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ELVAE: Evidential Learning-Based Variational Autoencoder for Uncertainty-Aware Generation
Ge Wang
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 8/13/2026, 5:24:03 AM
Summary
The paper introduces ELVAE, an evidential learning-based Variational Autoencoder that models latent coordinates using a Normal-Inverse-Gamma (NIG) posterior. This approach explicitly separates epistemic uncertainty (uncertainty about the latent location) from aleatoric variability. The authors demonstrate that this uncertainty metric (u_epi) can stratify the reliability of generated samples on MNIST, where high-uncertainty anchors lead to lower semantic fidelity. The study highlights that direct regularization of the NIG hierarchy is required to identify this uncertainty decomposition, which is not possible with marginalized Student-t distributions alone.
Entities (6)
Relation Signals (5)
ELVAE → evaluatedon → MNIST
confidence 98% · In a 10,000-image MNIST pilot, samples were ranked by within-class u_epi
ELVAE → uses → Normal-Inverse-Gamma
confidence 95% · We formulate ELVAE, an evidential learning-based VAE in which each latent coordinate is governed by an input-dependent normal-inverse-gamma (NIG) posterior.
ELVAE → produces → Epistemic Uncertainty
confidence 92% · The hierarchy yields an explicit latent-location uncertainty, u_epi... that can stratify posterior anchors
ELVAE → optimizes → ELBO
confidence 90% · The objective is an exact evidence lower bound
Epistemic Uncertainty → stratifies → MNIST
confidence 85% · samples were ranked by within-class u_epi and evaluated with a classifier... Classifier error increased... in the top 20%
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Abstract
Abstract:Variational autoencoders generate samples from probabilistic latent representations but do not distinguish uncertainty about the latent location from variability around it. We formulate ELVAE, an evidential learning-based VAE in which each latent coordinate is governed by an input-dependent normal-inverse-gamma posterior. This hierarchy yields an explicit latent-location uncertainty that can be used during generation, not merely reported after inference: low-uncertainty anchors support more reliable synthetic samples, while high-uncertainty anchors can be deliberately exploited for stress testing. The objective is an exact evidence lower bound, and we show that direct regularization of the full hierarchy is required, since the marginalized latent law alone cannot identify the uncertainty decomposition. In an MNIST generation pilot with a frozen external classifier, this uncertainty clearly stratified the semantic reliability of generated digits. A zero-displacement control revealed that most of the effect reflects how reliably an anchor can be re-generated, while a smaller but distinct component is attributable to uncertainty-scaled perturbation itself. The effect holds only under within-class uncertainty ranking, and its magnitude varies across seeds. These findings support the learned latent-location uncertainty as a practical control variable for uncertainty-aware generation, separating anchor reliability from perturbation-induced failure.
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- Source: https://arxiv.org/abs/2608.10398v1
- Canonical: https://arxiv.org/abs/2608.10398v1
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ELVAE: Evidential Learning-Based Variational Autoencoder for Uncertainty-Aware Generation Ge Wang Biomedical Imaging Center, Rensselaer Polytechnic Institute, Troy, New York, USA Abstract: Variational autoencoders (VAEs) generate samples from probabilistic latent representations but do not explicitly distinguish uncertainty in the latent location from variability around that location. We formulate ELVAE, an evidential learning-based VAE in which each latent coordinate is governed by an input-dependent normal-inverse-gamma (NIG) posterior. The hierarchy yields an explicit latent-location uncertainty, uepi=β/[ν(α−1)]u_epi=β/[ν(α-1)], that can stratify posterior anchors and modulate generation. In a 10,000-image MNIST pilot, samples were ranked by within-class uepiu_epi and evaluated with a classifier trained only on real MNIST. Classifier error increased from 26.30% in the bottom 20% of uncertainty to 37.80% in the top 20% (1.437×, 95% bootstrap CI 1.33–1.58). A zero-displacement control, z=γz=γ, retained most of this contrast (1.395×), showing that the dominant effect reflects anchor re-generation reliability. Among anchors correctly re-generated at z=γz=γ, an uncertainty-scaled