1 Conditional Probability and Update Logic

Dynamic update of information states is the dernier cri in logical semantics. And it is old hat in Bayesian probabilistic reasoning. This note brings the two perspectives together, and proposes a mechanism for updating probabilities while changing the informational state spaces. 1 Tree diagrams for probability Many textbooks use a perspicuous tree format for simple probability spaces. Branches are histories of successive events. Going down the tree, actions generate new probability spaces, with the current space being more or less the current tree level. Arrows downward from a node are labeled with probabilities, summing to 1. By way of illustration, take the perennial Monty Hall puzzle. First, Nature puts a car behind one of three doors (the quizmaster knows which, you do not), then you choose a door, and finally, the quizmaster opens a door not chosen by you which has no car behind it. This involves a tree-diagram like the following. Of course, which actions you put in precisely is a matter of picking the right level of detail. Nature acts 1/3 1/3 1/3 car behind 1 car behind 2 car behind 3 1 1 1 I choose 1 I choose 1 I choose 1 1/2 1/2 1 1 Q opens 2 Q opens 3 Q opens 3 Q opens 2 Let's say I chose door 1, Monty opened door 3. Should I switch or not? We must find the right conditional probability for the car being behind door 1, given all that has passed. If we conditionalize on 'the car is not behind 3', we find a probability of 1/2. But, if we do the job well, we will pick up the more informative true proposition A = 'Monty opened door 3' to compute P('car behind 1'|A) = 1/3 – and conclude that we should switch. Is luck needed in picking the right A , or is there a systematic principle at work? In this note we will analyze this process, which is close to current dynamic update logics, with information flowing down the tree.