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The Epistemic Support-Point Filter: Jaynesian Maximum Entropy Meets Popperian Falsification

Moriba Kemessia Jah

Abstract

This paper proves that the Epistemic Support-Point Filter (ESPF) is the unique optimal recursive estimator within the class of epistemically admissible evidence-only filters. Where Bayesian filters minimize mean squared error and are driven toward an assumed truth, the ESPF minimizes maximum entropy and surfaces what has not been proven impossible -- a fundamentally different epistemic commitment with fundamentally different failure modes. Two results locate this theorem within the broader landscape of estimation theory. The first is a unification: the ESPF's optimality criterion is the log-geometric mean of the alpha-cut volume family in the Holder mean hierarchy. The Popperian minimax bound and the Kalman MMSE criterion occupy the p=+inf and p=0 positions on the same curve. Possibility and probability are not competing frameworks: they are the same ignorance functional evaluated under different alpha-cut geometries. The Kalman filter is the Gaussian specialization of the ESPF's optimality criterion, not a separate invention. The second result is a diagnostic: numerical validation over a 2-day, 877-step Smolyak Level-3 orbital tracking run shows that possibilistic stress manifests through necessity saturation and surprisal escalation rather than MVEE sign change -- a direct consequence of the Holder ordering, not an empirical observation. Three lemmas establish the result: the Possibilistic Entropy Lemma decomposes the ignorance functional; the Possibilistic Cramer-Rao Bound limits entropy reduction per measurement; the Evidence-Optimality Lemma proves minimum-q selection is the unique minimizer and that any rule incorporating prior possibility risks race-to-bottom bias.

The Epistemic Support-Point Filter: Jaynesian Maximum Entropy Meets Popperian Falsification

Abstract

This paper proves that the Epistemic Support-Point Filter (ESPF) is the unique optimal recursive estimator within the class of epistemically admissible evidence-only filters. Where Bayesian filters minimize mean squared error and are driven toward an assumed truth, the ESPF minimizes maximum entropy and surfaces what has not been proven impossible -- a fundamentally different epistemic commitment with fundamentally different failure modes. Two results locate this theorem within the broader landscape of estimation theory. The first is a unification: the ESPF's optimality criterion is the log-geometric mean of the alpha-cut volume family in the Holder mean hierarchy. The Popperian minimax bound and the Kalman MMSE criterion occupy the p=+inf and p=0 positions on the same curve. Possibility and probability are not competing frameworks: they are the same ignorance functional evaluated under different alpha-cut geometries. The Kalman filter is the Gaussian specialization of the ESPF's optimality criterion, not a separate invention. The second result is a diagnostic: numerical validation over a 2-day, 877-step Smolyak Level-3 orbital tracking run shows that possibilistic stress manifests through necessity saturation and surprisal escalation rather than MVEE sign change -- a direct consequence of the Holder ordering, not an empirical observation. Three lemmas establish the result: the Possibilistic Entropy Lemma decomposes the ignorance functional; the Possibilistic Cramer-Rao Bound limits entropy reduction per measurement; the Evidence-Optimality Lemma proves minimum-q selection is the unique minimizer and that any rule incorporating prior possibility risks race-to-bottom bias.
Paper Structure (30 sections, 7 theorems, 29 equations, 2 tables)

This paper contains 30 sections, 7 theorems, 29 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Two Principles, One Filter
  3. The Minimax Entropy Principle
  4. The Two Regimes
  5. The Correct Entropy Functional
  6. Structure of the Paper
  7. Setup
  8. Possibility Distributions, $\alpha$-Cuts, and Possibilistic Entropy
  9. Epistemically Admissible Filters
  10. Innovation Geometry and Possibilistic Information Content
  11. Three Lemmas
  12. Lemma 1: Possibilistic Entropy as Ignorance Functional
  13. Lemma 2: Possibilistic Cramér--Rao Bound
  14. Lemma 3: Minimax Entropy and Evidence-Only Selection
  15. Main Theorem
  16. ...and 15 more sections

Key Result

Lemma 3.1

Let $\pi$ be a normalized possibility distribution over $\{\chi^{(i)}\}^M_{i=1} \subset \mathbb{R}^n$ with $\pi^{(i)} \in (0,1]$ and $\max_i \pi^{(i)} = 1$. The possibilistic entropy $H_\pi$ satisfies:

Theorems & Definitions (33)

  • Definition 2.1: Possibility distribution and $\alpha$-cuts
  • Definition 2.2: Possibilistic entropy
  • Remark 2.3: Uniform special case
  • Remark 2.4: Relation to U-uncertainty and origin of the logarithm
  • Definition 2.5: ESPF possibility assignment
  • Remark 2.6: Non-uniformity of the ESPF distribution
  • Definition 2.7: Epistemically admissible filter
  • Remark 2.8: Why evidence-referencing is the right admissibility condition
  • Definition 2.9: Innovation geometry
  • Definition 2.10: Possibilistic information content
  • ...and 23 more