perturbation induced semantic failure in 1.97% of the low-uepiu_epi group versus 5.92% of the high-uepiu_epi group (3.01×, 95% CI 2.03–4.75). Two caveats are reported alongside the headline: under unnormalized global uepiu_epi ranking the contrast is essentially null (1.015×), so within-class normalization is not cosmetic, and across three random seeds the headline ratio ranges from 1.126 to 1.437. The formulation is also mathematically well posed: the ELVAE objective is an exact ELBO for the corresponding hierarchical generative model, and direct NIG-to-NIG regularization identifies an uncertainty decomposition that the marginalized Student-t latent law alone does not. These results support uepiu_epi as a useful variable for posterior-anchored uncertainty-aware generation, while distinguishing anchor reliability from perturbation-attributable failure. Keywords: evidential learning; variational autoencoder; uncertainty-aware generation; synthetic data; stress testing; normal-inverse-gamma; epistemic uncertainty; generative AI. 1 Introduction Generative AI is typically evaluated by realism, fidelity, diversity, or downstream usefulness. For training-data generation and robustness testing, however, a second question is equally important: how uncertain is the model about the latent state from which a particular synthetic image is generated? Two generated images can both look plausible while one lies in a well-constrained latent region and the other is produced from a latent location for which the model has weak evidence. A conventional VAE models qϕ(z∣y)=(μϕ(y),diagσϕ2(y)),q_φ(z y)=N\! ( _φ(y),diag _φ^2(y) ), (1) where μϕ _φ and σϕ2 _φ^2 are deterministic network outputs [2]. This provides stochastic generation, but the model has only one level of latent uncertainty. Evidential learning suggests a richer construction: place a distribution over the latent mean and variance themselves [4]. We call the resulting model ELVAE. ELVAE turns uncertainty into a generation control variable: low uncertainty can identify more reliable synthetic samples, while high uncertainty can be deliberately retained to create difficult stress-test samples. This is particularly attractive for scientific and medical generation, where synthetic images may be used to enlarge scarce training sets or to probe failure modes of a downstream network. Relation to prior evidential and autoencoder work. Autoencoders have been used for representation learning since [1], and the variational formulation of [2] made the latent code an explicit probabilistic object. Evidential learning replaces a point prediction with parameters of a higher-order distribution, first for classification through a Dirichlet output [3] and then for regression through the NIG family [4]; the latter supplies the hierarchy used here. Itkina et al. [7] apply an evidential treatment to the discrete latent distribution of a conditional VAE to prune implausible modes, while Baykal et al. [8] use a Dirichlet evidential distribution over discrete VQ-VAE codebook assignments to mitigate codebook collapse. Catoni et al. [9] study uncertainty representations in continuous VAEs and introduce an Explaining-Away VAE with an additional global scaling latent variable; they evaluate latent uncertainty on natural, MNIST, and medical-image domains. ELVAE addresses a different question: it places a coordinate-wise NIG hierarchy directly over continuous latent location and variance, identifies the higher-order uncertainty decomposition through NIG-to-NIG regularization, and exposes the latent-location variance uepiu_epi as a variable for posterior-anchored generation. Known pathologies of evidential regression objectives [5, 6] remain relevant and motivate direct regularization of the full NIG hierarchy. Figure 1 summarizes the training and generation pipeline. The present study makes four contributions. First, we formulate a continuous NIG hierarchy for VAE latent variables and define an explicit latent-location uncertainty uepiu_epi. Second, we show that direct NIG regularization is required to identify the uncertainty decomposition and that the resulting training objective is an exact ELBO. Third, we perform a direct generation pilot: ELVAE generates labeled MNIST variants from posterior anchors, low- and high-uepiu_epi images are displayed, and one frozen classifier quantifies whether high-uepiu_epi generated samples are more likely to lose their intended semantic class. Fourth, we separate that effect into an anchor-quality component and a generation-attributable component by means of a z=γz=γ control, and report both. 2 