## The Foundations of Inference

7:14 pm

In October 2009, Kevin Knuth (President, Autonomous Exploration, Inc.) was invited to present a talk titled “Foundations of Inference” at the Perimeter Institute in Waterloo, CANADA.  This theoretical talk discussed the mathematical foundation of inference and its relation to order theory and quantum mechanics.  Due to the Quantum to Cosmos Festival which ran at the same time, this talk was not videotaped.

An abstract follows below:

### Foundations of Inference

In our mutual desire to develop an intuitive understanding of quantum mechanics, John Skilling and I have been working to clarify the foundations of inference.

In this talk I introduce lattice theory as a new foundation for a rational description of the world around us.  I will show that fundamental mathematical fields such as measure theory, probability theory and information theory are based on basic symmetries of the lattice and can be easily derived.

Our rational description relies on a partially ordered set (poset) of states.  The process of expansion (lattice exponentiation) generates a distributive lattice from a poset.  I show that successive expansions applied to the poset of states results in a lattice of all the possible statements that can be made about the system, and a lattice of all the possible questions that can be asked of a system.

Quantification is performed by introducing real-valued functions called valuations.  The symmetries of the lattice lead to a unique calculus that obeys a summation property, which is the basis of addition as the fundamental operation in measure theory.  Introduction of the notion of context via functions called bi-valuations leads to a product rule for context-based measures.  On the lattice of statements, this unique calculus is the probability calculus with its sum and product rule.  An additional constraint relating the relevance of a question to the probabilities of the statements that answer it leads to entropy as the basic measure of a question.  The unique calculus on the lattice of questions is a new context-based generalization of information theory.

I will finish by introducing a new picture of quantum mechanics where experimental states are partially ordered to form a poset.  The valuations on this poset are the quantum amplitudes with Feynman path integrals representing the summation property.  Expansion of the state poset to a lattice of statements leads to probabilities whose values depend on the quantum amplitudes.  In this way probability theory naturally enters the quantum picture as inference.