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 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 encodes 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 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: Calibration Preservation (ECE <= epsilon + C/sqrt(K_eff)); Monte Carlo Error decaying as O(1/sqrt(M)); a non-vacuous PAC-Bayes bound with train-test gap of 0.0085 at N=4200; operation within 1.12x of the information-theoretic lower bound; graceful degradation as O(epsilon*delta*sqrt(K)) under corruption, maintaining 88% performance with half of evidence adversarially replaced; O(1/sqrt(K)) calibration decay with R^2=0.849; and exact epistemic-aleatoric uncertainty decomposition with error below 0.002%. All theorems are empirically validated on controlled datasets spanning up to 4,200 training examples. Our theoretical framework establishes LPF as a foundation for trustworthy multi-evidence AI in safety-critical applications.

翻译：本文提出了潜在后验因素（LPF）的完整理论表征，这是一个在概率预测任务中聚合多个异构证据项的原则性框架。多证据推理广泛存在于医疗诊断、金融风险评估、法律案件分析和法规合规等高风险领域，然而现有方法要么缺乏形式化保证，要么在架构上无法处理多证据场景。LPF通过变分自编码器将每个证据项编码为高斯潜在后验，利用蒙特卡罗边缘化将后验转换为软因子，并通过精确的和积网络推理（LPF-SPN）或学习的神经聚合器（LPF-Learned）聚合因子。我们证明了七项形式化保证，涵盖可信AI的关键需求：校准保持（ECE ≤ epsilon + C/√K_eff）；蒙特卡罗误差以O(1/√M)衰减；非退化的PAC-Bayes界，在N=4200时训练-测试差距为0.0085；操作在信息论下界的1.12倍以内；在污染下以O(epsilon*delta*√K)优雅降级，在半数证据被对抗性替换时仍保持88%性能；以R²=0.849实现O(1/√K)的校准衰减；以及误差低于0.002%的精确认知-偶然不确定性分解。所有定理在多达4200个训练样本的受控数据集上得到经验验证。我们的理论框架确立了LPF作为安全关键应用中可信多证据AI的基础。