H$^2$EDL: Hyper Evidential Deep Learning for Hierarchical Classification

# H2EDL: Hyper Evidential Deep Learning for Hierarchical Classification
Source: [https://arxiv.org/html/2608.18185](https://arxiv.org/html/2608.18185)
###### Abstract

Fine\-grained recognition often involves hierarchical label spaces, where a model may be confident about a coarse semantic concept while remaining uncertain among its descendant classes\. Such structured ambiguity requires uncertainty representations that capture both fine\-grained classes and intermediate concepts\. However, existing tools each capture only half of it: flat evidential classifiers quantify total ignorance with a single vacuity on the leaf frame, and hierarchical classifiers propagate point probabilities with no notion of evidence\. Hyper\-opinions would unify the two, but their general form is exponential in the label count, and existing hyper\-evidential networks either require composite labels to be supplied in the training data or read them off an unstructured weight pattern, with no principled notion of which composites deserve mass\. We observe that the taxonomy itself is the missing hyperdomain\. Its subtrees and leaf singletons form a linear\-size focal family, and one local Dirichlet opinion per branching node induces every composite mass in closed form\. The resulting model, H2EDL, can be interpreted in two complementary ways using the same set of parameters\. From a prediction perspective, it functions as a hierarchical classifier that preserves consistency across different levels of the label tree\. From a probabilistic perspective, it defines a valid tree\-structured hyper\-opinion, where the mass assigned to each node represents the belief that reaches that node but does not provide sufficient confidence to further specialize into its descendants\. On FGVC\-Aircraft and DERM12345, H2EDL reduces calibration error by approximately half compared with cross\-entropy baselines, with the improvement becoming more pronounced at deeper hierarchy levels and under larger training budgets\. Despite achieving similar leaf\-level accuracy, it preserves the correct coarse category 19% more often when making fine\-grained mistakes\. Code will be released\.

## Introduction

Uncertainty\-aware classification asks a model to report not only a label but how much its prediction can be trusted, and a substantial literature now supplies that number through calibration, Bayesian approximation, or evidential parameterizations\([6](https://arxiv.org/html/2608.18185#bib.bib15);[5](https://arxiv.org/html/2608.18185#bib.bib17);[10](https://arxiv.org/html/2608.18185#bib.bib18);[16](https://arxiv.org/html/2608.18185#bib.bib1)\)\. When the label space is a hierarchy, however, a single number cannot carry that message\. Consider a model that classifies aircraft assigning0\.40\.4to the Boeing 737\-800 \(Fig\.[1](https://arxiv.org/html/2608.18185#Sx1.F1)\)\. It may be in either of two very different states: it may have settled that the aircraft is a 737 and be undecided only among the \-700, \-800 and \-900 variants, or it may be spreading the remaining mass across unrelated manufacturers\. The confidence is identical; the two predictions are not\. What separates them is not how much uncertainty the model holds but where that uncertainty sits\.

![Refer to caption](https://arxiv.org/html/2608.18185v1/teaser_V3.png)Figure 1:Uncertainty on a label hierarchy has a location\. H2EDL carries a local opinion at every branching decision, so it reports a separate vacuity for each level: the evidence is decisive about the manufacturer \(u1=0\.2u^\{1\}=0\.2\), already thinner about the family \(u2=0\.5u^\{2\}=0\.5\), and runs out among the737737variants \(u3=0\.7u^\{3\}=0\.7\), where a flat evidential model has one vacuity for the entire leaf level\.Flat evidential deep learning \(EDL\)\([16](https://arxiv.org/html/2608.18185#bib.bib1)\)can estimate the uncertainty, but it attaches to the leaf level only, where one global scalar cannot depict hierarchy\. Hierarchical classifiers\([17](https://arxiv.org/html/2608.18185#bib.bib12);[1](https://arxiv.org/html/2608.18185#bib.bib13)\)incorporate the label tree into prediction, but typically still rely on flat probability normalization within local decisions\. Subjective Logic supplies the object that would unify them, the*hyper\-opinion*\([8](https://arxiv.org/html/2608.18185#bib.bib2)\), which assigns mass to composite subsets\. Its general form admits up to2L−12^\{L\}\-1focal sets\. Existing hyper\-evidential networks rely on composite labels that must be manually specified in the training set, despite the absence of an inherent structure to define such labels\([11](https://arxiv.org/html/2608.18185#bib.bib7)\), or by letting the composites fall out of the network’s own weight structure\([15](https://arxiv.org/html/2608.18185#bib.bib19)\)\. Both operate on a flat frame, which offers no principled way to say which composites deserve mass\.

A taxonomy naturally provides the required hyperdomain\. The sets of descendant leaves associated with internal nodes, together with leaf singletons, define a focal family with size at mostL\+\|ℬ\|L\+\|\\mathcal\{B\}\|, which grows linearly with the tree structure rather than exponentially with the number of labels\. Moreover, each element of this family corresponds to a meaningful concept already represented in the taxonomy\. Additionally, a local Dirichlet opinion at each branching node determines the complete hyper\-opinion in closed form\. Standard leaf supervision naturally trains these local opinions, since predicting a leaf label is equivalent to making a sequence of decisions along the tree\. Thus, the taxonomy simultaneously provides the hyperdomain, the parameterization, and the supervision signal\. No additional label construction, auxiliary prediction network, or post\-hoc calibration procedure is required\.

Concretely, H2EDL places an evidential head at each branching node and computes leaf probabilities by multiplying conditional means along root\-to\-leaf paths\. The same concentration parameters admit two complementary interpretations\. Predictively, they define a normalized and hierarchically consistent leaf classifier, from which predictions at any coarse level can be obtained directly\. From an uncertainty perspective, they induce a tree\-structured hyper\-opinion, where the mass at nodevvrepresents belief that reachesvvbut lacks sufficient evidence to descend further\.

We prove that this mass assignment is valid, that the predictive distribution lies within the belief–plausibility bounds of the induced hyper\-opinion for every subset of leaves, and that non\-specific mass can be decomposed exactly across hierarchy depths\. The model can therefore identify where evidence becomes insufficient along each prediction path without additional inference procedures\.

Unlike flat models that compare allLLleaf classes at once, H2EDL makes a sequence of local decisions\. Each decision is trained with all samples in its subtree, so uncertainty should increase gradually at deeper levels\. Ancestor decisions also restrict the possible error region, encouraging errors to remain within nearby subtrees\. We evaluate both properties experimentally\. We first examine whether existing benchmarks can distinguish hierarchical classifiers and find that their leaf accuracy is often dominated by a single coarse decision, which can obscure gains from fine\-grained hierarchical modeling\.

