Theoretical Study on the Evidential Learning-based Variational Autoencoder

arXiv:2609.27853v1 Announce Type: cross Abstract: A normal--inverse-gamma (NIG) latent hierarchy has four parameters, but its induced latent law does not identify all four. For sigma^2simInvGamma(alpha,eta), mumidsigma^2simN(gamma,sigma^2/nu), and zmidmu,sigma^2simN(mu,sigma^2), the marginal law of z depends on (nu,eta) only through c=eta(1+1/nu). Hence the reconstruction-visible parameter space is the three-dimensional quotient (gamma,alpha,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 rho_0=2eta_0/nu_0 and T=c/{alpha[(gamma-gamma_0)^2+rho_0]}, we show that inverse canonical allocation 1/nu_{rm can} is an explicit strictly increasing function of T. For alpha>1, the ratio u_{rm epi}/u_{rm var}=1/nu_{rm 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: (gamma_0,rho_0) govern ordinal dependence, while (nu_0,alpha_0) determine numerical calibration and the analytic ceiling of 1/nu_{rm can}.

Source: arXiv cs.LG | 2026-09-24