Collaborative Research: Updating Ambiguous Beliefs: Bayesian and Non-Bayesian Learning

不确定情况下的贝叶斯决策方法规定，决策者具有唯一的先验概率和效用函数，从而做出决策以最大化预期效用。这一范式得到了公理的证明，但受到了几类证据的挑战：实验结果与贝叶斯方法所暗示的一致性条件不相容，人们难以确定先验概率，以及经济现象与理论结果相矛盾。在许多建议推广贝叶斯方法的模型中，有非加性概率模型和多先验模型。这些模型已经通过了一项重要的考验：它们已经公理化。然而，这些一次性决策模型几乎没有阐明随着新信息的收集而动态更新概率的问题。那么，自然的问题是：随着时间的推移，经济主体更新信念的合理方式是什么？本研究将公理地处理这一问题，并提出合理的更新规则，这些规则满足一些基本要求。将提出和研究一族伪贝叶斯规则，每一类规则都可以被认为是贝叶斯规则对于唯一可加先验的推广。也将研究一族经典的更新规则，每一种规则都从给定的一组先验开始，在面对新信息时可能会排除其中的一些，并继续对其余的进行贝叶斯更新。假设一个唯一的给定先验概率显然是不现实的。新的政治环境、技术进步、新公司和新产品都是反例，在这些反例中，假设一个独特的先例是不合理的。因此，这项研究将为我们理解经济主体对新现象或新信息的适应提供重要的理论基础。

The Bayesian approach to decision making under uncertainty prescribes that a decision maker have a unique prior probability and a utility function such that decisions are made so as to maximize the expected utility. This paradigm was justified by axiomatic treatments but challenged by several classes of evidence: experimental results incompatible with the consistency condition implied by the Bayesian approach, difficulties people have in specifying a prior probability, and economic phenomena contradictory to the theoretical results. Among many models suggested to generalize the Bayesian approach are the nonadditive probabilities model and the multiple priors model. These models have passed an important test: they have been axiomatized. However, these one-shot decision models shed little light on the problem of dynamically updating probabilities as new information is gathered. The natural question then is: what are the reasonable ways for economic agents to update beliefs over time? This study will deal with this problem axiomatically and suggest plausible updating rules which satisfy some basic requirements. A family of pseudo-Bayesian rules, each of which may be considered a generalization of Bayes' rule for a unique additive prior, will be presented and studied. A family of classical updating rules, each of which starts out with a given set of priors, possibly rules some of them out in the face of new information, and continues with the Bayesian updates of the remaining ones, will also be studied. The assumption of a unique given prior probability is clearly unrealistic. New political circumstances, technological advances, new firms, and new products are all counter-examples in which the assumption of a unique prior is unreasonable. Consequently, this study will provide important theoretical foundation upon which we build our understanding of the adjustment of economic agents to new phenomena or new information.