Contributions\.\(i\) A tractable hyper\-evidential formulation, in which a taxonomy supplies a linear\-size focal family on which composite belief is induced rather than predicted\. This removes the composite\-label requirement that has confined prior hyper\-evidential networks to prescribed label groups\. \(ii\) An operational classifier that is hierarchically coherent by construction and exposes a vacuity at every individual decision\. \(iii\) Theory linking the two readings: validity of the induced hyper\-opinion \(Prop\. 1\), containment of the predictor in its credal set \(Prop\. 2\), and an exact depth decomposition of non\-specific mass \(Prop\. 3\)\. \(iv\) An evaluation matched to the problem\. Having established that leaf accuracy measures one coarse decision common to all methods, we score every level, and find H2EDL ahead of all baselines at both intermediate DERM12345 levels\.

## Related Work

#### Hierarchical classification\.

Label taxonomies\([17](https://arxiv.org/html/2608.18185#bib.bib12)\)are exploited by encoding the tree in the loss or in embedding geometry\([1](https://arxiv.org/html/2608.18185#bib.bib13)\), by minimizing a hierarchical risk post hoc over an unchanged probability vector\([9](https://arxiv.org/html/2608.18185#bib.bib11)\), or by hedging a prediction upward to an ancestor\([4](https://arxiv.org/html/2608.18185#bib.bib14)\)\. All of these act on point probabilities\. The tree constrains or rescores a distribution that contains no notion of evidence, so ambiguity confined to one subtree and ambiguity spread across the frame remain the same object\. Our aim is to give the prediction a structured uncertainty that the tree makes both meaningful and cheap\.

#### Evidential deep learning\.

EDL\([16](https://arxiv.org/html/2608.18185#bib.bib1)\)predicts Dirichlet concentrations and obtains closed\-form epistemic uncertainty in a single forward pass\. Prior networks\([14](https://arxiv.org/html/2608.18185#bib.bib10)\), the family surveyed by[18](https://arxiv.org/html/2608.18185#bib.bib16), and refinements that relax the prior weight and variance term\([2](https://arxiv.org/html/2608.18185#bib.bib9)\)or reweight by Fisher information\([3](https://arxiv.org/html/2608.18185#bib.bib8)\)all share that design, and all keep a flat frame with a single vacuity for the entire label set\. Closest to us, HENN\([11](https://arxiv.org/html/2608.18185#bib.bib7)\)reaches hyper\-opinions through grouped Dirichlets over a prescribed collection of composite labels, which must be given in the training data; HEDL\([15](https://arxiv.org/html/2608.18185#bib.bib19)\)instead lets the composites emerge from the sign pattern of the final linear layer, and then projects the hyper\-opinion back to a flat multinomial opinion for training and prediction\. Both methods operate on an unstructured class frame, where no mechanism determines which composite hypotheses should receive mass\. Consequently, composite masses merely reflect uncertainty over arbitrary class subsets without revealing what those subsets represent\.

## Method

H2EDL assigns every branching node of the taxonomy a local Dirichlet opinion over its children and multiplies these opinions along root\-to\-leaf paths\. One parameter set then supports two readings: an operational classifier whose levels cannot disagree, and a tree\-structured hyper\-opinion in which composite mass is induced rather than predicted\. After fixing notation we develop the two readings in turn\.

### Notation and Evidential Background

We classify over a label hierarchy given as a rooted tree𝒯\\mathcal\{T\}with rootrr\. Leavesℒ\\mathcal\{L\}\(\|ℒ\|=L\|\\mathcal\{L\}\|=L\) are the fine\-grained targets; internal nodes are coarser concepts\. Nodevvhas children𝒞⁡\(v\)\\mathcal\{C\}\(v\),Kv=\|𝒞⁡\(v\)\|K\_\{v\}=\|\\mathcal\{C\}\(v\)\|; it is*branching*ifKv≥2K\_\{v\}\\geq 2and*pass\-through*ifKv=1K\_\{v\}=1, andℬ\\mathcal\{B\}denotes the branching nodes\. Each leafℓ\\ellhas a unique root\-to\-leaf pathπ⁡\(ℓ\)\\pi\(\\ell\)inducing a set of*branching decisions*𝒟⁡\(ℓ\)=\{\(v,cv\):v∈ℬ∩π⁡\(ℓ\)\}\\mathcal\{D\}\(\\ell\)=\\\{\(v,c\_\{v\}\):v\\in\\mathcal\{B\}\\cap\\pi\(\\ell\)\\\}, wherecvc\_\{v\}indexes the child ofvvon the path; pass\-through nodes carry no decision\.

Evidential Deep Learning\([16](https://arxiv.org/html/2608.18185#bib.bib1)\)treats aKK\-way prediction as a Dirichlet over the simplex: the network emits non\-negative evidencee∈ℝ≥0Ke\\in\\mathbb\{R\}^\{K\}\_\{\\geq 0\}and forms𝜶=e\+W​𝒂\\boldsymbol\{\\alpha\}=e\+W\\boldsymbol\{a\}with prior weightW\>0W\>0, base rate𝒂∈ΔK−1\\boldsymbol\{a\}\\in\\Delta^\{K\-1\}and strengthS=∑kαkS=\\sum\_\{k\}\\alpha\_\{k\}\. In Subjective Logic\([8](https://arxiv.org/html/2608.18185#bib.bib2)\)this is an*opinion*\(b,u,a\)\(b,u,a\)with

bk=ekS,u=WS∈\(0,1\],p^k=αkS=bk\+ak​u,b\_\{k\}=\\frac\{e\_\{k\}\}\{S\},\\qquad u=\\frac\{W\}\{S\}\\in\(0,1\],\\qquad\\hat\{p\}\_\{k\}=\\frac\{\\alpha\_\{k\}\}\{S\}=b\_\{k\}\+a\_\{k\}u,\(1\)and∑kbk\+u=1\\sum\_\{k\}b\_\{k\}\+u=1\. Vacuityuuis large when total evidence is small: the model “does not know\.” Sensoy EDL is the caseW=KW\\\!=\\\!K,ak=1/Ka\_\{k\}\\\!=\\\!1/K\. We exploit vacuity per node to reason about granularity\.

