Grounding Predictive Reliability through Masked Observability

Shang Gao; Bernhard Sick; Franz Götz-Hahn

cs.LGOct 29, 2026

Grounding Predictive Reliability through Masked Observability

Authors: Shang Gao, Bernhard Sick, Franz Götz-Hahn

Organizations: University of Kassel, Germany

Abstract

An uncertainty estimate is meaningful only if its semantics can be tied to observable predictive behavior. Normal-Inverse-Gamma (NIG) evidential regression provides a structured decomposition of predictive variability into data-scale and predictive-mean components, but marginal likelihood does not identify their allocation. We address this semantic ambiguity in masked representation learning. MORSE (Masked Observability for Reliable Structured Encoding) predicts the same hidden target under controlled changes in visible context and uses normalized cross-context variation to ground predictive-mean uncertainty, while a Gaussian objective learns the conditional mean and data scale. The normalized variation plays the role of the NIG factor $1/ν$ , which the marginal leaves unidentified; this yields a policy-relative observability variance that measures dependence on visible evidence, distinct from variation induced by resampling the training set. On a Gaussian-process benchmark with a contextual reference built from analytic Bayes conditional means, MORSE has the lowest log-variance error among compared evidential, ensemble, and dropout estimators in all nine tested settings, while the estimator closest to training-set variation perturbs the model rather than its context in every setting, so the two references separate the estimators empirically. MORSE amortizes observability estimation into one forward pass per observation state. On a heteroscedastic pulse benchmark, its predictor's cross-context variance approaches a large-sample reference as training data increase. MORSE also improves low-label transfer on PTB-XL and in-distribution trajectory prediction on IMPTC.

Figures & tables

Figure 1: Marginal fitting leaves the internal allocation open. (A) Under standard DER, the four NIG outputs enter the marginal likelihood only through the location, scale, and degrees of freedom of a Student- $t$ distribution. (B) Its total predictive variance can be written as a data-scale component plus a predictive-mean component. (C) The same marginal is compatible with different internal allocations, so fitting it cannot determine which allocation is appropriate for a target. MORSE supplies the missing evidence through controlled changes in visible context.

Figure 2: MORSE overview. (a) Common targets $C$ remain hidden in all $K$ views, while view-specific masks $A_{k}$ alter their supporting context. (b) Cross-context predictive-mean variance is normalized by the local data scale to form the observability response, which takes the position of $U_{\mathrm{rel}}=1/\nu$ in the NIG decomposition. (c) The detached response conditions fitting, bounds the local magnitude of the relative mean-uncertainty factor, and supervises its ordering. All masked positions train reconstruction, whereas only $C$ provides matched $K$ -view supervision.

	LogVar MAE $\downarrow$
Method	Contextual	Training set
DER	$2.150\pm 0.105$	$3.432\pm 0.105$
RN-DER	$2.414\pm 0.071$	$3.704\pm 0.070$
Gaussian-MV	$4.336\pm 0.111$	$5.632\pm 0.111$
Deep ensemble (5)	$2.166\pm 0.140$	$0.915\pm 0.091$
Bootstrap ensemble (5)	$1.607\pm 0.092$	$\mathbf{0.548\pm 0.027}$

Table 1: Agreement with two distinct GP uncertainty targets (

N=2048

; five paired anchors; LogVar MAE, mean

\pm

sample SD; lower is better). Contextual: variance of analytic Bayes conditional means over 128 fixed masks; Appendix E.2 re-scores these checkpoints on an independent bank. Training set: variance across 20 independent first-order fits at fixed masks. All learned scores average the same four evaluation contexts; evidential heads, including MORSE, use one forward pass per context, MC dropout 50 draws, and each ensemble five members. Bold marks the best unrounded mean. The no-attention MC dropout row is retrained without attention-probability dropout (Appendix E.3 ).

