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
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.
Table 1: Epistemic width contraction under three sequential observations. [ak,bk] : flat-top support; δk : ramp half-width; Wk : normalized aggregate width from ( 38 ); Ck=∫0∞Πk{x≥t}dt : Choquet expectation of f(x)=x ; Lebk : Lebesgue expectation ∫xπ~kdμ under the normalized density π~k ; gap =∣Ck−Lebk∣ confirms Theorem 4.5 . All values are analytically exact.
Diagnostic
UKF
ESPF
Final RIC error (both cases)
≈1 m
≈1 m (comparable)
Stress-onset signal
NIS spike (retrospective)
Necessity + surprisal (earlier)
Post-recovery signal
None— Pk collapsed
Wep volatile, prune bursts persist
Nominal overconfidence
NIS exceeds 95% χ2 bound
Wep≈0.8 ; honest width reported
Covariance after stress
Indistinguishable from pre-stress
Support geometry retains stress memory
Epistemic audit trail
None
Wep , α^c , Ic , 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 π(x)
Probability density p(x)
Epistemic collapse: Π(A)=N(A)∀A
Possibility measure Π(A)
Probability measure P(A)
π(x)=p(x)μ -a.e.
Necessity measure N(A)
Complement 1−P(Ac)
Same as above
Credal set Pπ
Singleton {P∗}
W=0
Choquet/Sugeno expectation
Lebesgue expectation EP∗[f]
∣Pπ∣=1 and π=p∗
Epistemic width W
0 (zero credal variance)
W→0
Appendix
Table 3: Correspondence between possibilistic and probabilistic constructs and the conditions for their coincidence.
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.
Moriba Kemessia Jah
Black Swan Research Group, GaiaVerse, Ltd. · Jah Decision Intelligence Group, Aerospace Engineering & Engineering Mechanics, The University of Texas at Austin
The standard taxonomy of predictive uncertainty defines epistemic uncertainty as the part removable by collecting more data, while the standard measure identifies it with a mutual-information term. We prove the definition and the measure are extensionally inconsistent. On an explicit construction, the measure assigns all uncertainty to the epistemic class, yet no quantity of training data reduces it. Reducibility is instead a property of the pair (uncertainty, acquisition class), and the dichotomy resolves into three parts: aleatoric, sample-reducible epistemic, and mechanism-reducible epistemic uncertainty. An exact identity for the value of an observation shows that in-distribution data never reduces mechanism-irreducible uncertainty and generically increases it. Ensemble disagreement, the deployed epistemic estimate, tracks the training procedure rather than the epistemic term. It collapses to zero beneath a positive truth under consistent training, and equals hyperparameter-scaled initialization noise under interpolation. A finite-sample falsification test and seed-swept experiments confirm the theory.
Conformal prediction (CP) is a widely used frequentist framework to quantify uncertainty by constructing prediction sets with user-specified marginal coverage guarantees. In practice, CP is typically applied on top of probabilistic classifiers, which are able to express aleatoric but not epistemic uncertainty. In this paper, we consider the question of how to optimally employ CP on top of a more expressive formalism, namely credal sets, which can express both aleatoric and epistemic uncertainty. More specifically, we propose probabilistic Bernoulli prediction sets (BPS) and derive a variant that achieves conditional coverage for valid credal sets while remaining minimal in expected size. We then address the more realistic scenario in which the validity of the credal sets is not guaranteed. Assuming access to calibration data with ground-truth distributions over labels, we apply conformal risk control to BPS and derive a PAC-style guarantee: with high probability over the data, the achieved conditional coverage is at least the desired level. We validate our theoretical findings empirically over various datasets.
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