### Operational Hierarchical Classifier

Rather than one Dirichlet overLLleaves, H2EDL attaches a lightweight evidential head to every branching node\. From the shared backbone featuref=ϕ⁡\(x\)f=\\phi\(x\)each head produces a local opinion over theKvK\_\{v\}children,

ev=softplus⁡\(Wv⊤​f\+bv\),𝜶v=ev\+W​av,e\_\{v\}=\\operatorname\{softplus\}\\\!\\bigl\(W\_\{v\}^\{\\top\}f\+b\_\{v\}\\bigr\),\\qquad\\boldsymbol\{\\alpha\}\_\{v\}=e\_\{v\}\+W\\,a\_\{v\},\(2\)
withbv,k,uv,p^v,kb\_\{v,k\},u\_\{v\},\\hat\{p\}\_\{v,k\}following Eq\. \([1](https://arxiv.org/html/2608.18185#Sx3.E1)\)\. Crucially𝒑^v\\hat\{\\boldsymbol\{p\}\}\_\{v\}is read as the conditionalP⁡\(child∣v,x\)P\(\\text\{child\}\\mid v,x\)\. Settingav=1Kv​𝟏a\_\{v\}=\\frac\{1\}\{K\_\{v\}\}\\mathbf\{1\},W=KvW=K\_\{v\}recovers the standard EDL prior\. For imbalanced children we also allow a*tempered inverse\-frequency*base rate,av,k∝\(nv,k\+s\)−τa\_\{v,k\}\\propto\(n\_\{v,k\}\+s\)^\{\-\\tau\}over the training countsnv,kn\_\{v,k\}with smoothingss, temperatureτ∈\[0,1\]\\tau\\in\[0,1\]\(τ=0\\tau\\\!=\\\!0uniform\) andW=wscale​KvW=w\_\{\\mathrm\{scale\}\}K\_\{v\}: rarer children start from a higher baseline belief, so less evidence is needed to predict them\. It is a fixed buffer adding no parameters; we refer to it as the*base\-rate*variant\. Pass\-through nodes receive neither head nor loss\. Local opinions compose into a leaf distribution by multiplying projected means along each path,

P⁡\(ℓ∣x\)=∏\(v,c\)∈𝒟⁡\(ℓ\)p^v,c=∏\(v,c\)∈𝒟⁡\(ℓ\)αv,cSv,P\(\\ell\\mid x\)=\\\!\\\!\\prod\_\{\(v,c\)\\in\\mathcal\{D\}\(\\ell\)\}\\\!\\\!\\hat\{p\}\_\{v,c\}\\;=\\\!\\\!\\prod\_\{\(v,c\)\\in\\mathcal\{D\}\(\\ell\)\}\\\!\\\!\\frac\{\\alpha\_\{v,c\}\}\{S\_\{v\}\},\(3\)and the probability of any internal concept is recovered exactly by the telescoping partial product,

P⁡\(v∣x\)=∑ℓ∈leaves⁡\(v\)P⁡\(ℓ∣x\)=∏\(u,c\)∈𝒟⁡\(v\)p^u,c\.P\(v\\mid x\)=\\\!\\\!\\sum\_\{\\ell\\in\\mathrm\{leaves\}\(v\)\}\\\!\\\!P\(\\ell\\mid x\)=\\\!\\\!\\prod\_\{\(u,c\)\\in\\mathcal\{D\}\(v\)\}\\\!\\\!\\hat\{p\}\_\{u,c\}\.\(4\)
Proposition 0\(normalization & coherence\)\.For any local means𝒑^v∈ΔKv−1\\hat\{\\boldsymbol\{p\}\}\_\{v\}\\in\\Delta^\{K\_\{v\}\-1\}: \(i\)∑ℓP⁡\(ℓ∣x\)=1\\sum\_\{\\ell\}P\(\\ell\\mid x\)=1; \(ii\) coarse probabilities equal Eq\. \([4](https://arxiv.org/html/2608.18185#Sx3.E4)\); \(iii\) ifvvis an ancestor ofv′v^\{\\prime\}thenP⁡\(v′∣x\)≤P⁡\(v∣x\)P\(v^\{\\prime\}\\mid x\)\\leq P\(v\\mid x\)\.

Coarse\-level evaluation therefore needs no separate head, and the model is hierarchically coherent by construction\. We show next that\{𝜶v\}\\\{\\boldsymbol\{\\alpha\}\_\{v\}\\\}carries strictly more structure than the point distribution of Eq\. \([3](https://arxiv.org/html/2608.18185#Sx3.E3)\)\.

### The Induced Tree\-Structured Hyper\-Opinion

A hyper\-opinion\([8](https://arxiv.org/html/2608.18185#bib.bib2)\)assigns mass to singleton and composite subsets of a frame; a leaf frame admits2L−12^\{L\}\-1of them, impractical to parameterize or supervise\. H2EDL instead lets the taxonomy select the focal family

ℱ𝒯=\{\{ℓ\}:ℓ∈ℒ\}∪\{leaves⁡\(v\):v∈ℬ\},\\mathcal\{F\}\_\{\\mathcal\{T\}\}=\\bigl\\\{\\\{\\ell\\\}:\\ell\\in\\mathcal\{L\}\\bigr\\\}\\cup\\bigl\\\{\\mathrm\{leaves\}\(v\):v\\in\\mathcal\{B\}\\bigr\\\},\(5\)so\|ℱ𝒯\|≤L\+\|ℬ\|\|\\mathcal\{F\}\_\{\\mathcal\{T\}\}\|\\leq L\+\|\\mathcal\{B\}\|is linear in the tree and every focal set is a named concept\. Prior evidential work requires composite sets to be given as training labels on a flat frame\([11](https://arxiv.org/html/2608.18185#bib.bib7)\); here all masses are induced from ordinary leaf supervision, with zero mass outside Eq\. \([5](https://arxiv.org/html/2608.18185#Sx3.E5)\)\. Writing each local opinion in belief form and defining the*reach mass*mv=∏\(w,c\)∈𝒟⁡\(v\)bw,cm\_\{v\}=\\prod\_\{\(w,c\)\\in\\mathcal\{D\}\(v\)\}b\_\{w,c\}\(somr=1m\_\{r\}=1\), assign

m⁡\(\{ℓ\}\)=∏\(v,c\)∈𝒟⁡\(ℓ\)bv,c,m⁡\(leaves⁡\(v\)\)=mv​uv\.m\(\\\{\\ell\\\}\)=\\\!\\\!\\prod\_\{\(v,c\)\\in\\mathcal\{D\}\(\\ell\)\}\\\!\\\!b\_\{v,c\},\\qquad m\\bigl\(\\mathrm\{leaves\}\(v\)\\bigr\)=m\_\{v\}\\,u\_\{v\}\.\(6\)Each subtree massmv​uvm\_\{v\}u\_\{v\}is the share of belief that reachesvvbut cannot be committed to any child; at the rootm⁡\(ℒ\)=urm\(\\mathcal\{L\}\)=u\_\{r\}\. Pass\-through nodes create no focal set, having no uncertain decision\. Note the operational predictor multiplies means, whereas singleton mass here is a product of beliefs; the latter are smaller and need not sum to one over leaves\.