Methodology 2.1 Evidential latent hierarchy For each latent coordinate k=1,…,Kk=1,…,K, the encoder predicts four NIG parameters (γk,νk,αk,βk),νk>0,αk>1,βk>0,( _k, _k, _k, _k), _k>0,\ _k>1,\ _k>0, (2) which define σk2∣y∼InvGamma(αk,βk),μk∣σk2,y∼(γk,σk2νk),zk∣μk,σk2∼(μk,σk2). _k^2 y ( _k, _k), _k _k^2,y \! ( _k, _k^2 _k ), z_k _k, _k^2 ( _k, _k^2). (3) (a) TrainingyyEncoderqϕ(μ,σ2∣y)=NIG(γ,ν,α,β)q_φ(μ,σ^2 y)=NIG(γ,ν,α,β)z sampleDecodery yNIG prior p0p_0: LNIG=1K∑kDKL(qϕ∥p0)L_NIG= 1K _kD_KL(q_φ\|p_0)LreconL_recon(b) Generationanchor (yi,ci)(y_i,c_i)Encoderγi,uepi,i _i,\ u_epi,iuncertainty-scaled perturbationz=γi+τepiuepi,i⊙ϵz= _i+ _epi u_epi,i classifierci→c^ic_i→ c_i?xigenx_i^gencontrol: τepi=0⇒z=γi _epi=0 z= _iuepiu_epi retained for ranking Figure 1: ELVAE training and the posterior-anchored generation pilot. (a) Training: the encoder predicts a NIG posterior over latent location and variance, regularized toward a fixed NIG prior. (b) Generation: the pilot uses the NIG-derived uepiu_epi both to rank anchors and to scale a controlled Gaussian perturbation, and a frozen classifier tests whether the intended class survives. The dashed control branch sets τepi=0 _epi=0, yielding z=γiz= _i, while retaining uepiu_epi for ranking. Thus qϕ(μk,σk2∣y)=NIG(γk,νk,αk,βk).q_φ( _k, _k^2 y)=NIG( _k, _k, _k, _k). (4) We enforce positivity by softplus transforms, with the standard +1+1 offset on α so that [σ2]E[σ^2] exists [4]. The hierarchy separates two statistically different sources of latent spread: uvar,k u_var,k ≡[σk2∣y]=βkαk−1, [ _k^2 y]= _k _k-1, (5) uepi,k u_epi,k ≡Var(μk∣y)=βkνk(αk−1), ( _k y)= _k _k( _k-1), (6) Var(zk∣y) (z_k y) =uvar,k+uepi,k. =u_var,k+u_epi,k. (7) In this paper we call Eq. (6) epistemic latent uncertainty because it quantifies uncertainty in the latent location itself, while Eq. (5) quantifies variability around that location. For one image we summarize uepi(y)=1K∑k=1Kuepi,k(y).u_epi(y)= 1K _k=1^Ku_epi,k(y). (8) 2.2 Training objective With P input pixels and K latent coordinates, we use LELVAE(y)=Lrecon(y)+λNIGLNIG(y),L_ELVAE(y)=L_recon(y)+ _NIGL_NIG(y), (9) where Lrecon=1P‖y−gθ(z)‖22,L_recon= 1P\,E\|y-g_θ(z)\|_2^2, (10) and LNIG=1K∑k=1KDKL[NIG(γk,νk,αk,βk)∥NIG(γ0,ν0,α0,β0)].L_NIG= 1K _k=1^KD_KL\! [NIG( _k, _k, _k, _k)\,\|\,NIG( _0, _0, _0, _0) ]. (11) The NIG KL directly regularizes the higher-order quantities to which uncertainty meaning is assigned. Because LNIGL_NIG is the coordinate average, the total higher-order KL is KLNIGKL_NIG. For q=NIG(γ,ν,α,β)q=NIG(γ,ν,α,β) and p0=NIG(γ0,ν0,α0,β0)p_0=NIG( _0, _0, _0, _0) under the inverse-gamma parameterization of Eq. (3), the coordinate-wise divergence is DKL(q∥p0)= D_KL(q\|p_0)= α0logβ0−logΓ(α)+logΓ(α0)+(α−α0)ψ(α)−α+αβ0β _0 β _0- (α)+ ( _0)+(α- _0)ψ(α)-α+ α _0β +12[logν0+ν0ν−1+ν0α(γ−γ0)2β], + 12 [ ν _0+ _0ν-1+ _0α(γ- _0)^2β ], (12) where ψ is the digamma function. This is the KL between the inverse-gamma factors plus the expected KL between the conditional normal distributions of μ. 2.3 ELBO interpretation and determination of the NIG weight Starting directly from Eq. (9), the only apparent free balance is λNIG _NIG, which weights the NIG regularizer relative to the reconstruction term. This weight has a likelihood interpretation rather than being an arbitrary tuning parameter. Consider the hierarchical generative model p(μ,σ2)=p0,p(z∣μ,σ2)=(μ,σ2),p(y∣z)=(gθ(z),s2IP),p(μ,σ^2)=p_0, p(z μ,σ^2)=N(μ,σ^2), p(y z)=N\! (g_θ(z),s^2I_P ), (13) with inference model q(μ,σ2,z∣y)=qϕ(μ,σ2∣y)p(z∣μ,σ2)q(μ,σ^2,z y)=q_φ(μ,σ^2 y)p(z μ,σ^2). The scalar s2s^2 is the homoscedastic observation variance in image space. For this model, the negative ELBO is −ELBO(y)=P2log(2πs2)+P2s2Lrecon(y)+KLNIG(y).-ELBO(y)= P2 (2π s^2)+ P2s^2L_recon(y)+KL_NIG(y). (14) The chain-rule KL reduces to DKL[qϕ(μ,σ2∣y)∥p0]D_KL[q_φ(μ,σ^2 y)\|p_0] because q(z∣μ,σ2)q(z μ,σ^2) is chosen to equal p(z∣μ,σ2)p(z μ,σ^2). The Gaussian observation model supplies the first two terms in Eq. (14). Multiplying Eq. (9) by P/(2s2)P/(2s^2) gives P2s2LELVAE=P2s2Lrecon+PλNIG2s2LNIG. P2s^2L_ELVAE= P2s^2L_recon+ P _NIG2s^2L_NIG. (15) Matching the coefficient of LNIGL_NIG in Eqs. (14) and (15) gives λNIG=2s2KP,or equivalentlys2=PλNIG2K. _NIG= 2s^2KP, equivalently s^2= P _NIG2K. (16) Thus λNIG _NIG and the assumed image-space residual variance are two parameterizations of the same relative weighting. Fixing one determines the other. The observation variance can itself be estimated from reconstruction residuals. If s2s^2 is treated as an unknown scalar and Eq. (14) is minimized with respect to s2s^2 while the encoder and decoder are held fixed, then