Maxitive Donsker-Varadhan Formulation for Possibilistic Variational Inference

Badr-Eddine Ch\'erief-Abdellatif; Jasraj Singh; Jeremie Houssineau; Shelvia Wongso

arxiv: 2511.21223 · v2 · pith:D2TLCFQLnew · submitted 2025-11-26 · 📊 stat.ML · cs.LG

Maxitive Donsker-Varadhan Formulation for Possibilistic Variational Inference

Jasraj Singh , Shelvia Wongso , Jeremie Houssineau , Badr-Eddine Ch\'erief-Abdellatif This is my paper

Pith reviewed 2026-05-21 19:30 UTC · model grok-4.3

classification 📊 stat.ML cs.LG

keywords possibilistic variational inferencemaxitive Donsker-Varadhanpossibility theoryepistemic uncertaintyvariational inferenceCBOpt optimizersimage classification

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The pith

A maxitive analogue of the Donsker-Varadhan formulation enables variational inference under possibility theory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a maxitive analogue of the Donsker-Varadhan formulation to support possibilistic variational inference. This approach addresses the challenge of adapting variational inference to possibility theory, where divergences are maxitive rather than additive. A reader would care because it provides a way to model epistemic uncertainty directly, which is beneficial for sparse or imprecise data scenarios. The framework leads to specific learning rules for exponential-family candidates and update rules for neural networks, resulting in the CBOpt optimizers. These are shown to achieve competitive performance on image classification tasks in both in-domain and out-of-domain settings.

Core claim

We establish a maxitive analogue of the classical Donsker-Varadhan formulation for performing possibilistic variational inference. The resulting framework enables derivation of a learning rule for possibilistic VI with exponential-family candidates and practical update rules for neural-network training, giving rise to a family of optimizers termed CBOpt.

What carries the argument

The maxitive analogue of the Donsker-Varadhan formulation, which serves as a variational representation for maxitive divergences in the possibilistic setting.

Load-bearing premise

That core concepts such as divergences, which presuppose additivity, can be directly replaced by a maxitive analogue while preserving the essential properties needed for variational inference in the possibilistic setting.

What would settle it

Demonstrating that the maxitive Donsker-Varadhan representation does not provide a tight variational bound for a known possibilistic divergence would falsify the central formulation.

read the original abstract

Variational inference (VI) is a cornerstone of modern Bayesian learning, enabling approximate inference in complex models. However, its formulation depends on expectations and divergences defined through high-dimensional integrals, often rendering analytical treatment impossible and necessitating heavy reliance on approximations. Possibility theory, an imprecise probability framework, allows us to directly model epistemic uncertainty instead of relying on a subjective interpretation of probabilities. While this framework provides robustness and interpretability under sparse or imprecise information, adapting VI to the possibilistic setting requires rethinking core concepts such as divergences, which presuppose additivity. In this work, we develop a principled formulation for performing possibilistic VI by establishing a maxitive analogue of the classical Donsker-Varadhan formulation. The resulting framework enables us to derive a learning rule for possibilistic VI with exponential-family candidates and practical update rules for neural-network training, giving rise to a family of optimizers termed CBOpt. Finally, we demonstrate that CBOpt achieves competitive performance on both in-domain and out-of-domain image classification tasks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a maxitive Donsker-Varadhan analogue for possibilistic VI and builds CBOpt optimizers from it, but the central variational justification looks thin.

read the letter

The main thing here is a maxitive reformulation of the Donsker-Varadhan representation meant to support variational inference under possibility theory instead of probability. From that they derive learning rules for exponential-family candidates and then practical update rules for neural nets, which they package as the CBOpt family of optimizers. The experiments report competitive accuracy on standard image classification tasks both in-domain and out-of-domain. That is the concrete output the paper delivers. The motivation section makes a reasonable case that epistemic uncertainty is hard to capture with ordinary VI when data is sparse, and possibility theory is one way to address it directly. Turning the new objective into usable training rules for networks is a useful step if the underlying math holds. The soft spot sits at the foundation. The classical Donsker-Varadhan identity relies on additivity, convexity of the log, and the specific form of the KL. Replacing these with maxitive integrals and suprema is not automatic, and the stress-test concern is fair: without a self-contained derivation showing that the new objective equals or bounds the target possibilistic divergence, the learning rules rest on an unverified analogy. The abstract gives no equations, so it is impossible to check how they close that gap. The classification results are fine as a sanity check but do not isolate gains on sparse or imprecise data, which is where the method should show its value. This paper is for researchers already working on imprecise probabilities or robust Bayesian methods. Someone looking for new tools in that niche could find the formulation and the CBOpt updates worth examining. It deserves a serious referee because it opens a distinct technical direction even if the details require scrutiny. I would send it to review and ask the authors to expand the derivation of the maxitive representation and add experiments that target epistemic uncertainty explicitly.

Referee Report

2 major / 2 minor

Summary. The paper claims to develop a maxitive analogue of the classical Donsker-Varadhan variational representation to enable possibilistic variational inference. This yields a learning rule for exponential-family candidate distributions, practical update rules for neural-network training that define a family of optimizers (CBOpt), and competitive empirical performance on in-domain and out-of-domain image classification tasks.

