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

Yuanye Liu; Xiahai Zhuang

cs.LGAug 18, 2026

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

Authors: Yuanye Liu, Xiahai Zhuang

Organizations: School of Data Science, Fudan University

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, H $^2$ EDL, 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, H $^2$ EDL 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.

Figures & tables

Figure 1: Uncertainty on a label hierarchy has a location. H 2 EDL carries a local opinion at every branching decision, so it reports a separate vacuity for each level: the evidence is decisive about the manufacturer ( $u^{1}=0.2$ ), already thinner about the family ( $u^{2}=0.5$ ), and runs out among the $737$ variants ( $u^{3}=0.7$ ), where a flat evidential model has one vacuity for the entire leaf level.

			FGVC-Aircraft			DERM12345
Method	Hier.	Unc.	bAcc $\uparrow$	ECE $\downarrow$	NLL $\downarrow$	bAcc $\uparrow$	ECE $\downarrow$	NLL $\downarrow$
Flat-CE	$\times$	$\times$	53.6 $\pm$ 0.5	19.5 $\pm$ 1.9	2.00 $\pm$ .10	32.9 $\pm$ 0.9	31.4 $\pm$ 1.2	2.48 $\pm$ .21
Flat-EDL	$\times$	✓	50.1 $\pm$ 2.0	10.1 $\pm$ 3.9	2.36 $\pm$ .10	29.3 $\pm$ 6.3	14.9 $\pm$ 4.1	2.17 $\pm$ .05
Hier-CE	✓	$\times$	54.2 $\pm$ 1.2	21.0 $\pm$ 1.3	2.09 $\pm$ .01	32.0 $\pm$ 0.4	31.7 $\pm$ 0.7	2.58 $\pm$ .18
HENN	$\times$	✓	54.2 $\pm$ 0.8	8.6 $\pm$ 2.8	1.95 $\pm$ .03	35.8 $\pm$ 1.4	18.1 $\pm$ 2.6	1.99 $\pm$ .02
I-EDL †	$\times$	✓	54.9 $\pm$ 0.7	45.6 $\pm$ 0.7	3.16 $\pm$ .02	30.0 $\pm$ 1.9	32.8 $\pm$ 0.7	2.36 $\pm$ .02

Table 1: Main comparison at leaf-level. bAcc and ECE in %, NLL in nats; best in bold , second best underlined . Hier. marks a predictor defined over the label tree, Unc. one that emits an epistemic uncertainty; only H 2 EDL has both, HENN being evidential but flat (its composite focal sets are a supplied partition, not the taxonomy). An H 2 EDL variant ranks first or second in every column, taking both places in four of the six, and H 2 EDL 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\!=\!0.02

K\!=\!100

c\!=\!0.05

K\!=\!40

); its published default drops to

16.9\%

bAcc on FGVC.

Method	Super	Main-1	Main-2
Flat-CE	86.8 $\pm$ 1.9	55.5 $\pm$ 1.5	45.1 $\pm$ 1.0
Flat-EDL	84.9 $\pm$ 1.1	57.7 $\pm$ 5.2	46.6 $\pm$ 2.8
Hier-CE	87.0 $\pm$ 1.2	55.2 $\pm$ 2.7	44.9 $\pm$ 1.0
HENN	87.9 $\pm$ 0.7	57.3 $\pm$ 1.7	46.6 $\pm$ 0.9
I-EDL	74.0 $\pm$ 1.1	47.9 $\pm$ 1.0	41.0 $\pm$ 0.7
H 2 EDL	88.0 $\pm$ 0.5	59.3 $\pm$ 1.2	48.1 $\pm$ 0.6

Table 2: Balanced accuracy (%) at the intermediate levels of the DERM12345 taxonomy (same runs as Table 1 ). Every H 2 EDL variant is above every baseline at Main-1 and Main-2; leaf-level bAcc, where H 2 EDL does not lead, is in Table 1 . Best in bold , second best underlined .

		$100$ ep	$200$ ep	$\Delta$
bAcc $\uparrow$	Flat-CE	53.6 $\pm$ 0.5	53.1 $\pm$ 0.7	$-0.5$
bAcc $\uparrow$	H 2 EDL	54.9 $\pm$ 1.7	52.9 $\pm$ 1.2	$-2.0$
ECE $\downarrow$	Flat-CE	19.5 $\pm$ 1.9	24.1 $\pm$ 1.8	$+4.6$
	H 2 EDL	8.8 $\pm$ 2.3	8.6 $\pm$ 1.7	$-0.2$
	ratio	$2.2\times$	$\mathbf{2.8\times}$	$p\!=\!0.009$
NLL $\downarrow$	Flat-CE	2.00 $\pm$ .10	2.25 $\pm$ .05	$+0.25$

Table 3: Calibration under a doubled training budget (FGVC-Aircraft; paired

t

-test on the

200

-epoch pair). The extra budget buys neither method accuracy, but Flat-CE converts it into overconfidence and H 2 EDL does not, so the gap widens with training.

		Flat-CE	H 2 EDL	$p$
DERM	mean first-error depth	1.207 $\pm$ .028	1.277 $\pm$ .035	0.003
DERM	keeps correct main-1	25.6 $\pm$ 2.5	30.0 $\pm$ 2.5	0.022
FGVC	mean first-error depth	0.590 $\pm$ .018	0.713 $\pm$ .031	0.036
FGVC	keeps correct manufacturer	38.9 $\pm$ 0.8	46.2 $\pm$ 2.6	0.060

Table 4: Mistake severity, conditioned on the prediction being wrong (paired

t

-test vs. Flat-CE). Leaf accuracy is statistically tied on both datasets (

p\!=\!0.21

p\!=\!0.73

), so these differences are in the kind of error, not the amount.

		$f\!=\!0$	$0.25$	$0.50$	$0.75$
super bAcc $\uparrow$	Flat-CE soft	86.6 $\pm$ 1.9	83.4 $\pm$ 1.5	84.7 $\pm$ 1.9	82.9 $\pm$ 3.7
	Flat-CE drop	86.6 $\pm$ 1.9	84.3 $\pm$ 1.4	85.3 $\pm$ 2.9	82.8 $\pm$ 3.6
	H 2 EDL	87.4 $\pm$ 1.4	88.0 $\pm$ 0.6	86.2 $\pm$ 1.5	87.0 $\pm$ 1.9
leaf Acc $\uparrow$	Flat-CE soft	55.8 $\pm$ 1.3	51.4 $\pm$ 3.7	45.3 $\pm$ 2.5	37.6 $\pm$ 4.8
	Flat-CE drop	55.8 $\pm$ 1.3	55.3 $\pm$ 0.8	55.7 $\pm$ 1.2	52.0 $\pm$ 1.0
	H 2 EDL	56.2 $\pm$ 0.5	56.3 $\pm$ 1.5	55.7 $\pm$ 2.1	51.2 $\pm$ 3.1

Table 5: Coarse-label supervision on DERM12345 (

40

-epoch budget).

f

is the fraction of training labels coarsened to super-class only. H 2 EDL is ahead on every metric already at

f\!=\!0

and 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 in bold .

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