Figure 3: Different estimators align with different variance targets. Mean variance of each estimator within ten equal-count deciles of each target on the GP benchmark. Axes share a log scale. The deciles are formed separately in each panel from that panel’s target, and all estimators share them; lines are means over five anchors and bands are anchor standard deviations. (A) Contextual target: variance of analytic Bayes conditional means over 128 fixed masks. (B) Training-set target

V_{\mathrm{ref}}

: variance across 20 independently trained first-order models at fixed masks.

Method	AUROC $\uparrow$	Macro-F1 $\uparrow$	ECE $\downarrow$	AURC $\downarrow$
Supervised	.827 $\pm$ .005	.592 $\pm$ .011	.069 $\pm$ .010	.110 $\pm$ .011
DAE	.826 $\pm$ .006	.596 $\pm$ .012	.079 $\pm$ .016	.096 $\pm$ .008
MAE	.842 $\pm$ .006	.620 $\pm$ .005	.043 $\pm$ .020	.099 $\pm$ .008
PatchTST	.823 $\pm$ .011	.590 $\pm$ .015	.030 $\pm$ .016	.120 $\pm$ .014
SimMTM	.849 $\pm$ .005	.622 $\pm$ .008	.043 $\pm$ .013	.090 $\pm$ .007
ST-MEM	.835 $\pm$ .010	.604 $\pm$ .012	.052 $\pm$ .020	.106 $\pm$ .012

Table 2: PTB-XL low-label transfer at the 10% label budget (five-seed mean

\pm

sample standard deviation).

Appendix figures & tables35 assets

Supplementary material from the paper’s appendix.

Appendix

Setting	Synthetic	PTB-XL	IMPTC
Views $K$	4	4	4
Mask rate per view / shared-target rate	.50 / .25	.50 / .25	.50 / .25
$\lambda_{\mathrm{cond}}$ / $\lambda_{\nu}$	1 / 0	1 / 0	1 / 0
$\lambda_{\mathrm{ref}}$ / $\lambda_{\mathrm{rank}}$	.01 / .05	.05 / .05	.05 / .05
ObsRef margin $m$ / upper weight $\omega$	.25 / 1	1 / .5	1 / .5
Target-floor quantile $f$	0	.10	.10

Appendix

Table 3: Confirmed frozen MORSE settings. All reported main-method runs use the decoupled objective, so

\lambda_{\nu}=0

Figure 4: Exact fixed-pulse diagnostic. From left to right, the panels show the data-generating process with predictive means and 50 independently fitted DER, RN-DER, and Gaussian-MV models. Blue is the predictive data-width proxy

\sqrt{\beta(1+\nu)/(\alpha\nu)}

and orange is the relative mean-uncertainty width

1/\sqrt{\nu}

, matching the original diagnostic’s plotting convention; its legend names them aleatoric and epistemic proxies. This is an implementation check; the scale-sensitive metrics in Table 4 prevent a visually flat response from being treated as sufficient evidence of a correct allocation. The supplied diagnostic’s compact Eq.11/Eq.12 labels map to RN-DER/Gaussian-MV, respectively.

Method	MSE $\downarrow$	NLL $\downarrow$	$L_{1/\sqrt{\nu}}\downarrow$	$L_{\mathrm{rel}}\downarrow$	$L_{\mathrm{mean}}\downarrow$
DER	.0498 $\pm$ .0008	$-2.0920$ $\pm$ .1483	2.2526 $\pm$ 1.1065	4.4823 $\pm$ 2.2005	6.1481 $\pm$ 2.0911
RN-DER	.0488 $\pm$ .0020	$-2.0164\pm$ .1641	1.3639 $\pm$ .9395	2.7457 $\pm$ 1.8722	5.0433 $\pm$ 1.7440
Gaussian-MV	.0362 $\pm$ .0163	$-1.9573\pm$ .1258	1.4991 $\pm$ 1.0410	3.0724 $\pm$ 2.0066	7.4254 $\pm$ 2.1064

Appendix

Table 4: Fixed-pulse direct diagnostic, 50 independent fits (mean

\pm

sample standard deviation). Leakage metrics are accompanied by external utility and interval diagnostics because a constant score can obtain low leakage.

