Theoretical Foundations of Latent Posterior Factors: Formal Guarantees for Multi-Evidence Reasoning

Formal guarantees enable trustworthy deployment in high-stakes domains: The paper proves seven specific mathematical guarantees for the LPF framework across healthcare diagnosis, financial risk assessment, legal analysis, and regulatory compliance—domains where AI decisions directly affect people's lives. This formal theoretical foundation addresses a critical gap where "existing approaches either lack formal guarantees or fail to handle multi-evidence scenarios."

Calibration preservation ensures reliable confidence estimates: The framework maintains calibration (Expected Calibration Error ≤ epsilon + C/√K_eff), meaning the system's stated confidence levels accurately reflect actual accuracy. This is essential for responsible AI because decision-makers need to know when the system is uncertain, particularly in healthcare and legal contexts where overconfident wrong answers are dangerous.

Robustness to corrupted evidence protects against manipulation: LPF gracefully degrades under adversarial attack, maintaining 88% performance even when half the evidence is adversarially replaced. This formal guarantee (O(epsilondelta√K) degradation) demonstrates the system won't catastrophically fail if some evidence is tampered with or unreliable—a critical consideration for regulatory compliance and financial risk assessment.

Uncertainty is precisely decomposed and quantified: The framework achieves exact epistemic-aleatoric uncertainty decomposition with error below 0.002%, meaning it can distinguish between uncertainty from lack of knowledge versus inherent randomness. This transparency helps stakeholders understand why the system is uncertain, supporting more informed decision-making.

Theoretical bounds are empirically validated at realistic scale: All guarantees are validated on datasets up to 4,200 examples with specific performance metrics (e.g., PAC-Bayes train-test gap of 0.0085), demonstrating the theory translates to practice rather than remaining purely abstract—a key