s^2=1Py,z‖y−gθ(z)‖22=Lrecon, s^2= 1PE_y,z\|y-g_θ(z)\|_2^2=L_recon, (17) where LreconL_recon denotes the reconstruction MSE averaged over the data distribution and latent sampling. Substituting Eq. (17) into Eq. (16) yields the practical calibration λ^NIG=2KPLrecon. λ_NIG= 2KPL_recon. (18) Therefore, once the reconstruction MSE is measured, λNIG _NIG can be determined from the likelihood model instead of selected independently. The relevant quantity is the dataset-averaged, approximately converged reconstruction MSE, not the error of one image or one mini-batch. Because λNIG _NIG also affects training, a fully self-consistent implementation can use a short warm-up, estimate LreconL_recon, update λNIG _NIG using Eq. (18), and optionally repeat; an even cleaner alternative is to learn logs2 s^2 jointly in the unscaled ELBO of Eq. (14). Because Eq. (18) is a testable identity rather than a recommendation, we report the test. On the trained pilot model the held-out reconstruction MSE under the sampled hierarchy is Lrecon=0.0408L_recon=0.0408, so Eq. (18) gives λ^NIG=8.33×10−4 λ_NIG=8.33× 10^-4, whereas the fixed value used for training is 5×10−45× 10^-4 (equivalently s2=0.0245s^2=0.0245). The fixed weight is therefore roughly a factor of 1.67 smaller than its own likelihood-consistent value: the pilot places somewhat less weight on the NIG regularizer than the observation model would imply. We retain the fixed setting used in the reported pilot and flag the discrepancy explicitly rather than leaving Eq. (18) as an unverified claim. Closing the gap by one fixed-point iteration is an important next step for a study in which λNIG _NIG is varied deliberately. 2.4 Why the full NIG hierarchy must be regularized Marginalizing (μ,σ2)(μ,σ^2) yields a Student-t latent distribution. However, that marginal does not identify the decomposition in Eqs. (5)–(6). Proposition 1 (Marginal non-identifiability). Under Eq. (3), zkz_k is Student-t with 2α2α degrees of freedom, location γ, and squared scale β(1+1/ν)/αβ(1+1/ν)/α. Therefore the marginal depends on (ν,β)(ν,β) only through c=β(1+1/ν)c=β(1+1/ν). Along the curve β(1+1/ν)=cβ(1+1/ν)=c, the marginal distribution of z is unchanged while uvar=cα−1ν1+ν,uepi=cα−111+νu_var= cα-1 ν1+ν, u_epi= cα-1 11+ν (19) can trade continuously against one another. The practical consequence is important for generation: if uepiu_epi is to control which images are treated as reliable or challenging, the hierarchy that defines uepiu_epi must itself be constrained. A loss written only on the marginalized p(z)p(z) cannot supply that identification [5, 6]. 2.5 Prior and pilot architecture We use (γ0,ν0,α0,β0)=(0,1,3,1).( _0, _0, _0, _0)=(0,1,3,1). (20) This gives [σ2]=1/2E[σ^2]=1/2, Var(μ)=1/2Var(μ)=1/2, and hence Var(z)=1Var(z)=1, preserving the familiar centered unit-variance latent convention while explicitly separating the two components. The pilot ELVAE uses an MLP encoder 784→128→64784→ 128→ 64, latent dimension K=8K=8, and a mirrored decoder. It is trained for four epochs with Adam, learning rate 10−310^-3, batch size 1024, and λNIG=5×10−4 _NIG=5× 10^-4. With P=784P=784 and K=8K=8, Eq. (16) shows that this fixed weight corresponds to an assumed observation variance s2=0.0245s^2=0.0245. The goal is not maximum MNIST generation quality; it is a compact test of whether the learned uepiu_epi provides useful generation stratification. 3 Uncertainty-Aware Generation 3.1 Uncertainty-scaled posterior-anchored generation For a labeled anchor (yi,ci)(y_i,c_i), the encoder gives (γi,νi,αi,βi)( _i, _i, _i, _i) and the coordinate-wise uncertainty vector uepi,i=βi⊘[νi⊙(αi−1)],u_epi,i= _i [ _i ( _i-1)], (21) where ⊘ and ⊙ denote elementwise division and multiplication. The exact NIG marginal of the latent location μi _i is Student-t. To isolate uepiu_epi as a generation-control amplitude without introducing the ordinary z∣μ,σ2z μ,σ^2 variability, the pilot instead uses the variance-matched Gaussian perturbation ziepi=γi+uepi,i⊙ϵ,ϵ∼(0,I),z_i^epi= _i+ u_epi,i ε, ε (0,I), (22) followed by xigen=gθ(ziepi).x_i^gen=g_θ(z_i^epi). (23) Equation (22) is therefore not an exact sample from the NIG/Student-t posterior. It is a controlled Gaussian perturbation centered at γi _i whose coordinate-wise variance matches Var(μi∣yi)=uepi,iVar( _i y_i)=u_epi,i. This deliberate construction tests whether the learned uncertainty is useful as a generation variable while keeping the reported pilot simple and interpretable. For exact posterior generation one can instead draw σ2∼InvGamma(α,β)σ^2 (α,β), then μ∼(γ,σ2/ν)μ (γ,σ^2/ν) and z∼(μ,σ2)z (μ,σ^2). For control-oriented generation, the variance-matched rule can be generalized to z=γ+τepiuepi⊙ϵepi+τvaruvar⊙ϵvar,z=γ+ _epi u_epi _epi+ _var u_var _var, (24) with two interpretable amplitudes. This second expression is likewise a variance-matched control parameterization rather than an exact draw from the hierarchical posterior. Low τepi _epi favors conservative posterior-anchored variation, whereas increasing τepi _epi deliberately probes sensitivity to latent-location uncertainty. 