Variational Neurons and Bayesian Neural Networks: Distinct Probabilistic Objects, Inference Loci, and Computational Granularity
Abstract
Probabilistic neural architectures can reuse priors, Kullback–Leibler (KL) terms, reparameterized samples, Monte Carlo evaluation, and predictive distributions while assigning them to different random objects and computational locations. This article introduces an operational framework that separates four non-equivalent axes—Bayesianity, variationality, stochasticity, and distribution-valued computation—and uses six questions to identify the random object, inference or regularization location, conditioning information, sampling location, propagated representation, and prediction rule. It formalizes Definition 1, a generic unit-level specification criterion for a variational neuron: a hidden computational unit containing an input-conditioned local variational latent variable, a declared prior or reference distribution, a locally scoped forward map, a separately computable local cost, and diagnostics at the same declared unit index. Clause-level applications to published third-party architectures illustrate how the criterion distinguishes example-, representation-, weight-, output-, and unit-level probabilistic loci. EVE (Elemental Variational Expanse) provides the controlled worked implementation. Its mechanistic experiments show fixed-checkpoint functional sensitivity, latent-allocation changes with dimensionality, and covariance tracking under controlled relocation of observation noise. A locked fresh 50-seed, exact 2,472-parameter comparison with a heteroscedastic output model yields paired MSE, NLL, CRPS, and MACE intervals including zero, showing closely matched predictive performance at the attained sensitivity. Together, the framework and worked evidence provide a traceable method for separating structural specification, functional use, causal relevance, semantic sensitivity, and predictive utility in probabilistic neural architectures.
What this paper formalizes
The paper separates four non-equivalent dimensions of probabilistic neural computation—Bayesianity, variationality, stochasticity and distribution-valued computation—and organizes architectural reporting around six questions: the random object, inference or regularization location, conditioning evidence, sampling location, propagated representation and prediction rule.
Definition 1 then gives a generic unit-level specification criterion for a variational neuron, requiring an explicitly indexed local latent variable, a declared prior or reference distribution, a predeclared unit partition, a locally scoped forward map before downstream mixing, a separately computable local variational cost, unit-indexed diagnostics, and declared latent-dimension and parameter-sharing conventions.
Controlled EVE evidence
EVE (Elemental Variational Expanse) is the worked implementation. The reported tests separate structural specification, functional utilization, same-checkpoint causal relevance, semantic sensitivity and predictive utility. The fresh exact-capacity follow-up compares a Gaussian-affine EVE unit with a heteroscedastic output model at exactly 2,472 trainable parameters over 50 paired fresh splits; paired MSE, NLL, CRPS and MACE confidence intervals all include zero, supporting closely matched predictive performance at the attained sensitivity.
Keywords
- Bayesian neural networks
- variational inference
- variational neurons
- probabilistic computation
- computational granularity
- uncertainty quantification
- EVE
- Elemental Variational Expanse
Citation
Ruffenach, Y. (2026). Variational Neurons and Bayesian Neural Networks: Distinct Probabilistic Objects, Inference Loci, and Computational Granularity. Zenodo. DOI: 10.5281/zenodo.21870089.
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