In standard Bayesian probability revision, the adoption of full beliefs (propositions with probability 1) is irreversible. Once an agent has full belief in a proposition, no subsequent revision can remove that belief. This is an unrealistic feature, and it also makes probability revision incompatible with belief change theory, which focuses on how the set of full beliefs is modified through both additions and retractions. This problem in probability theory can be solved in a model that (i) lets the codomain of the probability function be a hyperreal-valued rather than the real-valued closed interval [0, 1], and (ii) identifies the full beliefs as the propositions whose probability is either 1 or infinitesimally smaller than 1. In this model, changes in the probability function will result in changes in the set of full beliefs (belief set), which constitutes a submodel that can be conceived as the “tip of the iceberg” within the larger model that also contains beliefs on lower levels of probability. The patterns of change in the set of full beliefs in this modified Bayesian model coincides with the corresponding pattern in a slightly modified version of AGM revision, which is commonly conceived as the gold standard of (dichotomous) belief change. The modification only concerns the marginal case of revision by an inconsistent input sentence. These results show that probability revision and dichotomous belief change can be unified in one and the same framework, or – if we so wish – that belief change theory can be subsumed under a modified version of probability revision that allows for iterated change and for the removal of full beliefs.

在标准贝叶斯概率修正中，完全信念（概率为 1 的命题）的采用是不可逆转的。一旦代理人完全相信某个提议，后续的修改就无法消除这种信念。这是一个不切实际的特征，它也使得概率修正与信念改变理论不相容，信念改变理论关注如何通过添加和撤回来修改完整信念集。概率论中的这个问题可以在一个模型中解决，该模型 (i) 让概率函数的余域为超实值闭区间而不是实值闭区间 [0, 1]，并且 (ii) 将完整信念标识为概率为 1 或无限小于 1 的命题。在该模型中，概率函数的变化将导致完全信念集合（belief set）的变化，它构成了一个子模型，可以将其视为较大模型中的“冰山一角”，该模型还包含对较低概率水平的信念。此修改后的贝叶斯模型中完整信念集的变化模式与年度股东大会修订版的稍微修改版本中的相应模式一致，后者通常被认为是（二分法）信念变化的黄金标准。修改仅涉及输入句子不一致而进行修改的边缘情况。这些结果表明，概率修正和二分信念改变可以统一在同一个框架中，或者——如果我们愿意的话——信念改变理论可以包含在概率修正的修改版本中，该版本允许迭代改变和删除的完整信念。