The Geometry of Knowing: From Possibilistic Ignorance to Probabilistic Certainty -- A Measure-Theoretic Framework for Epistemic Convergence

Moriba Kemessia Jah

cs.AIMar 13, 2026

The Geometry of Knowing: From Possibilistic Ignorance to Probabilistic Certainty -- A Measure-Theoretic Framework for Epistemic Convergence

Authors: Moriba Kemessia Jah

Organizations: Black Swan Research Group, GaiaVerse, Ltd., Austin, TX 78728, USA · Jah Decision Intelligence Group, Aerospace Engineering & Engineering Mechanics The University of Texas at Austin, Austin, TX 78712

Abstract

This paper develops a measure-theoretic framework establishing when and how a possibilistic representation of incomplete knowledge contracts into a probabilistic representation of intrinsic stochastic variability. Epistemic uncertainty is encoded by a possibility distribution and its dual necessity measure, defining a credal set bounding all probability measures consistent with current evidence. As evidence accumulates, the credal set contracts. The epistemic collapse condition marks the transition: the Choquet integral converges to the Lebesgue integral over the unique limiting density. We prove this rigorously (Theorem 4.5), with all assumptions explicit and a full treatment of the non-consonant case. We introduce the aggregate epistemic width W, establish its axiomatic properties, provide a canonical normalization, and give a feasible online proxy resolving a circularity in prior formulations. Section 7 develops the dynamics of epistemic contraction: evidence induces compatibility, compatibility performs falsification, posterior possibility is the min-intersection of prior possibility and compatibility, and a credibility-directed flow governs support geometry contraction. This is not belief updating. It is knowledge contraction. Probability theory is the limiting geometry of that process. The UKF and ESPF solve different problems by different mechanisms. The UKF minimizes MSE, asserts truth, and requires a valid generative model. The ESPF minimizes maximum entropy and surfaces what evidence has not ruled out. When the world is Gaussian and the model valid, both reach the same estimate by entirely different routes -- convergent optimality, not hierarchical containment. We prove this (Theorem 9.1) and compare both on a 2-day, 877-step orbital tracking scenario. Both achieve 1-meter accuracy. The UKF is accurate but epistemically silent. The ESPF is accurate and epistemically honest.

Figures & tables

\mathrm{NEES}_{t}=\frac{1}{m}\sum_{k=t-L+1}^{t}\tilde{y}_{k}^{\top}S_{k}^{-1}\tilde{y}_{k},\quad\widehat{W}_{t}=1-\exp(-\kappa_{W}\cdot\mathrm{NEES}_{t}).

Algorithm 1 Adaptive Possibilistic–Probabilistic Filtering

Step $k$	$[a_{k},b_{k}]$	$\delta_{k}$	$\overline{W}_{k}$	$\widehat{C}_{k}$	$\mathrm{Leb}_{k}$	gap
0	$[0,\,1]$	—	$1.000$	$1.000$	$0.500$	$0.500$
1	$[0.14,\,0.50]$	$0.14$	$0.219$	$0.570$	$0.320$	$0.250$
2	$[0.22,\,0.42]$	$0.10$	$0.150$	$0.470$	$0.320$	$0.150$
3	$[0.26,\,0.36]$	$0.05$	$0.075$	$0.385$	$0.310$	$0.075$

Table 1: Epistemic width contraction under three sequential observations.

[a_{k},b_{k}]

: flat-top support;

\delta_{k}

: ramp half-width;

\overline{W}_{k}

: normalized aggregate width from ( 38 );

\widehat{C}_{k}=\int_{0}^{\infty}\Pi_{k}\{x\geq t\}\,dt

: Choquet expectation of

f(x)=x

;

\mathrm{Leb}_{k}

: Lebesgue expectation

\int x\,\tilde{\pi}_{k}\,d\mu

under the normalized density

\tilde{\pi}_{k}

; gap

=|\widehat{C}_{k}-\mathrm{Leb}_{k}|

confirms Theorem 4.5 . All values are analytically exact.

Diagnostic	UKF	ESPF
Final RIC error (both cases)	$\approx 1$ m	$\approx 1$ m (comparable)
Stress-onset signal	NIS spike (retrospective)	Necessity + surprisal (earlier)
Post-recovery signal	None— $P_{k}$ collapsed	$\overline{W}_{\mathrm{ep}}$ volatile, prune bursts persist
Nominal overconfidence	NIS exceeds 95% $\chi^{2}$ bound	$\overline{W}_{\mathrm{ep}}\approx 0.8$ ; honest width reported
Covariance after stress	Indistinguishable from pre-stress	Support geometry retains stress memory
Epistemic audit trail	None	$\overline{W}_{\mathrm{ep}}$ , $\hat{\alpha}_{c}$ , $I_{c}$ , necessity, surprisal

Table 2: ESPF vs. UKF: empirical comparison on the orbital tracking scenario. Both filters achieve comparable final accuracy; the distinction is epistemic.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Possibilistic construct	Probabilistic analogue	Coincidence condition
Possibility distribution $\pi(x)$	Probability density $p(x)$	Epistemic collapse: $\Pi(A)=N(A)\;\forall A$
Possibility measure $\Pi(A)$	Probability measure $P(A)$	$\pi(x)=p(x)$ $\mu$ -a.e.
Necessity measure $N(A)$	Complement $1-P(A^{c})$	Same as above
Credal set $\mathcal{P}_{\pi}$	Singleton $\{P^{*}\}$	$\overline{W}=0$
Choquet/Sugeno expectation	Lebesgue expectation $\mathbb{E}_{P^{*}}[f]$	$\|\mathcal{P}_{\pi}\|=1$ and $\pi=p^{*}$
Epistemic width $\overline{W}$	$0$ (zero credal variance)	$\overline{W}\to 0$

Appendix

Table 3: Correspondence between possibilistic and probabilistic constructs and the conditions for their coincidence.

Imprecise Probability

See all 9 papers

The Epistemic Support-Point Filter: Jaynesian Maximum Entropy Meets Popperian Falsification
Mar 10, 2026Moriba Kemessia JahEpistemic Uncertainty Minimax Estimation
Black Swan Research Group, GaiaVerse, Ltd. · Jah Decision Intelligence Group, Aerospace Engineering & Engineering Mechanics, The University of Texas at Austin
Epistemic Uncertainty Is Not the Reducible Kind
Jun 10, 2026Robin YoungUncertainty Quantification Uncertainty Decomposition
University of Cambridge
Optimal Conformal Prediction under Epistemic Uncertainty
May 25, 2025Alireza Javanmardi, Soroush H. Zargarbashi, Santo M. A. R. Thies +3Conformal Prediction Conformal Risk Control
1LMU Munich · 2MCML · 3CISPA +3

Mar 10, 2026J/K move · Enter open · S save

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

Moriba Kemessia Jah

Black Swan Research Group, GaiaVerse, Ltd. · Jah Decision Intelligence Group, Aerospace Engineering & Engineering Mechanics, The University of Texas at Austin

Epistemic Uncertainty Minimax Estimation

Open paper AskPDF arXiv

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.

Abstract

Imprecise Probability

Explore similar work

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

Epistemic Uncertainty Is Not the Reducible Kind

Optimal Conformal Prediction under Epistemic Uncertainty