Foundations of Inference
Professors Caticha, Earle, Goyal and Knuth
Inference is the process of making reasonable inferences from limited information. But what is "reasonable"? The Foundations of Inference is concerned with the codification of principles which embody what we mean by "reasonable" and developing them into mathematical inferential frameworks, with the comparative study of various inferential principles, and with understanding the interrelationships and domains of validity of existing inferential frameworks (such as Bayesian inference and the Principle of Maximum Entropy).
In our group, we have developed a new derivation of probability theory based on lattice theory, developed a new inquiry calculus (developing earlier ideas of Cox), and have developed a new framework within which Bayes' rule is seen to be a direct consequence of the Principle of Maximum Entropy.
There has been a close and very fruitful interplay between our work in inference and the Foundations of Quantum Theory, which has (for example) led to a derivation of Feynman's rules of quantum theory which adapts the method that Cox used to derive probability theory.
The members of our group are closely involved with the yearly "International Workshop on Bayesian Inference and Maximum Entropy methods in Science and Engineering", which we have organized in 2005, 2007, and 2011.