Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Aliyu Agboola Alege

Abstract

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.

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

Abstract

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.
Paper Structure (52 sections, 14 theorems, 59 equations, 8 figures, 10 tables)

This paper contains 52 sections, 14 theorems, 59 equations, 8 figures, 10 tables.

Table of Contents

  1. Problem Setting and Formal Framework
  2. Multi-Evidence Prediction Problem
  3. LPF Architecture
  4. Empirical Validation
  5. Core Assumptions
  6. Core Theorems
  7. Theorem 1: SPN Calibration Preservation
  8. Theorem 2: Monte Carlo Error Bounds
  9. Theorem 3: Learned Aggregator Generalization Bound
  10. Theorem 4: Information-Theoretic Lower Bound
  11. Theorem 5: Robustness to Evidence Corruption
  12. Theorem 6: Sample Complexity and Data Efficiency
  13. Theorem 7: Uncertainty Quantification Quality
  14. Formal Dependency Structure
  15. Implementation Alignment
  16. ...and 37 more sections

Key Result

Theorem 3.1

Suppose each individual soft factor $\Phi_i(y)$ is $\epsilon$-calibrated, i.e., for all confidence levels $c \in [0,1]$: Then under Assumptions ass:independence--ass:valid-spn, the aggregated distribution $P_{\mathrm{SPN}}(y \mid \mathcal{E})$ satisfies: with probability at least $1 - \delta$, where is the effective sample size Kish1965SurveySampling and $C(\delta, |\mathcal{Y}|) = \sqrt{2\log(

Figures (8)

  • Figure 1: Dependency graph of LPF theoretical results. Assumptions (top) support lemmas and intermediate results, which enable the seven main theorems. Arrows indicate logical dependence. Note that different theorems use different subsets of assumptions: Theorem \ref{['thm:generalization']} (Generalization) is data-dependent and does not directly rely on Assumptions A1--A6, while Theorems \ref{['thm:sample-complexity']} and \ref{['thm:uncertainty']} build on the results of Theorems \ref{['thm:calibration']}, \ref{['thm:mc-error']}, and \ref{['thm:info-lower-bound']} rather than their assumptions alone.
  • Figure 2: Calibration verification (Theorem 1). Left: ECE for individual evidence ($0.140$), LPF-SPN ($0.185$), and LPF-Learned ($0.058$), with Hoeffding ($0.772$) and Bernstein ($0.459$) tight bounds annotated. Centre and right: reliability diagrams for LPF-SPN and LPF-Learned showing confidence vs. accuracy against the perfect-calibration diagonal.
  • Figure 3: Monte Carlo error bounds (Theorem 2). Left: log-log plot of mean error, 95th-percentile error, and theoretical bound vs. $M \in \{4,8,16,32,64\}$; all empirical curves remain well below the bound. Right: normalised error scaling confirms the empirical rate closely tracks $O(1/\sqrt{M})$ theory.
  • Figure 4: Generalization bound verification (Theorem 3). Top-left: train and test loss learning curves with confidence intervals across $N \in \{2002,3003,4200\}$. Top-right: empirical gap (near zero) vs. VC bound (loose) and data-dependent PAC-Bayes bound (tight, $0.228$ at $N=4200$). Bottom-left: bound-to-gap ratio on a log scale. Bottom-right: test loss vs. $N$ with effective dimension $d_{\mathrm{eff}}=1335$ marked.
  • Figure 5: Information-theoretic lower bound (Theorem 4). Top-left: decomposition of total uncertainty $H(Y)=1.399$ bits into evidence information $I(E;Y)=1.399$ and residual $H(Y|E)\approx 0$. Top-right: ECE comparison --- theoretical lower bound ($0.158$), achievable bound including MC term ($0.317$), and LPF-SPN empirical ECE ($0.178$). Bottom-left: evidence quality distribution (mean $\approx 1.0$). Bottom-right: scatter of calibration error vs. evidence conflict (KL divergence), with trend $y=0.248x+0.137$.
  • ...and 3 more figures

Theorems & Definitions (29)

  • Theorem 3.1: SPN Calibration Preservation
  • Remark 1
  • Theorem 3.2: Monte Carlo Error Bounds
  • Theorem 3.3: Learned Aggregator Generalization
  • Theorem 3.4: Information-Theoretic Lower Bound
  • Theorem 3.5: Robustness to Evidence Corruption
  • Theorem 3.6: Sample Complexity
  • Theorem 3.7: Uncertainty Decomposition
  • Lemma A.1: Monte Carlo Unbiasedness
  • proof
  • ...and 19 more