A General Theory of Correlation Among Multiple Variables: Statistical Connectivity Spaces

Abstract

Abstract This monograph develops a measure-theoretic theory of dependence for collections of internal variables observed under external background information. The primitive objects are projectively compatible regular conditional probability kernels on standard Borel spaces. For a finite internal set \(S\), statistical connectivity is the relative entropy of the joint law from the product of its coordinate marginals. For a background set \(B\), the pointwise field \(\mathfrak C_{B,S}(z_B)\) and its integrated value \(\overline{\mathfrak C}(B,S)\) are treated as distinct objects. The term connectivity refers to statistical dependence and not to connectedness of a graph. The finite-set theory establishes nonnegativity, the mutual-independence equality condition, coordinate invariance, data processing, independent-block additivity, and monotonicity under variable addition. Every variable ordering gives an exact mutual-information path decomposition, and permutation averaging yields the Shapley allocation. Adjacent path swaps, order spectra, Möbius dividends, allocation stability, and finite-game core constraints describe how the same total dependence can be redistributed without changing its endpoint value. Gaussian determinant and regression formulas, singular examples, parity laws, redundancy models, weak-coupling expansions, and maximum-entropy connected-information layers provide specializations and counterexamples. Background refinement is governed by an exact balance between information about the joint vector and information about its individual coordinates. Its sign is not fixed. Signed background paths, contextual Shapley responses, and variable–background mixed differences are defined only where the required finite contrasts are meaningful. Binary event-state mixtures yield exact total-variation and average-KL identities, sharp complementary-branch reconstruction bounds, and data-processing results for common Markov readouts. None of these conditional quantities is assigned a causal or temporal interpretation without additional structure. The theory extends to countable projective systems through finite-coordinate limits and process-level relative entropy. Local-to-global constructions distinguish probability fields, coordinate sheaves, Čech cohomology, support homology, numerical gluing residuals, and electrical graph summaries. Linear local sections admit Gauss–Markov and Tikhonov reconstruction, while probability-valued local models are compared with the compact image of admissible global laws. After statistical calibration, a path defect also has an operational classification role: orders whose terminal outputs are equivalent within a declared tolerance define the same order subtype, whereas a defect separated from that tolerance distinguishes two order-sensitive subtypes. Confidence bounds, rather than an isolated point estimate, determine this decision. These constructions are auxiliary structures and do not redefine statistical connectivity. For finite joint tables, the monograph develops plug-in consistency, regular delta-method limits, independence-boundary likelihood-ratio limits, joint inference for subset connectivities and Shapley values, smoothing and Dirichlet targets, and regular inference for maximum-entropy layers. Further chapters treat weighted laws, finite populations, stratified and clustered samples, unequal-probability inclusion, finite-state Markov chains, dependency-graph concentration, missing-data identification, validation designs, measurement channels, growing candidate families, and finite-state conditional fields over continuous backgrounds. Continuous internal variables are handled through explicitly distinguished density, variational, classification-ratio, copula, Wasserstein, and kernel targets. A public-data application uses three NHANES cycles from 2005–2010 to illustrate the complete-case, survey-weighted finite-state analysis of nine binary variables. Population targets, controlled-law calculations, real-data estimates, regularization sensitivity, and external-validity limits are reported separately. The closing chapter states open problems concerning nonregular boundaries, growing systems, adaptive conditional fields, complex sampling, incomplete observation, and numerical optimization with error bounds. Keywords Total correlation; relative entropy; regular conditional probability; projective system; conditional mutual information; Shapley value; context lattice; statistical connectivity.