3.2 Separating anchor reliability from perturbation-attributable effects Equation (22) makes uepiu_epi serve two distinct roles: it is (i) an uncertainty score computed from the NIG posterior and used to rank anchors, and (i) the scale of the random displacement applied to γ. A raw comparison of classifier error between low- and high-uepiu_epi populations therefore cannot say which role produced the difference. Anchors with large uepiu_epi may simply be atypical digits that the encoder–decoder pair represents poorly, in which case the decoded image can lose its class even with no uncertainty-scaled displacement. The distinction can be written explicitly as uepi=βν(α−1)⏟estimated for every anchorand retained for rankingversusτepiuepiϵ⏟use of that uncertaintyin perturbation u_epi= βν(α-1)_ subarrayc estimated for every anchor\\[-1.0pt] and retained for ranking subarray _epi u_epi\,ε βν(α-1)_ subarrayc use of that uncertainty\\[-1.0pt] in perturbation subarray Setting z=γz=γ switches off only the second quantity by setting τepi=0 _epi=0. It does not set uepi=0u_epi=0, and it does not prevent the decoder from generating gθ(γ)g_θ(γ). We therefore use three conditions, all sharing one trained ELVAE, one frozen classifier, and one realization of ϵε: (A) Uncertainty-scaled generation. z=γ+uepi⊙ϵz=γ+ u_epi ε, i.e. Eq. (22). This is the primary uncertainty-scaled generation condition. (C) Zero-displacement control. z=γz=γ, equivalently τepi=0 _epi=0 in Eq. (24) (with τvar=0 _var=0 in this pilot). The encoder still produces uepiu_epi for each anchor and the samples are still ranked by uepiu_epi; only uncertainty-scaled displacement is disabled. Any low/high stratification surviving here is therefore associated with anchor representation/re-generation reliability rather than with uncertainty-scaled perturbation. (I) Generation-attributable failure. Restrict attention to anchors whose condition-(C) generation is already classified correctly, and measure the failure rate under condition (A). Every failure counted here was induced by uncertainty-scaled perturbation. Condition (C) is the required zero-displacement baseline. Statistic (I) isolates the part of the result attributable to the uncertainty-scaled perturbation. 3.3 Low-uncertainty augmentation and high-uncertainty stress testing For a class-conditional or posterior-anchored application, generated samples can be divided by uepiu_epi: Generation region Interpretation Intended use Low uepiu_epi latent location is comparatively well determined higher-confidence synthetic augmentation after task-specific validity checks Intermediate uepiu_epi moderate uncertainty/diversity exploratory generation and data enrichment High uepiu_epi latent location is weakly determined stress testing, failure analysis, and hard-example generation High-uepiu_epi images are not automatically “bad” images. They are images generated from an anchor whose latent location ELVAE estimates less precisely. Some remain semantically correct; others cross a task boundary. The value of uepiu_epi is therefore statistical stratification, not deterministic rejection. Because much of the low/high contrast is inherited from the anchor, the table is best read as a policy over anchors to be re-generated, with the additional sampling-induced risk quantified separately by statistic (I). 4 Generation Pilot on MNIST 4.1 Study design The 70,000 MNIST digit images [10] were pooled and partitioned into 60,000 training images and 10,000 held-out images. ELVAE is trained without using digit labels. The labels are used only to define the intended semantic identity of each posterior anchor and to evaluate the generated images. A separate MLP classifier 784→256→128→10784→ 256→ 128→ 10 is trained only on the real 60,000-image training set for three epochs and then frozen. It achieves 96.85% accuracy on the held-out real images. No generated image is used to train or tune this classifier. For each held-out anchor, one image is generated using Eq. (22). The scalar uepiu_epi is the mean of the eight coordinate-wise uncertainties from Eq. (8). A generation error occurs when the frozen classifier prediction