Dynamic Update with Probabilities

Conditional probabilities Pi(φ | A) describe how agent i’s probability distributions for propositions φ change as new information A comes in. The standard probabilistic calculus describing such changes revolves around Bayes’ Law in case the new information A is factual, concerning some actual situation under investigation. But there are also proposed mechanisms in the literature that deal with non-factual new information A, such as the Jeffrey Update Rule for probabilistic information of the form “Pi(A) = x”. Current dynamic-epistemic logics manipulate formulas [!A]Kiφ describing what agents know or believe after a proposition A has been communicated. Here A may be either about the real world or about information that other agents have. And the most sophisticated modern update systems can even deal with a much greater variety of informative events, such as partial observation, whispers, or lies. Thus, it seems of interest to combine the two perspectives – for reasons of mutual benefit. As always, there are two aspects to the task at hand, which can be called ‘modelling’ and ‘reasoning’. Logical models are designed to capture the informational essence of some informally presented situation. Especially in the area of probability where ‘choosing the right model’ is a task of recognized difficulty, formal guidance can be helpful. Using logical models, we can then describe information update and other processes of interest. Often intertwined with this, agents also invoke general inferences, involving validity in some larger class of situations than just the current one. We will also look at logical validities of the latter sort, high-lighting some general principles of probabilistic reasoning. The paper is organized as follows. The first two sections cover our point of departure: in Section 2 we present static epistemic-probabilistic logic, and in Section 3 we present dynamic epistemic logics. Combinations of probabilistic and dynamic epistemic logics have been proposed before. The systems by Kooi and van Benthem are presented in Section 4. In Section 5 we present the way we would like to model probabilistic updates with varying degrees of generality. In Section 6 we turn to reasoning about probability.