Proposition 1\(validity\)\.All masses in Eq\. \([6](https://arxiv.org/html/2608.18185#Sx3.E6)\) are nonnegative and∑ℓm⁡\(\{ℓ\}\)\+∑v∈ℬmv​uv=1\\sum\_\{\\ell\}m\(\\\{\\ell\\\}\)\+\\sum\_\{v\\in\\mathcal\{B\}\}m\_\{v\}u\_\{v\}=1; i\.e\.\{𝜶v\}\\\{\\boldsymbol\{\\alpha\}\_\{v\}\\\}induces a valid, tree\-supported belief mass assignment, which is a sparse Subjective\-Logic hyper\-opinion onℱ𝒯\\mathcal\{F\}\_\{\\mathcal\{T\}\}\.

Proposition 2\(credal containment\)\.Withbel⁡\(A\)=∑F⊆Am⁡\(F\)\\mathrm\{bel\}\(A\)=\\\!\\sum\_\{F\\subseteq A\}m\(F\)andpl⁡\(A\)=∑F∩A≠∅m⁡\(F\)\\mathrm\{pl\}\(A\)=\\\!\\sum\_\{F\\cap A\\neq\\varnothing\}m\(F\), both summed overF∈ℱ𝒯F\\in\\mathcal\{F\}\_\{\\mathcal\{T\}\}, the H2EDL distribution satisfiesbel⁡\(A\)≤P⁡\(A∣x\)≤pl⁡\(A\)\\mathrm\{bel\}\(A\)\\leq P\(A\\mid x\)\\leq\\mathrm\{pl\}\(A\)for*every*A⊆ℒA\\subseteq\\mathcal\{L\}: it lies in the credal set of the induced hyper\-opinion\. TakingA=\{ℓ\}A=\\\{\\ell\\\}givesbel⁡\(ℓ\)=m⁡\(\{ℓ\}\)\\mathrm\{bel\}\(\\ell\)=m\(\\\{\\ell\\\}\)andpl\(ℓ\)=m\(\{ℓ\}\)\+∑v:ℓ∈leaves⁡\(v\)mvuv\\mathrm\{pl\}\(\\ell\)=m\(\\\{\\ell\\\}\)\+\\sum\_\{v:\\,\\ell\\in\\mathrm\{leaves\}\(v\)\}m\_\{v\}u\_\{v\}\.

Proposition 3\(depth decomposition\)\.The total non\-specific massU=1−∑ℓbel⁡\(ℓ\)U=1\-\\sum\_\{\\ell\}\\mathrm\{bel\}\(\\ell\)decomposes exactly by depth:U=∑dUdU=\\sum\_\{d\}U\_\{d\}withUd=∑v∈ℬ:d⁡\(v\)=dmvuvU\_\{d\}=\\sum\_\{v\\in\\mathcal\{B\}:\\,d\(v\)=d\}m\_\{v\}u\_\{v\}\.

Proposition 2 turns Eq\. \([3](https://arxiv.org/html/2608.18185#Sx3.E3)\) into a projection of the hyper\-opinion: the per\-leaf imprecision bandpl⁡\(ℓ\)−bel⁡\(ℓ\)=∑v∋ℓmv​uv\\mathrm\{pl\}\(\\ell\)\-\\mathrm\{bel\}\(\\ell\)=\\sum\_\{v\\ni\\ell\}m\_\{v\}u\_\{v\}is driven by ancestral vacuities weighted by reach mass, where a flat evidential model has one global widthuu\. Proposition 3 gives a depth\-resolved profile:UdU\_\{d\}is the mass that refuses to descend past depthdd, computed per sample at no extra cost\. It is the formal counterpart of “how far down the tree did the evidence carry?”, and we treat it as a structural decomposition of where belief stops rather than a calibrated epistemic\-uncertainty estimate\.

### Hierarchical Evidential Objective

We supervise only the branching nodes on each sample’s ground\-truth path, mirroring the conditional factorization of Eq\. \([3](https://arxiv.org/html/2608.18185#Sx3.E3)\)\. For nodevvwith one\-hot child targetyvy\_\{v\}we use the Bayes\-risk EDL loss\([16](https://arxiv.org/html/2608.18185#bib.bib1)\),ℒvedl=∑k\(yv,k−p^v,k\)2\+∑kp^v,k​\(1−p^v,k\)/\(Sv\+1\)\\mathcal\{L\}^\{\\mathrm\{edl\}\}\_\{v\}=\\sum\_\{k\}\(y\_\{v,k\}\-\\hat\{p\}\_\{v,k\}\)^\{2\}\+\\sum\_\{k\}\\hat\{p\}\_\{v,k\}\(1\-\\hat\{p\}\_\{v,k\}\)/\(S\_\{v\}\+1\)\. To suppress evidence on wrong children we add an annealed KL term that pulls the off\-target Dirichlet toward the prior: masking the target class to its prior value,𝜶~v=yv⊙\(W​av\)\+\(1−yv\)⊙𝜶v\\tilde\{\\boldsymbol\{\\alpha\}\}\_\{v\}=y\_\{v\}\\odot\(Wa\_\{v\}\)\+\(1\-y\_\{v\}\)\\odot\\boldsymbol\{\\alpha\}\_\{v\}, we penalizeℒvkl=KL\(Dir\(𝜶~v\)∥Dir\(Wav\)\)\\mathcal\{L\}^\{\\mathrm\{kl\}\}\_\{v\}=\\mathrm\{KL\}\(\\mathrm\{Dir\}\(\\tilde\{\\boldsymbol\{\\alpha\}\}\_\{v\}\)\\\|\\mathrm\{Dir\}\(Wa\_\{v\}\)\), which reduces to the standard Sensoy regularizer for uniformava\_\{v\}\. The per\-sample hierarchical loss sums over the ground\-truth path with depth weightswdw\_\{d\}\(we usewd=1w\_\{d\}\\\!=\\\!1\):

