Describing realistic states of knowledge with exact probabilities

Abstract

Given the locations of N known points, where is an unknown point? We believe that each configuration of N points should be perfectly described by exactly one probability distribution (a "predictive distribution") over the location of an unknown point. Here we introduce this 'N-point problem' in the context of the unified theory of knowledge and reason (information and logic) advocated by Jaynes [1]. We designed the N-point problem to be both primitive and realistic. Unlike most work with probabilities, we assume no knowledge (hypothesis or model) of how the points (data) were generated, and therefore this is a problem of inference from data alone. We believe that this simplicity should allow us to derive probabilities that are exact and perfect descriptions of a configuration of N points. We also believe that the configuration of N points is realistic in its relation to physical states of knowledge that actually exist (the locations of N particles, or N data points). In contrast, most or all previous probability distributions have been approximate descriptions of real states of knowledge, or exact descriptions of unrealistic states. We briefly summarize the initial steps that we think will allow for an exact solution to the N-point problem. Finally, we briefly discuss our work in relation to our ultimate interest in the physical basis of information and logic, and its relation to Bayesian theories of brain function.