Panel of Figure 5 — Figure 5: Parameterized direct diagnostic. From upper left to lower center: prediction aligned by latent pulse center, data-scale response aligned by the independently sampled noise boundary, and relative mean-uncertainty response aligned by that boundary. Curves show the median and bands the interquartile range across 50 tasks. Alignment is used only for visualization, not for training or metric computation. Panel titles keep the original diagnostic’s aleatoric and epistemic proxy names. The supplied diagnostic’s compact Eq.11/Eq.12 labels map to RN-DER/Gaussian-MV, respectively.

Method	MSE $\downarrow$	NLL $\downarrow$	$L_{1/\sqrt{\nu}}\downarrow$	$L_{\mathrm{rel}}\downarrow$	$L_{\mathrm{mean}}\downarrow$
DER	.0591 $\pm$ .0282	$-2.0797$ $\pm$ .3317	.9234 $\pm$ .7862	1.8467 $\pm$ 1.5562	3.2412 $\pm$ 1.7635
RN-DER	.0573 $\pm$ .0283	$-1.9756\pm$ .3310	1.0276 $\pm$ .8912	2.0126 $\pm$ 1.7041	3.9321 $\pm$ 1.7180
Gaussian-MV	.0422 $\pm$ .0283	$-1.9377\pm$ .3587	1.2887 $\pm$ 1.0253	2.5494 $\pm$ 1.9830	5.5316 $\pm$ 2.7207

Appendix

Table 5: Parameterized direct diagnostic over 50 independently sampled tasks (mean

\pm

sample standard deviation).

Figure 6: Common-target observation policy. One sequence is shown, with the legend below. Red circles mark the common targets $C$ , hidden in all four views; the four rows of dots beneath the signal mark each view’s additional hidden context $A_{k}$ . The pulse and noise boundary vary independently. Every cross-view prediction therefore refers to the same target while its supporting evidence changes.

Method	MSE $\downarrow$	NLL $\downarrow$	$L_{\mathrm{rel}}$ $\downarrow$	$L_{\mathrm{mean}}$ $\downarrow$	Err. AUROC $\uparrow$	Sel. excess $\downarrow$	Cov. gap $\downarrow$
Masked DER	.6188 $\pm$ .1811	$-.2508\pm$ .0516	1.4947 $\pm$ .2270	.5562 $\pm$ .4302	.8271 $\pm$ .0228	.0895 $\pm$ .0470	.0460 $\pm$ .0166
RN-DER	.5151 $\pm$ .0588	$-.2565$ $\pm$ .0268	1.1061 $\pm$ .2679	.3149 $\pm$ .4609	.8319 $\pm$ .0087	.0360 $\pm$ .0040	.0512 $\pm$ .0085
Gaussian-MV	.5416 $\pm$ .0423	$-.0996\pm$ .0499	.5343 $\pm$ .2257	.5772 $\pm$ .1008	.8785 $\pm$ .0168	.0289 $\pm$ .0025	.0934 $\pm$ .0020
MORSE	.5331 $\pm$ .0371	$-.1681\pm$ .0298	.3424 $\pm$ .0906	.1253 $\pm$ .1092	.8705 $\pm$ .0070	.0295 $\pm$ .0034	.0176 $\pm$ .0086

Appendix

Table 6: Parameterized pulse-window backbone comparison over ten paired seeds (mean

\pm

sample standard deviation).

