Theoretical Study on the Evidential Learning-based Variational Autoencoder

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), μmidσ2~N(γ,σ2/nu), and zmidμ,σ2~N(μ,σ2), the marginal law of z depends on (nu,β) only through c=β(1+1/nu). 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/nu0 and T=c/α[(γ-γ0)2+ρ0], we show that inverse canonical allocation 1/nurm can is an explicit strictly increasing function of T. For α>1, the ratio urm epi/urm var=1/nurm 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: (γ0,ρ0) govern ordinal dependence, while (nu0,α0) determine numerical calibration and the analytic ceiling of 1/nurm can.