Dynamical-State Predictive Directed Association: A General Framework for Incremental Value

Prediction is fundamentally an information problem. A predictive system does not operate on a target alone, but on an information structure available to it. In modern statistical learning and artificial intelligence, however, an increasingly important question is not only how to improve prediction from existing information, but also which additional information is worth obtaining when prediction remains uncertain. This raises a basic mathematical question: how should the incremental predictive value of an information source be defined independently of any particular predictive criterion? We develop a general mathematical framework for incremental predictive value based on information refinement and predictive risk. Rather than taking a particular loss, divergence, or causal criterion as primitive, we define predictive value through the reduction in optimal predictive risk induced by enlarging the information structure. For a target X, baseline in- formation F, and additional information source S, the resulting quantity is the predictive-risk reduction associated with the refinement F ∨ GS . This formulation separates the information being evaluated from the criterion used to measure predictive improvement. The framework is deliberately non-causal: predictive improvement may arise from causal mechanisms, common causes, latent dynamics, temporal dependence, state dependence, or other statistical structure. It therefore provides a common mathematical language for both causal and non-causal prediction. At the foundational level, the principal requirement is refinement mono- tonicity: enlarging the information available to an optimal predictor cannot increase predictive risk. We establish this principle abstractly and verify it for an important class of proper scoring rules through the convexity of the associated Bayes self-score. The formulation also provides a mathematical basis for AI systems that evaluate informa- tion by its expected incremental predictive value. Given a prediction task and a collection of candidate information sources, the framework assigns each source a predictive value relative to the current information structure, thereby providing a principled criterion for comparing het- erogeneous sources of additional information. In this sense, the framework shifts attention from prediction using information to the quantitative value of information for prediction. We further develop a dynamical-state-conditioned specialization, showing how state-dependent predictive association can be formulated within the same information-refinement framework. The resulting hierarchy separates the prediction target, information structure, predictive-risk criterion, and dynamic conditioning mechanism, allowing different predictive objectives and ap- plications to be treated as distinct realizations of a common notion of incremental predictive value.