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Progressive Bayesian Confidence Architectures for Cold-Start Personal Health Analytics: Formalizing Early Insight Through Posterior Contraction and Risk-Aware Interpretation

Personal health analytics systems face a persistent cold-start dilemma: users expect meaningful insights early in data collection, while conventional statistical inference requires data volumes that often exceed engagement horizons. Existing approaches either delay inference until fixed statistical thresholds are met -- leading to user disengagement -- or surface heuristic insights without formal uncertainty quantification, risking false confidence. We propose a progressive Bayesian confidence architecture that formalizes early-stage inference through phased interpretation of posterior uncertainty. Drawing on Bayesian updating and epistemic strategies from financial risk modeling under sparse observations, we map posterior contraction to interpretable tiers of insight, ranging from exploratory directional evidence to robust associative inference. We demonstrate the framework's performance through controlled experimentation with synthetic N-of-1 health data, showing that calibrated early insights can be generated within 5--7 days while maintaining explicit epistemic humility. Compared to fixed-threshold baselines requiring 30+ days of data, the proposed approach yields earlier directional signals (mean: 5.3 vs 31.7 days, p<0.001) while controlling false discovery rates below 6% (5.9% at day 30) despite 26-day earlier detection, compared to 0% FDR for fixed-threshold baselines that delay insights by 30 days. In addition, we show strong uncertainty calibration (76% credible interval coverage for ground-truth correlations at day 90). This work contributes a methodological framework for uncertainty-aware early inference in personalized health analytics that bridges the gap between user engagement requirements and statistical rigor.