differs from the anchor label. Because the baseline scale of uepiu_epi differs across digit classes, uncertainty is ranked within the intended digit class before pooling. This prevents an intrinsically high-uncertainty but easy class from dominating the high-uepiu_epi group. If uepiu_epi is instead ranked globally across all 10,000 held-out images, the bottom-versus-top 20% contrast is 26.70% versus 27.10%, a ratio of 1.015×—essentially no effect. All of the reported stratification therefore lives within digit classes and none of it survives pooling across classes. The reason is that the class-level mean of uepiu_epi and class-level digit difficulty are not aligned, so global ranking mixes an easy high-uepiu_epi class into the high-uncertainty group and cancels the within-class trend. This is a legitimate use of stratification rather than a selection effect, but it means the unconditional model produces a uepiu_epi scale that is only comparable within a class. A conditional ELVAE would be required for applications in which anchors of different classes must be ranked against each other. The primary quantitative comparison is classifier error in the bottom and top 20% of within-class uepiu_epi. A class-stratified bootstrap with 800 replicates gives confidence intervals. We also report error across ten within-class uncertainty deciles. Every quantity is additionally reported under conditions (A), (C), and (I) of Sec. 3.2, and the whole pilot is repeated for three random seeds so that run-to-run spread is visible. Figure 2: Visualization of posterior-anchored uncertainty-aware generation. Top row: condition (C), the measured zero-displacement control z=γz=γ with τepi=0 _epi=0; the displayed uepiu_epi values remain the NIG-derived uncertainty scores used for ranking. The top-row anchors are at the median within-class uepiu_epi of each class. Middle row: approximately the 10th within-class uepiu_epi percentile. Bottom row: approximately the 90th percentile. Each image reports intended class and frozen-classifier result as c→c^c→ c, together with its uepiu_epi value; red titles mark generations whose class was lost. 4.2 Qualitative visualization Figure 2 shows three rows of empirical ELVAE generations. The top row is the τepi=0 _epi=0 zero-displacement control: the decoder is evaluated at z=γz=γ for an anchor at the median within-class uepiu_epi of each digit class. The encoder still computes a nonzero uepiu_epi for each anchor; that score is shown above each image but is not used to perturb z. At z=γz=γ the frozen classifier errs on 28.28% of the 10,000 anchors, reflecting information loss in the deliberately small MLP encoder–decoder. The second and third rows are empirical ELVAE generations selected at approximately the 10th and 90th within-class uepiu_epi percentiles. Their titles give intended class → frozen-classifier prediction and the absolute uepiu_epi value. These rows are selected by uncertainty percentile only, not by classifier outcome. 4.3 Quantitative result with a fixed classifier Table 1 collects the pilot results. Across all 10,000 generated images, the frozen classifier error was 29.06%. Uncertainty stratified this error. In the bottom 20% of within-class uepiu_epi, classifier error was 26.30%; in the top 20%, it was 37.80%. Thus the high-uepiu_epi group had an 11.50 percentage-point higher error, or a 1.437× error rate. The class-stratified bootstrap gave a 95% CI of 8.99–14.53 percentage points for the difference and 1.33–1.58 for the ratio. A two-proportion chi-square test gave p=6.56×10−15p=6.56× 10^-15. Table 1: Generation pilot result. Quantity Result Frozen classifier accuracy on held-out real MNIST 96.85% Classifier error on all ELVAE generations 29.06% Bottom 20% within-class uepiu_epi: classifier error 26.30% Top 20% within-class uepiu_epi: classifier error 37.80% High/low error ratio 1.437× (95% CI 1.33–1.58) High-minus-low absolute increase 11.50 points (95% CI 8.99–14.53) Top uepiu_epi decile error 43.00% AUROC of within-class uepiu_epi percentile for failure 0.556 Classifier error at z=γz=γ (control C) 28.28% Global (unnormalized) uepiu_epi high/low ratio 1.015× Held-out reconstruction MSE LreconL_recon 0.0408 λ^NIG λ_NIG from Eq. (18) 8.33×10−48.33× 10^-4 Table 2 places the result next to the controls and changes how it should be read. Under condition (C), with uncertainty-scaled displacement removed entirely, the bottom-versus-top contrast is 26.30% versus 36.70%, a ratio of 1.395×. Almost the whole condition-(A) stratification is therefore already present before uncertainty-scaled perturbation is applied. This does