Theoretical Study on the Evidential Learning-based Variational Autoencoder
Abstract
A normal--inverse-gamma (NIG) latent hierarchy has four parameters, but its induced latent law does not identify all four. For σ2∼InvGamma(α,β), μ|σ2∼𝒩(γ,σ2/ν), and z|μ,σ2∼𝒩(μ,σ2), the marginal law of z depends on (ν,β) only through c=β(1+1/ν). Hence the reconstruction-visible parameter space is the three-dimensional quotient (γ,α,c), with a one-dimensional fiber degree of freedom. For a fixed hierarchical variational objective, exact partial minimization of the forward KL divergence to a complete NIG prior selects a unique prior-relative representative on each fiber, yielding an exact three-coordinate reduction with the same optimum as the four-coordinate objective. Writing ρ0=2β0/ν0 and T=c/{α[(γ−γ0)2+ρ0]}, we show that inverse canonical allocation 1/νcan is an explicit strictly increasing function of T. For α>1, the ratio uepi/uvar=1/νcan is therefore determined by the quotient state and prior; for rank-only use under a common calibration, T contains the same coordinatewise ordinal information. The residual prior gauge is characterized rather than eliminated: (γ0,ρ0) govern ordinal dependence, while (ν0,α0) determine numerical calibration and the analytic ceiling of 1/νcan.
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BibTeX
@misc{wang2026theoretical,
title = {{Theoretical Study on the Evidential Learning-based Variational Autoencoder}},
author = {Ge Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27853},
eprint = {2609.27853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27853},
}
RIS
TY - UNPB
TI - Theoretical Study on the Evidential Learning-based Variational Autoencoder
AU - Ge Wang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27853
UR - https://arxiv.org/abs/2609.27853
AB - A normal--inverse-gamma (NIG) latent hierarchy has four parameters, but its induced latent law does not identify all four. For σ^2∼InvGamma(α,β), μ|σ^2∼𝒩(γ,σ^2/ν), and z|μ,σ^2∼𝒩(μ,σ^2), the marginal law of z depends on (ν,β) only through c=β(1+1/ν). Hence the reconstruction-visible parameter space is the three-dimensional quotient (γ,α,c), with a one-dimensional fiber degree of freedom. For a fixed hierarchical variational objective, exact partial minimization of the forward KL divergence to a complete NIG prior selects a unique prior-relative representative on each fiber, yielding an exact three-coordinate reduction with the same optimum as the four-coordinate objective. Writing ρ_0=2β_0/ν_0 and T=c/{α[(γ−γ_0)^2+ρ_0]}, we show that inverse canonical allocation 1/ν_can is an explicit strictly increasing function of T. For α>1, the ratio u_epi/u_var=1/ν_can is therefore determined by the quotient state and prior; for rank-only use under a common calibration, T contains the same coordinatewise ordinal information. The residual prior gauge is characterized rather than eliminated: (γ_0,ρ_0) govern ordinal dependence, while (ν_0,α_0) determine numerical calibration and the analytic ceiling of 1/ν_can.
ER -