ℒpath=∑\(v,c\)∈𝒟⁡\(ℓ⋆\)wd⁡\(v\)​\(ℒvedl\+λt​ℒvkl\),\\mathcal\{L\}^\{\\mathrm\{path\}\}=\\\!\\\!\\sum\_\{\(v,c\)\\in\\mathcal\{D\}\(\\ell^\{\\star\}\)\}\\\!\\\!w\_\{d\(v\)\}\\bigl\(\\mathcal\{L\}^\{\\mathrm\{edl\}\}\_\{v\}\+\\lambda\_\{t\}\\,\\mathcal\{L\}^\{\\mathrm\{kl\}\}\_\{v\}\\bigr\),\(7\)withλt\\lambda\_\{t\}annealed from00toλmax\\lambda\_\{\\max\}so evidence can accumulate before regularization\. We additionally optimize the quantity the model is evaluated on by adding a negative log\-likelihood on the differentiable path product,ℒleaf=−logP\(ℓ⋆∣x\)=∑\(v,c\)∈𝒟⁡\(ℓ⋆\)−logp^v,c\\mathcal\{L\}^\{\\mathrm\{leaf\}\}=\-\\log P\(\\ell^\{\\star\}\\mid x\)=\\sum\_\{\(v,c\)\\in\\mathcal\{D\}\(\\ell^\{\\star\}\)\}\-\\log\\hat\{p\}\_\{v,c\}\. The identity makes its role precise: the leaf NLL is separable into per\-node log\-losses, so it adds no interaction between heads beyond the shared backbone, only a cross\-entropy on each on\-pathp^v,c\\hat\{p\}\_\{v,c\}alongside the Bayes\-risk term\. The objective isℒ=ℒpath\+β​ℒleaf\\mathcal\{L\}=\\mathcal\{L\}^\{\\mathrm\{path\}\}\+\\beta\\mathcal\{L\}^\{\\mathrm\{leaf\}\}withβ=0\.1\\beta\\\!=\\\!0\.1\. An optional tree\-consistency regularizer, penalizing vacuity that increases from a parent to a confidently selected child, is referred to as the*consistency*variant\.

#### Partial hierarchical supervision\.

Because supervision enters Eq\. \([7](https://arxiv.org/html/2608.18185#Sx3.E7)\) as a set of decisions rather than a leaf index, an annotation naming only an ancestorv⋆v^\{\\star\}is not a degraded label but a shorter one: the sum runs over𝒟⁡\(v⋆\)\\mathcal\{D\}\(v^\{\\star\}\)\. No architectural or objective change is needed, and heads belowv⋆v^\{\\star\}stay free to express vacuity rather than being taught a fabricated target\. A flat model, by contrast, must either discard the sample or invent a distribution overv⋆v^\{\\star\}’s descendants\.

### What the Factorization Predicts

Two consequences follow, and we test both\. Each path factor is fit against a decision that every sample in its subtree supervises, whereas a flat softmax resolves the wholeLL\-way competition at once; miscalibration should therefore accumulate gradually along a path rather than appear all at once at the leaves, so the gap over flat models should widen with depth\. Each ancestor factor gates its entire subtree, so by Prop\. 0\(iii\) no leaf can exceed its ancestor’s mass and errors should stay within\-subtree\. One prediction is negative: evidential shrinkage movesp^v,k\\hat\{p\}\_\{v,k\}towardav,ka\_\{v,k\}without reordering samples, so ranking\-based uses of confidence such as selective prediction should not improve, and we claim no gain there\.

Table 1:Main comparison at leaf\-level\. bAcc and ECE in %, NLL in nats; best inbold, second bestunderlined\.*Hier\.*marks a predictor defined over the label tree,*Unc\.*one that emits an epistemic uncertainty; only H2EDL has both, HENN being evidential but flat \(its composite focal sets are a supplied partition, not the taxonomy\)\. An H2EDL variant ranks first or second in every column, taking both places in four of the six, and H2EDL is the only method top\-two in bAcc on both datasets; each baseline instead falls away on at least one axis\. The bAcc margins sit inside seed noise, whereas ECE and NLL separate decisively\.†I\-EDL uses its validation\-retuned Fisher weight \(c=0\.02c\\\!=\\\!0\.02atK=100K\\\!=\\\!100,c=0\.05c\\\!=\\\!0\.05atK=40K\\\!=\\\!40\); its published default drops to16\.9%16\.9\\%bAcc on FGVC\.

## Experiments

### Experimental Setup

We evaluate on two hierarchies of very different character\. DERM12345\([19](https://arxiv.org/html/2608.18185#bib.bib6)\)is a long\-tailed dermatoscopic benchmark with a44\-level diagnostic taxonomy,4040leaves and1515branching nodes\. We use its official patient\-disjoint split and hold out10%10\\%of patients for validation\. FGVC\-Aircraft\([13](https://arxiv.org/html/2608.18185#bib.bib5)\)is a33\-level fine\-grained benchmark with3030manufacturers,7070families and100100variants on its official split\. All models share an ImageNet\-pretrained ResNet\-50\([7](https://arxiv.org/html/2608.18185#bib.bib3)\)and an identical budget of100100AdamW epochs\([12](https://arxiv.org/html/2608.18185#bib.bib4)\), are selected on validation leaf balanced accuracy, and are scored once on test\. We report mean±\\pmstd over33seeds with pairedtt\-tests, and coarse\-level metrics always aggregate leaf probabilities through Eq\. \([4](https://arxiv.org/html/2608.18185#Sx3.E4)\)\. The baselines share the backbone and the recipe\. Flat\-CE and Flat\-EDL\([16](https://arxiv.org/html/2608.18185#bib.bib1)\)are the two flat references\. Hier\-CE is the softmax twin of H2EDL, with one local softmax per branching node and the same path product, which isolates the evidential machinery from the hierarchy itself\. HENN\([11](https://arxiv.org/html/2608.18185#bib.bib7)\)is a flat Group\-Dirichlet hyper\-evidential model over singletons plus a disjoint composite partition\. I\-EDL\([3](https://arxiv.org/html/2608.18185#bib.bib8)\)is a popular flat\-EDL variant\.

#### Overview\.

The evaluation proceeds in five steps, in the order of the subsections below\.\(i\) Classification accuracyscores the predictor at the leaf and at the intermediate levels\.\(ii\) Calibrationand\(iii\) mistake severitytest the two positive predictions of the previous section, and carry most of the empirical weight\.\(iv\) Partial supervisionexercises the setting the path parameterization was built for, in which labels stop at an interior node\.\(v\) An ablationisolates the contribution of each component and of the baselines’ published constants\.