Model comparison
	Predictive fit		Mean uncertainty	Error ranking		Intervals
Method	MSE $\downarrow$	NLL $\downarrow$	$L_{\mathrm{mean}}$ $\downarrow$	AUROC $\uparrow$	SelEx $\downarrow$	CovGap $\downarrow$	Width †
Masked DER	$.619\pm.181$	$-.251\pm.052$	$.556\pm.430$	$.827\pm.023$	$.0895\pm.0470$	$.0460\pm.0166$	$18.93\pm 35.84$
RN-DER	$\mathbf{.515\pm.059}$	$\mathbf{-.256\pm.027}$	$.315\pm.461$	$.832\pm.009$	$.0360\pm.0040$	$.0512\pm.0085$	$2.36\pm 1.44$
Gaussian-MV	$.542\pm.042$	$-.100\pm.050$	$.577\pm.101$	$\mathbf{.879\pm.017}$	$\mathbf{.0289\pm.0025}$	$.0934\pm.0020$	$2.79\pm.30$
MORSE	$.533\pm.037$	$-.168\pm.030$	$\mathbf{.125\pm.109}$	$.870\pm.007$	$.0295\pm.0034$	$\mathbf{.0176\pm.0086}$	$\mathbf{1.45\pm.07}$

Appendix

Table 7: Parameterized pulse-window results (ten-seed mean

\pm

sample standard deviation). MORSE-RC restores the direct residual-to-

\nu

gradient.

L_{\mathrm{mean}}

measures pooled high/low-noise sensitivity, and Width † is the mean 90% moment-interval width, to be read jointly with CovGap. Bold marks the best mean within each block.

Variant	MSE $\downarrow$	NLL $\downarrow$	$L_{\mathrm{rel}}$ $\downarrow$	$L_{\mathrm{mean}}$ $\downarrow$	Err. AUROC $\uparrow$	Sel. excess $\downarrow$	Cov. gap $\downarrow$
MORSE	.5331 $\pm$ .0371	$-.1681$ $\pm$ .0298	.3424 $\pm$ .0906	.1253 $\pm$ .1092	.8705 $\pm$ .0070	.0295 $\pm$ .0034	.0176 $\pm$ .0086
MORSE without ObsRef	.5440 $\pm$ .0408	1.4182 $\pm$ .1172	.8044 $\pm$ .1085	.2531 $\pm$ .1233	.8664 $\pm$ .0095	.0364 $\pm$ .0040	.1000 $\pm$ .0001
MORSE without Rank	.5411 $\pm$ .0450	$-.1059\pm$ .0338	.0114 $\pm$ .0050	.3191 $\pm$ .1597	.8737 $\pm$ .0117	.0292 $\pm$ .0032	.0768 $\pm$ .0040

Appendix

Table 8: Core MORSE ablations over ten paired seeds (mean

\pm

sample standard deviation; the same component cohort as Table 7 ).

Variant	MSE	NLL	$L_{\mathrm{rel}}$	$L_{\mathrm{mean}}$	AUROC	Sel.	Cov. gap	Width
$U_{\mathrm{data}}$ base	.5474 $\pm$ .0266	$-.1098\pm$ .0406	.1650 $\pm$ .0167	.2632 $\pm$ .0519	.8732 $\pm$ .0081	.0293 $\pm$ .0027	.0779 $\pm$ .0041	2.0594 $\pm$ .1027
Condition only	.5384 $\pm$ .0393	2.5312 $\pm$ .1932	.4731 $\pm$ .3790	.8549 $\pm$ .3924	.8360 $\pm$ .0171	.0729 $\pm$ .0319	.1000 $\pm$ .0000	39.4471 $\pm$ 10.3392

Appendix

Table 9: Extended mechanism controls over ten paired seeds (mean

\pm

sample standard deviation; the same component cohort as Table 7 ).

Metric	MORSE-RC	MORSE
MSE	.5320 $\pm$ .0198	.5618 $\pm$ .0787
NLL	$-.1110\pm$ .0376	$-.1519\pm$ .0509
Error AUROC	.8721 $\pm$ .0082	.8692 $\pm$ .0125
Selective excess	.0281 $\pm$ .0018	.0302 $\pm$ .0044
Coverage gap	.0762 $\pm$ .0035	.0188 $\pm$ .0100
Interval width	2.0153 $\pm$ .1033	1.4804 $\pm$ .0708

Appendix

Table 10: MORSE versus residual-coupled MORSE-RC over 20 paired seeds (mean

\pm

sample standard deviation). Lower is better except Error AUROC.

