Circulatory Fidelity: A Relational Theory of Information Flow in Hierarchical Models

Circulatory Fidelity: A Relational Theory of Information Flow in Hierarchical Models Circulatory Fidelity (CF) is a diagnostic framework for predicting when Mean-Field Variational Inference (MFVI) will fail in hierarchical Bayesian models—before computational resources are committed. While post-hoc diagnostics like PSIS-k̂ and simulation-based calibration assess inference quality after fitting, CF operates at the model specification stage, enabling practitioners to make informed methodological choices a priori. Theoretical Foundation The framework is grounded in the precision matrix decomposition Λ = D + R, where: D (diagonal) captures nodal structure—individual variable properties retained by MFVI R (off-diagonal) captures relational structure—coupling between variables that MFVI discards The primary diagnostic, Inference Coupling (IC), measures the normalized magnitude of this discarded relational structure. For Gaussian systems, IC equals the absolute Pearson correlation (IC = |ρ|), which is equivalent to the Linfoot informational correlation. The Relational Invariance Theorem establishes that mutual information I(Z;X) = -½log(1-ρ²) depends only on correlation—not on marginal variances—proving that IC captures exactly what matters for inference quality. Key Contributions The Computational Synergy Principle: Synergistic dependencies (where pairwise IC = 0 but joint information exists) arise precisely when the generative function is affine over GF(2). This enables a two-stage diagnostic protocol that detects both standard coupling and XOR-type algebraic structure. The Dependency Asymmetry: High IC predicts MFVI failure in filtering models (constitutive coupling) but hierarchy redundancy in pooling models (inductive coupling)—resolved through a taxonomy distinguishing model types. The Proximal Dominance Principle: In deep hierarchies, proximal coupling dominates—causing up to 40× degradation while distal coupling causes exactly 1.0× (mathematically guaranteed). This reduces diagnostic complexity from O(L²) to O(1). The Maximal Coupling Rule: For non-stationary time series, MFVI suitability depends on IC_max, not the global average. Validation The framework is validated across stochastic volatility filters (r = 0.86 correlation with log-likelihood gap, N = 8,000), hierarchical linear models, and three-layer hierarchies (N = 32,000+ total simulations), yielding calibrated thresholds and a practical workflow. Repository Contents Manuscript: Complete LaTeX source and compiled PDF Python module (circulatory_fidelity.py): Production implementation with support for NumPy, PyTorch, TensorFlow, and JAX Julia module (CirculatoryFidelity.jl): Native Julia implementation Jupyter notebooks: Six case study notebooks reproducing all manuscript figures and validation results Validation data: CSV files containing all simulation results Version History v1.1.0 (January 13, 2026) This release represents a significant theoretical clarification and refinement: Conceptual Updates: IC = |ρ| as primary diagnostic: The original v1.0 formulation used entropy-normalized CF = I(Z;X)/min(H(Z),H(X)). The Relational Invariance Theorem proves this normalization unnecessary for inference diagnostics—marginal entropies cancel in Gaussian mutual information. The v1.1 formulation uses IC = |ρ|, equivalent to the Linfoot correlation. Precision matrix framing: Explicit grounding in Λ = D + R decomposition clarifies exactly what MFVI discards. Three-metric framework: IC (inference), Balance Factor (architecture), and Control Coupling (intervention) are clearly distinguished, with IC identified as the sole predictor of MFVI failure. New Content: Fluid–crystalline distinction in synergy analysis (Rule 30 vs. Rule 150) "Cost of Connection" discussion addressing the physical substrate of relational structure "Broader applicability" section on extensions to discrete systems Enhanced bioelectric and gap junction connections Technical Improvements: Copula-based estimation recommended over k-NN methods (which exhibit 30-45% negative bias) Comprehensive validation appendix with detailed protocols Updated figure numbering and notation consistency (CF → IC throughout) v1.0.0 (January 1, 2026) Initial release establishing the diagnostic framework, empirical validation, and practical workflow. Citation: Lowry, A. (2026). Circulatory Fidelity: A Relational Theory of Information Flow in Hierarchical Models. Version 1.1.0. Zenodo. https://doi.org/10.5281/zenodo.XXXXXXX