Predictive Directed Association: Information Refinement and Incremental Predictive Value

How much predictive value is gained when an information set is refined? Many predictive- dependence frameworks emphasize whether additional information provides statistically de- tectable predictive improvement, while the magnitude of the resulting predictive value can be treated as a primary mathematical object in its own right. This paper develops Predictive Directed Association (PDA) from this perspective. For nested information sets, PDA is defined as the expected squared distance between the corresponding Bayes-optimal predictors. We prove that this quantity exactly equals the reduc- tion in Bayes-optimal prediction risk obtained by incorporating the richer information set in place of the baseline information set. Thus, PDA is simultaneously a probabilistic quantity, a projection distance, and a decision-theoretic measure of incremental predictive value. We establish a mathematical foundation consisting of nonnegativity and boundedness, a structural characterization of zero association, information monotonicity, an exact chain rule for nested information refinements, decision-theoretic monotonicity, and estimation stability. We further derive a conditional orthogonal decomposition and develop a dynamical-state-conditioned extension in which predictive value is evaluated within specified dynamical states. An exact construction demonstrates an important phenomenon: global predictive association can vanish completely while predictive value remains strictly positive within every state. This provides a rigorous witness that marginal predictive analysis can conceal state-conditioned predictive structure. For statistical implementation, we develop a blocked estimation framework under geomet- rically mixing observations and establish consistency and rate-transfer results under explicit empirical-process and nonparametric estimation conditions. The resulting framework is model- agnostic at its theoretical core and does not require linearity, Gaussianity, or a specific dynamical model. Rather than proposing a pointwise replacement for existing predictive-dependence measures, PDA takes information refinement and incremental predictive value as its primitive objects. This viewpoint provides a common mathematical interface connecting conditional expectation, statistical decision theory, predictive dependence, state-conditioned analysis, and nonparamet- ric estimation, while opening a route toward broader theories of decision value, distributional predictive value, and general information value.