### Hierarchical Classification Accuracy

To demonstrate that a set of local evidential opinions yields an effective hierarchical classifier, we compare against the five baselines at the leaf level \(Table[1](https://arxiv.org/html/2608.18185#Sx3.T1)\) and at the intermediate levels of the DERM12345 taxonomy \(Table[2](https://arxiv.org/html/2608.18185#Sx4.T2)\)\.

As shown in Table[1](https://arxiv.org/html/2608.18185#Sx3.T1), at the leaf level no method separates itself by a significant margin, but H2EDL is the only method that is top\-two in bAcc on both datasets, and an H2EDL variant places first or second in every one of the six columns of Table[1](https://arxiv.org/html/2608.18185#Sx3.T1), holding both places in four of them\. Each baseline instead has an axis on which it falls away: HENN leads DERM bAcc but is fourth on FGVC, I\-EDL ties for second on FGVC bAcc at45\.6%45\.6\\%ECE, and Flat\-EDL pairs the lowest DERM ECE with the lowest accuracy of any method\. On FGVC\-Aircraft H2EDL\+leaf\-path\+base\-rate reaches55\.3%55\.3\\%bAcc, nominally\+1\.7\+1\.7over Flat\-CE and\+1\.1\+1\.1over Hier\-CE, with a sixth of the seed variance of the plain model \(±0\.3\\pm 0\.3against±1\.7\\pm 1\.7\), so the base\-rate prior is stabilizing training and not only shifting the mean\. HENN is nonetheless indistinguishable from either plain H2EDL \(p=0\.47p\\\!=\\\!0\.47\) or Flat\-CE \(p=0\.27p\\\!=\\\!0\.27\)\. On DERM12345 HENN attains the highest bAcc \(35\.8±1\.435\.8\\pm 1\.4\), exceeding plain H2EDL \(p=0\.021p\\\!=\\\!0\.021\) though not Flat\-CE \(p=0\.062p\\\!=\\\!0\.062\), and our best variant at34\.5%34\.5\\%stays tied with Flat\-CE \(p=0\.10p\\\!=\\\!0\.10\)\. Two facts here are worth naming\. The softmax hierarchy alone buys nothing: Hier\-CE reaches32\.032\.0against Flat\-CE’s32\.932\.9\. And on the shallow, near\-balanced FGVC taxonomy a well\-tuned flat evidential head is already an excellent recognizer: retuned I\-EDL matches our plain model at54\.954\.9\. This parity is a property of the benchmarks rather than of any method: forcing the ground\-truth branching decision down to depthddand re\-taking the argmax within the surviving subtree lifts DERM12345 leaf accuracy from∼0\.57\{\\sim\}0\.57to∼0\.79\{\\sim\}0\.79the moment the super\-class decision is supplied, while at any fixed depth the methods differ by at most0\.0130\.013\. One44\-way decision commits7070–74%74\\%of all leaf errors, so leaf accuracy on these benchmarks is largely a proxy for a coarse routing decision that no fine\-level machinery moves\.

Table 2:Balanced accuracy \(%\) at the intermediate levels of the DERM12345 taxonomy \(same runs as Table[1](https://arxiv.org/html/2608.18185#Sx3.T1)\)\. Every H2EDL variant is above every baseline at Main\-1 and Main\-2; leaf\-level bAcc, where H2EDL does not lead, is in Table[1](https://arxiv.org/html/2608.18185#Sx3.T1)\. Best inbold, second bestunderlined\.Scored where the factorization actually operates \(Table[2](https://arxiv.org/html/2608.18185#Sx4.T2)\), DERM12345 separates cleanly: every H2EDL variant beats every baseline at both intermediate levels\. The consistency variant leads Main\-2 at48\.8%48\.8\\%against the best baseline’s46\.646\.6and Main\-1 at59\.759\.7against57\.757\.7, with\+2\.2\+2\.2and\+2\.0\+2\.0over the strongest competitor at each level,\+3\.7\+3\.7and\+4\.2\+4\.2over Flat\-CE, clearingp<0\.05p\\\!<\\\!0\.05against Flat\-CE \(p=0\.028p\\\!=\\\!0\.028and0\.0120\.012\), Hier\-CE \(p=0\.005p\\\!=\\\!0\.005\) and I\-EDL \(p=0\.002p\\\!=\\\!0\.002\)\. The two remaining comparisons are limited by the baselines’ own instability: HENN trails by the same2\.22\.2atp=0\.082p\\\!=\\\!0\.082, and Flat\-EDL’s seed variance at these levels is four to seven times ours \(±5\.2\\pm 5\.2against±0\.7\\pm 0\.7at Main\-1\), so an equal gap over it cannot be resolved\. Consistency across seeds is itself part of what the factorization buys\. At the super\-class level three of four variants still top every baseline \(88\.688\.6,88\.288\.2and88\.088\.0against HENN’s87\.987\.9\), on a level where everything above I\-EDL already sits near88%88\\%and the margins are correspondingly inside seed noise \(p≥0\.12p\\\!\\geq\\\!0\.12\)\.

On FGVC\-Aircraft we remain ahead at both coarse levels \(78\.078\.0against Flat\-CE’s77\.177\.1at manufacturer,65\.365\.3against64\.064\.0at family\)\.

### Calibration Across Hierarchy Levels

To test the first prediction of the factorization, that the advantage over flat models has a specific shape rather than merely an existence, we report ECE at every level of both taxonomies, and under a doubled training budget\. Each path factor is supervised by every sample in its subtree, whereas a flat softmax resolves the wholeLL\-way competition at once\. The gap should therefore be small where the decision is easy and well populated, and should grow as the tree deepens and the evidence thins\.

