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

We present a complete theoretical characterization of Latent Posterior Factors (LPF), a principled framework for aggregating multiple heterogeneous evidence items in probabilistic prediction tasks. Multi-evidence reasoning—where a prediction must be formed from several noisy, potentially contradictory sources—arises pervasively in high-stakes domains including healthcare diagnosis, financial risk assessment, legal case analysis, and regulatory compliance. Yet existing approaches either lack formal guarantees or fail to handle multi-evidence scenarios architecturally. LPF addresses this gap by encoding each evidence item into a Gaussian latent posterior via a variational autoencoder, converting posteriors to soft factors through Monte Carlo marginalization, and aggregating factors via either exact Sum-Product Network inference (LPF-SPN) or a learned neural aggregator (LPF-Learned).

We prove seven formal guarantees spanning the key desiderata for trustworthy AI. Theorem 1 (Calibration Preservation) establishes that LPF-SPN preserves individual evidence calibration under aggregation, with Expected Calibration Error bounded as ECE ≤ ε + C/√K_eff. Theorem 2 (Monte Carlo Error) shows that factor approximation error decays as O(1/√M), verified across five sample sizes. Theorem 3 (Generalization) provides a non-vacuous PAC-Bayes bound for the learned aggregator, achieving a train-test gap of 0.0085 against a bound of 0.228 at N=4200. Theorem 4 (Information-Theoretic Optimality) demonstrates that LPF-SPN operates within 1.12× of the information-theoretic lower bound on calibration error. Theorem 5 (Robustness) proves graceful degradation as O(εδ√K) under evidence corruption, maintaining 88% performance even when half of all evidence is adversarially replaced. Theorem 6 (Sample Complexity) establishes O(1/√K) calibration decay with evidence count, with empirical fit R² = 0.849. Theorem 7 (Uncertainty Decomposition) proves exact separation of epistemic from aleatoric uncertainty with decomposition error below 0.002%, enabling statistically rigorous confidence reporting.

All theorems are empirically validated on controlled datasets spanning up to 4,200 training examples and eight evaluation domains. Companion empirical results demonstrate mean accuracy of 99.3% and ECE of 1.5% across eight diverse domains, with consistent improvements over neural baselines, uncertainty quantification methods, and large language models. Our theoretical framework establishes LPF as a foundation for trustworthy multi-evidence AI in safety-critical applications.