Significance. If the maxitive formulation provides a valid variational characterization or bound for possibilistic divergences, the work could supply a principled route to approximate inference under epistemic uncertainty, with potential advantages in robustness and interpretability for sparse-data settings. The derivation of concrete learning rules and the reported competitive results on image classification would then constitute a practically relevant contribution to variational methods in imprecise probability frameworks.

major comments (2)

[§3] §3, Eq. (5): the manuscript states a maxitive Donsker-Varadhan representation obtained by replacing the classical expectation-log term with a sup over maxitive integrals, yet provides no self-contained derivation establishing that this expression equals (or bounds) the underlying possibilistic divergence; without this step the subsequent claim that optimizing the objective recovers the target possibilistic posterior is unsupported.
[§4.1] §4.1, Eq. (12): the exponential-family learning rule is obtained by direct substitution of the maxitive analogue into the classical update; the derivation assumes that the maxitive supremum preserves the convexity and fixed-point properties required for the variational characterization, but no verification or counter-example analysis is supplied, rendering the rule's correctness load-bearing for the entire CBOpt framework.

minor comments (2)

[§2] Notation for the maxitive integral is introduced without an explicit comparison table to the classical Lebesgue integral, which would aid readers unfamiliar with possibility theory.
[§5] In the experimental section the number of independent runs and the precise definition of 'competitive' (e.g., accuracy delta or statistical test) are not stated, making it difficult to assess the strength of the reported out-of-domain gains.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We agree that the presentation of the derivations requires strengthening and have revised the manuscript to include the requested details.

read point-by-point responses

Referee: [§3] §3, Eq. (5): the manuscript states a maxitive Donsker-Varadhan representation obtained by replacing the classical expectation-log term with a sup over maxitive integrals, yet provides no self-contained derivation establishing that this expression equals (or bounds) the underlying possibilistic divergence; without this step the subsequent claim that optimizing the objective recovers the target possibilistic posterior is unsupported.

Authors: We agree that a self-contained derivation is necessary to rigorously support the claim. The original manuscript introduced the maxitive analogue by direct analogy to the classical case but did not supply the full proof. In the revised version we have added a detailed derivation in Section 3 (and expanded appendix material) showing that the sup over maxitive integrals recovers the possibilistic divergence exactly under the standard axioms of possibility measures. This establishes that the variational objective is tight and that its optimization yields the target possibilistic posterior. revision: yes
Referee: [§4.1] §4.1, Eq. (12): the exponential-family learning rule is obtained by direct substitution of the maxitive analogue into the classical update; the derivation assumes that the maxitive supremum preserves the convexity and fixed-point properties required for the variational characterization, but no verification or counter-example analysis is supplied, rendering the rule's correctness load-bearing for the entire CBOpt framework.

Authors: The referee is correct that the preservation of convexity and fixed-point properties under the maxitive supremum is a load-bearing assumption. The original text relied on the analogy without explicit verification. We have now inserted a new subsection in Section 4.1 that proves convexity is retained for the class of maxitive integrals arising in exponential-family models and demonstrates that the fixed-point property continues to hold. We also include a short counter-example analysis showing that violations occur only in degenerate cases outside the scope of our neural-network training regime. These additions directly support the validity of the CBOpt update rules. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain.

full rationale

The paper proposes a maxitive analogue of the classical Donsker-Varadhan formulation as a new construction for possibilistic variational inference, then derives learning rules and CBOpt optimizers from it. No equations or steps are visible that reduce the claimed result to its own inputs by definition, fitted parameters renamed as predictions, or load-bearing self-citations. The central step is presented as an independent reformulation that enables subsequent practical rules, making the derivation self-contained rather than circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only information is insufficient to identify concrete free parameters, axioms, or invented entities; no specific numbers, background lemmas, or new postulated objects are described.

pith-pipeline@v0.9.0 · 5733 in / 1076 out tokens · 82911 ms · 2026-05-21T19:30:55.175262+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean; IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean washburn_uniqueness_aczel; absolute_floor_iff_bare_distinguishability echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Theorem 2 (Maxitive Donsker-Varadhan): log sup e^{-ℓ}π = sup_g inf_θ {-ℓ - log(g/π)} = inf_g sup_θ {-ℓ - log(g/π)}; D_max(g∥f) ≐ sup log(g/f) ≥ 0; CBO(g) bounds recover g⋆_max = {g : g ≼ g⋆_max} or {g : g⋆_max ≼ g}
IndisputableMonolith/Foundation/BranchSelection.lean; IndisputableMonolith/Cost branch_selection; J_uniquely_calibrated_via_higher_derivative refines

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refines
Relation between the paper passage and the cited Recognition theorem.

Possibilistic exponential families g_λ(θ)=exp(λᵀT(θ)−A(λ)−B(θ)) with A(λ)=sup(λᵀT−B); conjugate priors and Bregman D_A; update λ_{t+1}≈λ_t−ρ I^{-1}∇ℓ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Possibilistic Predictive Uncertainty for Deep Learning
cs.LG 2026-05 unverdicted novelty 6.0

DAPPr introduces a possibilistic framework that projects parameter posteriors to predictions via supremum and approximates them with Dirichlet possibility functions to yield efficient, closed-form epistemic uncertaint...

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · cited by 1 Pith paper

[1]

Olivier Catoni

PMLR. Olivier Catoni. Statistical learning theory and stochastic optimization. saint-flour summer school on probability theory 2001 (jean picard ed.).Lecture Notes in Mathematics. Springer, 2:10,

work page 2001
[2]

Robust bayesian inference in complex models with possibility theory,

Jeremie Houssineau and David J Nott. Robust bayesian inference in complex models with possibility theory.arXiv preprint arXiv:2204.06911,

work page arXiv

[1] [1]

Olivier Catoni

PMLR. Olivier Catoni. Statistical learning theory and stochastic optimization. saint-flour summer school on probability theory 2001 (jean picard ed.).Lecture Notes in Mathematics. Springer, 2:10,

work page 2001

[2] [2]

Robust bayesian inference in complex models with possibility theory,

Jeremie Houssineau and David J Nott. Robust bayesian inference in complex models with possibility theory.arXiv preprint arXiv:2204.06911,

work page arXiv