Table[3](https://arxiv.org/html/2608.18185#Sx4.T3)puts the two models side by side on FGVC\-Aircraft at both budgets\. H2EDL is better calibrated at the leaf level by2\.2×2\.2\\timesat the standard schedule \(8\.88\.8against19\.5%19\.5\\%ECE\) and by2\.8×2\.8\\timeswhen it is doubled \(8\.68\.6against24\.124\.1,p=0\.009p\\\!=\\\!0\.009\), with the same ordering on NLL \(1\.951\.95against2\.002\.00, then1\.991\.99against2\.252\.25,p=0\.006p\\\!=\\\!0\.006\)\. The extra budget buys neither method accuracy, but Flat\-CE converts it into4\.64\.6points of ECE and0\.250\.25nats, while H2EDL moves by−0\.2\-0\.2and\+0\.04\+0\.04: the gap widens with training rather than closing\. This matters for durability\. An advantage that shrank as the baseline trained longer would be an artifact of an under\-trained comparison, whereas one that grows reflects a structural difference in what the two objectives do with extra capacity to fit\. The vacuity term gives each local decision a data\-dependent floor on how confident it may become; cross\-entropy has no such floor and spends the budget sharpening leaf logits it already gets right\.

Table 3:Calibration under a doubled training budget \(FGVC\-Aircraft; pairedtt\-test on the200200\-epoch pair\)\. The extra budget buys neither method accuracy, but Flat\-CE converts it into overconfidence and H2EDL does not, so the gap widens with training\.100100ep200200epΔ\\DeltabAcc↑\\uparrowFlat\-CE53\.6±\\pm0\.553\.1±\\pm0\.7−0\.5\-0\.5H2EDL54\.9±\\pm1\.752\.9±\\pm1\.2−2\.0\-2\.0ECE↓\\downarrowFlat\-CE19\.5±\\pm1\.924\.1±\\pm1\.8\+4\.6\+4\.6H2EDL8\.8±\\pm2\.38\.6±\\pm1\.7−0\.2\-0\.2*ratio*2\.2×2\.2\\times2\.8×\\mathbf\{2\.8\\times\}p=0\.009p\\\!=\\\!0\.009NLL↓\\downarrowFlat\-CE2\.00±\\pm\.102\.25±\\pm\.05\+0\.25\+0\.25H2EDL1\.95±\\pm\.061\.99±\\pm\.04\+0\.04\+0\.04*gap*0\.050\.050\.26\\mathbf\{0\.26\}p=0\.006p\\\!=\\\!0\.006That advantage resolves into a depth profile rather than an offset\. At the coarsest level of either tree, where the decision is easy and well populated, the two models are statistically indistinguishable \(p≥0\.20p\\\!\\geq\\\!0\.20on FGVC\-Aircraft\), and they separate only as the evidence thins:2\.8×2\.8\\timesat FGVC’s finest level \(Table[3](https://arxiv.org/html/2608.18185#Sx4.T3)\), and on DERM12345 a leaf ECE of31\.431\.4and31\.7%31\.7\\%for Flat\-CE and Hier\-CE against our16\.416\.4\(Table[1](https://arxiv.org/html/2608.18185#Sx3.T1)\)\. Every variant roughly halves the CE models’ leaf ECE on that dataset and cuts their NLL by about30%30\\%, for all three seeds\. A uniform advantage, or any constant post\-hoc temperature, would have shown up at all levels alike\.

Against the flat evidential baselines the result is carried by the proper scoring rule\. H2EDL has the best leaf NLL on both datasets \(1\.781\.78against HENN’s1\.991\.99and Flat\-EDL’s2\.172\.17on DERM12345,1\.911\.91against1\.951\.95on FGVC\-Aircraft\), and all four variants beat all five baselines on super\-class ECE\. At the two intermediate levels, Flat\-EDL and HENN are better calibrated\. On leaf ECE Flat\-EDL is lower,14\.914\.9against our16\.016\.0\. That comparison should be read with what it costs: ECE is minimized by never becoming confident, and Flat\-EDL buys its calibration at29\.3%29\.3\\%leaf bAcc, the lowest of any method here, carrying the largest seed variance in Table[1](https://arxiv.org/html/2608.18185#Sx3.T1)on both axes \(±6\.3\\pm 6\.3and±4\.1\\pm 4\.1\)\. NLL prices miscalibration and vagueness together, and on it the ordering reverses by a wide margin\. We therefore claim the depth profile against the cross\-entropy baselines and the proper\-scoring\-rule result against the flat evidential ones — and neither flat model supplies a coherent coarse read\-out at any level\.

### Severity of the Remaining Errors

To test the second prediction of the factorization, that ancestor gating confines errors to the correct coarse region, we measure two quantities, both conditioned on the prediction being wrong so that neither restates accuracy: the depth of the first incorrect branching decision, and the share of errors that still retain the correct ancestor at a given level\([1](https://arxiv.org/html/2608.18185#bib.bib13)\)\.

The motivation is that leaf accuracy treats every error identically and a taxonomy does not\. Confusing two nevus subtypes and calling a melanoma benign are the same event to a0/10/1loss and quite different events in a clinic\. Since the number of errors is fixed by a coarse decision no method improves, what is left to differ is the kind\.

Table 4:Mistake severity, conditioned on the prediction being wrong \(pairedtt\-test vs\. Flat\-CE\)\. Leaf accuracy is statistically tied on both datasets \(p=0\.21p\\\!=\\\!0\.21/p=0\.73p\\\!=\\\!0\.73\), so these differences are in the kind of error, not the amount\.The effect \(Table[4](https://arxiv.org/html/2608.18185#Sx4.T4)\) holds on both datasets, and it is larger on the bigger label set\. On FGVC\-Aircraft,46\.2%46\.2\\%of H2EDL’s errors still name the correct manufacturer against38\.9%38\.9\\%for Flat\-CE, a19%19\\%relative improvement in how recoverable its mistakes are, at indistinguishable leaf accuracy\. This is the mechanism of the factorization showing up in the errors\. Each ancestor factor gates its whole subtree at once, so a leaf outside the plausible region has to overcome a penalty at every level it crosses\. A flat softmax couples nothing, and its objective contains no term that distinguishes a near miss from a distant one, so the wider the label set the more room its argmax has to wander out of the correct coarse concept\.

### Learning from Partially Labeled Hierarchies

To demonstrate that the path parameterization absorbs coarse annotations without modification, we coarsen a fractionffof DERM12345 training labels to super\-class only and compare against the two standard flat fallbacks\.

Everything so far assumes every training image carries a leaf\. Dermatology archives do not work that way: a case confirmed as melanocytic but never subtyped is a real annotation, not a missing one\. Supervision enters Eq\. \([7](https://arxiv.org/html/2608.18185#Sx3.E7)\) as a set of decisions, so a label naming only an ancestorv⋆v^\{\\star\}shortens the sum instead of degrading it, and the heads below stay free to express vacuity\. No architectural change is involved\. We choose the coarsened subset by a hash of each sample’s stable identifier so that it is identical across methods, seeds and fractions and nested asffgrows\. Any difference is then attributable to how a coarse label is used, not to which samples were coarsened\. A flat model has two standard options and we run both:*soft*, training on the uniform distribution overv⋆v^\{\\star\}’s descendants, and*drop*, discarding the sample\.

