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Maxitive Donsker-Varadhan Formulation for Possibilistic Variational Inference

NeurIPS 2026 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Variational Bayesian deep learning · 0 citations (OpenAlex)

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Derives a maxitive (possibilistic) Donsker-Varadhan variational formula with lower and upper consistency bounds, and uses it to build CBOpt, a family of Newton-like optimizers for neural network training evaluated on image classification with OOD detection.

Problem

Variational inference relies on additive measures (expectations, KL divergence, Shannon entropy), so it does not carry over directly to possibility theory, an imprecise probability framework for epistemic uncertainty.

Approach

A maxitive analogue of the Donsker-Varadhan formula gives two dual consistency bounds (lower and upper CBO) over possibility functions, whose optima bound the possibilistic posterior; restricting to exponential-family (Gaussian) candidates yields a signed Newton-like update applied to neural network training.

Key results

• On Fashion-MNIST, lCBOpt-adapt reaches the best accuracy of 0.913 and NLL of 0.252 among compared optimizers.
• On CIFAR-100, all three CBOpt variants outperform the baselines on accuracy, NLL and OOD metrics, with uCBOpt reaching 0.671 accuracy versus 0.649 for SGD and MC-D.
• On CIFAR-10, CBOpt variants reach around 0.910 accuracy, comparable to SGD and Laplace, but do not surpass SWAG or MC Dropout on OOD detection.

Surprising findings

• On CIFAR-10, CBOpt does not surpass SWAG or MC Dropout on OOD detection despite competitive in-domain accuracy.
• IVON, MC Dropout and CBOpt variants have better ECE than the CBOpt variants on CIFAR-100, where IVON and MC dropout achieve better calibration.

Limitations

• Newton-like update requires curvature condition H_s positive definite, which is hard to maintain pointwise for the lower CBO, motivating adaptive variants.
• Exponential-family treatment assumes proper closed convex log-partition and excludes the fully uninformative limit.
• Empirically limited to isotropic Gaussian candidates with a diagonal empirical-Fisher surrogate.

Experimental setup

Datasets: Fashion-MNIST CIFAR-10 CIFAR-100

Benchmarks: Fashion-MNIST EMNIST CIFAR-10 SVHN CIFAR-100 TinyImageNet

Metrics: accuracy NLL ECE FPR@95 AUROC AUPR-Out

Baselines: AdamW SGD IVON MC Dropout Laplace SWAG

Builds on: Donsker-Varadhan variational formula IVON Adam Vprop generalized variational inference

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