A Set-Theoretic Approach to Bayesian Process Tracing

This article develops a set-theoretic approach to Bayes’s theorem and Bayesian process tracing. In the approach, hypothesis testing is the procedure whereby one updates beliefs by narrowing the range of states of the world that are regarded as possible, thus diminishing the domain in which the actual world can reside. By explicitly connecting Bayesian analysis to its set-theoretic foundations, the approach makes process tracing more intuitive and thus easier to apply for qualitative researchers. Moreover, the set-theoretic approach provides new tools for assessing both the consequentialness and expectedness of evidence when conducting process tracing. It also provides a new way to classify and interpret process-tracing tests, such as hoop tests and smoking gun tests, by viewing them as zones in a continuous space whose dimensions reflect the magnitude of changes in sets. The article shows that Bayesian process tracing and set-theoretic process tracing are not alternatives to each other but rather two sides of the same coin.