Table 5:Coarse\-label supervision on DERM12345 \(4040\-epoch budget\)\.ffis the fraction of training labels coarsened to super\-class only\. H2EDL is ahead on every metric already atf=0f\\\!=\\\!0and stays ahead as labels are coarsened; the quantity of interest is each method’s degradation from its own full\-supervision score, so read down each column\. Best inbold\.The effect appears where the parameterization says it should, at the level the retained prefix supervises \(Table[5](https://arxiv.org/html/2608.18185#Sx4.T5)\)\. Coarsening three quarters of the labels costs H2EDL0\.40\.4points of super\-class balanced accuracy \(87\.4→87\.087\.4\\rightarrow 87\.0\) against3\.73\.7for soft and3\.83\.8for drop\. A coarse annotation is a complete label for the super\-class decision, and only a model whose supervision is a path of decisions can spend it that way\. Against soft, the standard way a flat model absorbs a coarse label, we win every metric at everyf≥0\.25f\\\!\\geq\\\!0\.25and the gap grows withff: atf=0\.75f\\\!=\\\!0\.75we hold51\.2%51\.2\\%leaf accuracy against37\.6%37\.6\\%, having lost5\.05\.0points from our own full\-supervision score where soft lost18\.218\.2\. Fabricating a uniform target overv⋆v^\{\\star\}’s descendants corrupts the fine\-grained heads, and the damage compounds\. Calibration is the most robust effect: leaf ECE never exceeds12\.8%12\.8\\%and improves to7\.2%7\.2\\%atf=0\.75f\\\!=\\\!0\.75as vacuity absorbs the supervision that is no longer there, while drop never falls below27\.5%27\.5\\%\(p=0\.002p\\\!=\\\!0\.002\)\.

Drop is the harder baseline and we do not beat it on accuracy\. It matches us on leaf accuracy at every fraction and leads atf=0\.75f\\\!=\\\!0\.75\(52\.052\.0against51\.251\.2\)\. Training on a quarter of the leaf labels is evidently close to training on all of them at this scale, which caps what any hierarchy\-native use of the discarded labels can add\. What drop cannot do is stay calibrated, holding28\.5%28\.5\\%leaf ECE and2\.242\.24nats where we hold7\.2%7\.2\\%and1\.881\.88\. The narrow claim these numbers support is that a coarse annotation is a complete label for the level it names: coarsening three quarters of the leaves costs H2EDL0\.40\.4points of super\-class accuracy against3\.73\.7for the flat fallback that keeps those samples, while the fallback that matches our accuracy does so only by discarding them and never falls below27\.5%27\.5\\%leaf ECE where we reach7\.27\.2\.

### Ablation Study and Hyperparameter Transfer

To assess how the individual components contribute, we read Table[1](https://arxiv.org/html/2608.18185#Sx3.T1)as a component ablation, and then examine how each baseline’s published constants transfer across class counts\.

Which components help turns out to be dataset\-dependent in an informative way\. On FGVC\-Aircraft the base\-rate prior is the single most effective addition, giving the best accuracy, the best calibration and most of the variance reduction\. On DERM12345 it does not transfer\. There it raises seed variance \(bAcc±5\.1\\pm 5\.1against±1\.8\\pm 1\.8–2\.92\.9\) and worsens both ECE and NLL, which leaves the parameter\-free consistency variant as the strongest DERM configuration\. The mirror image holds: on FGVC\-Aircraft that same consistency term is the weakest H2EDL variant \(53\.0±0\.453\.0\\pm 0\.4bAcc, ECE10\.1±2\.110\.1\\pm 2\.1\), below the plain model on all three metrics and, notably, not even more path\-consistent at test time \(0\.9230\.923against0\.9310\.931without it;*path consistency*is the label\-free fraction of test samples whose argmax at every level agrees with the leaf argmax projected to that level\); neither component is a free addition\. We attribute the base\-rate reversal to DERM’s short, heavily long\-tailed frame, in which a tempered inverse\-frequency prior over\-corrects the rare leaves\. End\-to\-end leaf coupling is a mild but reliable gain on both datasets\.

A parallel lesson applies to the flat evidential references\. Flat\-EDL and HENN transfer with no retuning at all\. I\-EDL’s Fisher weight, by contrast, must scale with the class count: its publishedc=0\.05c\\\!=\\\!0\.05drives all evidence to zero atK=100K\\\!=\\\!100, leaving16\.9%16\.9\\%bAcc, and needsc=0\.02c\\\!=\\\!0\.02before it becomes competitive\. This is a systematic consequence of the class count\. Alongside our own base\-rate result it is a second reminder that constants published for one class count do not port to a4040\- or100100\-way fine\-grained tree\. The ablation therefore points one way: most of what the hierarchy has to offer is already supplied by the path factorization itself, the base\-rate prior and the consistency term each help on exactly one of the two benchmarks, and neither they nor a baseline’s published constant survives a change of class count unchanged\.

## Conclusion

H2EDL makes hyper\-evidential uncertainty tractable on tree\-structured label spaces, by observing that the taxonomy is already the right hyperdomain\. Its subtrees and leaf singletons form a linear\-size focal family on which the belief masses are induced in closed form, by one ordinary Dirichlet opinion per branching decision, instead of being predicted\. What comes out is a single parameter set with two readings: an operational classifier with exact coarse marginals, and a valid tree\-structured hyper\-opinion whose composite mass at a node is the belief that reached it and declined to descend, with a path\-local imprecision band \(Prop\. 2\) and a per\-sample depth profile\{Ud\}\\\{U\_\{d\}\\\}\(Prop\. 3\) that a flat evidential model cannot express\. No second network is required, nor composite annotations, nor a post\-hoc calibration stage\. Empirically, leaf accuracy on both benchmarks is dominated by one coarse decision that no method improves, so parity there says more about the data than about the model\. Scored at the levels the factorization governs, every variant leads all five baselines at both intermediate DERM12345 levels, the ECE of the cross\-entropy baselines is roughly halved by a margin that grows with depth and with training budget, and the remaining errors stay inside the correct coarse category far more often at indistinguishable accuracy\. Of these the depth profile is the stronger test, since a constant post\-hoc rescaling could reproduce the average but not the shape\.

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