Bidirectional General Graphs for inference. Principles and implications for medicine

Abstract

Probabilistic inference methods require a more general and realistic description of the world as a Bidirectional General Graph (BGG). While in its original form the Bayes Net (BN) has been promoted as a predictive tool, it is more immediately a way of testing a hypothesis or model about interactions in a system usually considered on a causal basis. Once established, the model can be used in a predictive way, but the problem here is that for a traditional BN the hypotheses or models that can be formed are limited to the Directed Acyclic Graph (DAG) by definition. Three interrelated features are highlighted that represent deficiencies of the DAG which are corrected by conversion to a method based on a BGG: (i) lack of intrinsic representation of coherence by Bayes' rule, (ii) relatedly the need to consider interdependence in parent nodes, and (iii) the need for management of a property called recurrence. These deficiencies can represent large errors in absolute estimates of probabilities, and while relative and renormalized probabilities ameliorate that, they can often make much of a net superfluous through cancelations by division. The Hyperbolic Dirac Net (HDN) based on Dirac's quantum mechanics is a solution that led naturally to avoiding these deficiencies. It encodes bidirectional probabilities in an h-complex value rediscovered by Dirac, i.e. with the imaginary number h such that hh = +1. Properties of the HDN described previously are reviewed (though emphasis is on descriptions in familiar probability terms), the issue of recurrence is introduced, methods of construction are simplified, and the severity of the quantitative differences between BNs and analogous HDNs are exemplified. There is also discussion of how results compare with other approaches in practice.

Keywords: Bayes net; Bayes' rule; Bidirectional general graph; Clinical decision support; Directed acyclic graph; Hyperbolic Dirac net; Inference net.