not mean “before generation” and does not mean uepi=0u_epi=0: the decoder still generates gθ(γ)g_θ(γ), and the encoder-derived uepiu_epi is still used to define the low/high groups. Uncertainty-scaled perturbation raises overall error only from 28.28% to 29.06%. The appropriate reading is that uepiu_epi ranks anchors by how faithfully the model can re-encode and re-decode them, which is a useful property of the evidential posterior but is not, by itself, a perturbation-induced effect. The generation-attributable statistic (I) isolates what uncertainty-scaled perturbation does. Among anchors that condition (C) already classifies correctly, the perturbation causes failure in 1.97% of the low-uepiu_epi group and 5.92% of the high-uepiu_epi group, a ratio of 3.01× (95% CI 2.03–4.75). The relative effect is larger than the headline ratio, but the base rate is small, so the number of images actually destabilized by uncertainty-scaled perturbation is modest. Figure 4 gives the corresponding decile trend. A stress-testing protocol built on Eq. (22) at τepi=1 _epi=1 will therefore produce hard examples at a low yield. Increasing τepi _epi is the natural next experiment. Table 2: The same trained model and frozen classifier under conditions (A), (C), and (I). Condition (C) keeps uepiu_epi as the ranking variable but disables its sampling role by setting τepi=0 _epi=0. Condition Bottom 20% Top 20% Ratio (A) Uncertainty-scaled generation z=γ+uepi⊙ϵz=γ+ u_epi ε 26.30% 37.80% 1.437× (C) z=γz=γ, τepi=0 _epi=0; uepiu_epi retained for ranking 26.30% 36.70% 1.395× (I) Generation-attributable failures 1.97% 5.92% 3.01× Figure 3 gives the full decile trend with the condition-(C) curve overlaid. The curve is not strictly monotone in every low-uncertainty bin, but the upper uncertainty range shows a clear increase and the highest decile has the largest error. The lowest decile is also elevated relative to deciles 2–4, and this shape reproduces across seeds; a plausible reading is that the very lowest-uepiu_epi coordinates are those closest to the prior mean, where the decoder has the least anchor-specific information to work with. Because decile 1 sits inside the bottom-20% group, it dilutes the reference level and makes the headline ratio conservative. On a sample-by-sample basis the association is modest: the AUROC of within-class uepiu_epi percentile for classifier failure is 0.556, and Spearman correlation with the binary failure indicator is 0.088. These values are safeguards against overclaiming. The result supports population-level uncertainty stratification, not a claim that uepiu_epi alone perfectly predicts which individual image will fail. Figure 3: Frozen-classifier error versus within-class ELVAE epistemic-uncertainty decile. The dashed control curve is z=γz=γ and tracks the uncertainty-scaled-generation curve closely. In this control τepi=0 _epi=0 while uepiu_epi is retained for ranking. The top uncertainty decile reaches 43.00% error. Table 3: Seed-to-seed variability of the pilot. Each row repeats the entire pipeline from a different random key. Random key Real acc. Error (A) Ratio (A) Ratio (C) Ratio (I) 20260809 96.85% 29.06% 1.437× 1.395× 3.01× 1 96.90% 30.40% 1.126× 1.097× 1.74× 2 96.96% 31.79% 1.209× 1.219× 1.59× 5 Discussion and Conclusion The pilot supports a focused and testable claim. ELVAE produces a continuous uncertainty variable that can be attached to the generation process, and populations selected by that variable differ substantially in semantic reliability. The z=γz=γ control shows that most of the low/high difference survives when uncertainty-scaled perturbation is switched off, even though uepiu_epi itself is still measured and used for ranking. Thus this dominant component is best described as an anchor representation/re-generation reliability effect, whereas only the smaller component isolated by statistic (I) is caused by uncertainty-scaled perturbation itself. Both components are useful: the anchor-level component can help decide which examples are safer to re-generate for augmentation, whereas the generation-attributable component informs how hard to push the sampler for stress testing. The low/high comparison should not be interpreted as a universal threshold. First, uepiu_epi is class dependent in the present unconditional model, which is why the pilot uses within-class rankings; the effect does not survive global ranking. A conditional ELVAE is therefore a prerequisite for cross-class use rather than a minor refinement. Second, the association is probabilistic and modest at the single-image level. Third, classifier error