebra of operators on a certain Hilbert space. Then, the choice of a norm or a divergence function on the latter algebra produces a theory of decision making and inference. Examples include models from the Choquet expected utility class, models from robust statistics, the smooth model, maxmin and maxmax (as limiting cases) as well as a novel theory. The paper also contributes to Bayesian theory, which obtains in correspondence to a Hilbert norm. It shows that Bayes’ theorem can be derived from the fundamental concept of conditional expectation and that it is the only updating rule for which the operations of updating and of calculating the predictive commute.