Mathematical foundations for probability and causality

Abstract

Event trees, and more generally, event spaces, can be used to provide a foundation for mathematical probability that includes a systematic understanding of causality. This foundation justifies the use of statistics in causal investigation and provides a rigorous semantics for causal reasoning. Causal reasoning, always important in applied statistics and increasingly important in artificial intelligence, has never been respectable in mathematical treatments of probability. But, as this article shows, a home can be made for causal reasoning in the very foundations of mathematical probability. The key is to bring the event tree, basic to the thinking of Pascal, Huygens, and other pioneers of probability, back into probability’s foundations. An event tree represents the possibilities for the step-by-step evolution of an observer’s knowledge. If that observer is nature, then the steps in the tree are causes. If we add branching probabilities, we obtain a probability tree, which can express nature’s limited ability to predict the effects of causes. As a foundation for the statistical investigation of causality, event and probability trees provide a language for causal explanation, which gives rigorous meaning to causal claims and clarifies the relevance of different kinds of evidence to those claims. As a foundation for probability theory, they allow an elementary treatment of martingales, 1991 Mathematics Subject Classification. Primary ; Secondary .