is only a semantic proxy for image utility; medical imaging applications would require task-specific validity checks, physics constraints, or expert review. Fourth, the current MLP generator is intentionally small and generates blurry digits, and its lossiness is exactly what makes the condition-(C) error rate as high as 28.28%; a stronger decoder would lower that baseline and would let condition (I) be measured on a much larger population of correctly re-generated anchors. A stronger decoder or a hybrid ELVAE-conditioning mechanism for modern diffusion or flow models could preserve the same uncertainty principle while greatly improving image fidelity. Fifth, as Table 3 shows, the headline ratio varies from 1.126 to 1.437 across three seeds, so single-seed reporting of a number like 1.437× overstates the precision of the pilot. The experiment also clarifies what high uncertainty should mean operationally. High-uepiu_epi images should not simply be discarded. If the objective is trusted augmentation, they can be down-weighted or rejected. If the objective is robustness analysis, those same samples are valuable because they are more likely to induce semantic instability. This turns one uncertainty quantity into two complementary data-generation policies. A natural next experiment is downstream retraining: train one classifier with an equal number of low-uepiu_epi synthetic images and another with unfiltered synthetic images, then compare real-test accuracy and robustness. A second extension is to vary τepi _epi in Eq. (24) and measure whether task error increases smoothly as uncertainty-scaled perturbation is strengthened; given the small base rate of condition (I) at τepi=1 _epi=1, this is the most informative single experiment remaining, since it directly tests whether uepiu_epi acts as a usable generation knob rather than only as an anchor-quality score. In medical imaging, the same framework could generate low-uncertainty anatomy for augmentation and high-uncertainty anatomy or acquisition conditions for controlled stress testing. ELVAE replaces the deterministic latent mean and variance outputs of a conventional VAE with an input-dependent NIG posterior over latent location and latent variance. This produces an explicit latent-location uncertainty uepi=β/[ν(α−1)]u_epi=β/[ν(α-1)] that is mathematically identified by direct higher-order regularization and can be used as a generation control variable. The likelihood analysis in Sec. 2.3 further shows that λNIG _NIG need not be regarded as an arbitrary balance parameter: under the homoscedastic Gaussian observation model it is determined by the residual variance, which can in turn be estimated by the converged reconstruction MSE. In conclusion, the MNIST pilot establishes two separable findings. First, uepiu_epi stratifies posterior anchors by re-generation reliability: the top 20% uncertainty group has 37.80% classification error against 26.30% in the bottom 20%, a 1.437× increase, of which a 1.395× ratio remains when τepi=0 _epi=0. In that control uepiu_epi is still present and used for ranking; only uncertainty-scaled perturbation is absent. Second, uncertainty-scaled perturbation itself induces semantic failure 3.01× more often in high-uepiu_epi anchors than in low-uepiu_epi anchors. The first finding is the larger anchor-reliability effect; the second is the specifically sampling-attributable generation effect. Together they support ELVAE as a framework that can stratify anchors and, to a lesser but measurable degree, control generation by model uncertainty. Figure 4: Generation-attributable failure rate by within-class uepiu_epi decile, condition (I). Only anchors already classified correctly at z=γz=γ are counted, so every failure shown was induced by uncertainty-scaled perturbation. The relative trend is steeper than in Fig. 3, but on a much smaller base rate. Reproducibility. The pilot uses the publicly available MNIST dataset [10], partitioned into 60,000 training and 10,000 held-out images. The ELVAE encoder is an MLP 784→128→64784→ 128→ 64 with K=8K=8 latent coordinates and a mirrored decoder, trained for four epochs with Adam, learning rate 10−310^-3, batch size 1024, and λNIG=5×10−4 _NIG=5× 10^-4. The frozen evaluation classifier is an MLP 784→256→128→10784→ 256→ 128→ 10 trained for three epochs on the real training split only. Confidence intervals use a class-stratified bootstrap with 800 replicates. Results in Tables 1 and 2 use random key 20260809; Table 3 repeats the entire pipeline for keys 1 and 2. Author–AI Collaboration. 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