Figure 7: Quality–cost trade-off for common-target views. Every point and its error bar show mean

\pm

standard deviation (divisor

n

) over ten paired seeds. Increasing

K

reduces reconstruction error but does not improve error ranking, while cost grows almost linearly;

K=4

is the prespecified compromise used throughout the paper.

$K$	MSE	NLL	Error AUROC	train s/run	passes
2	.5691 $\pm$ .0510	$-.1325\pm$ .0246	.8739 $\pm$ .0080	134.5 $\pm$ 8.9	3
4	.5331 $\pm$ .0371	$-.1681\pm$ .0298	.8705 $\pm$ .0070	213.1 $\pm$ 4.8	5
8	.5200 $\pm$ .0286	$-.1641\pm$ .0588	.8693 $\pm$ .0080	422.2 $\pm$ 38.7	9

Appendix

Table 11: Common-target view ablation over ten paired seeds (mean

\pm

sample standard deviation). Training time is total wall-clock seconds per run. Forward passes per step include one reference pass.

Head	MSE	NLL	Err. AUROC	Sel.	Cov. gap	Width
NIG-shaped	.5331	$-.1681$	.8705	.0295	.0176	1.4485
Explicit	.5140	$-.1509$	.8543	.0402	.0094	1.4328

Appendix

Table 12: NIG-shaped MORSE versus explicit independent heads over ten paired seeds.

Method	Center	Log-var	W1	Cov. gap	Width
DER	.2562 $\pm$ .0146	3.8535 $\pm$ .0897	.3757 $\pm$ .0333	.0323 $\pm$ .0023	6.1212 $\pm$ .5051
RN-DER	.2307 $\pm$ .0132	4.2056 $\pm$ .1369	.4418 $\pm$ .0269	.0471 $\pm$ .0012	8.3378 $\pm$ .5677
Gaussian-MV	.2354 $\pm$ .0814	4.6828 $\pm$ .1083	.5190 $\pm$ .0526	.0480 $\pm$ .0008	10.5144 $\pm$ 1.4901
MORSE	.1584 $\pm$ .0182	1.5496 $\pm$ .0513	.0785 $\pm$ .0086	.1203 $\pm$ .1085	1.3480 $\pm$ .1253

Appendix

Table 13: Strict Jürgens-aligned reference audit over five paired anchor datasets (mean

\pm

sample standard deviation). The empirical target is fixed and independent: 50 Gaussian first-order models trained on resampled datasets. Width ratio is ideal at one.

Baseline	Center benefit [95% CI]	Log-var benefit [95% CI]	W1 benefit [95% CI]
DER	.0978 [.0776,.1186]	2.3039 [2.1914,2.3870]	.2972 [.2631,.3273]
RN-DER	.0723 [.0548,.0898]	2.6560 [2.5830,2.7289]	.3632 [.3473,.3794]
Gaussian-MV	.0770 [.0011,.1515]	3.1332 [3.0243,3.2521]	.4405 [.4047,.4840]

Appendix

Table 14: Paired MORSE improvement over each baseline across the five anchor datasets. Intervals are paired 95% bootstrap confidence intervals (CIs). Positive differences favor MORSE for all three lower-is-better metrics. The smallest attainable two-sided exact sign-test

p

value with five consistently signed pairs is

1/16=.0625

; we therefore emphasize intervals and win counts.

Variant	LogVar	Spearman	Center RMSE	NLL	CovGap	Width
	$\downarrow$	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$
NIG C0B0O0	2.0910 $\pm$ .1165	.5932 $\pm$ .0074	.1395 $\pm$ .0018	.1702 $\pm$ .0214	.0806 $\pm$ .0031	1.4128 $\pm$ .0485
NIG C0B0O1	2.0641 $\pm$ .0798	.6246 $\pm$ .0041	.1416 $\pm$ .0047	.1701 $\pm$ .0159	.0809 $\pm$ .0022	1.3938 $\pm$ .0273
NIG C0B1O0	.7135 $\pm$ .0081	.5878 $\pm$ .0070	.1425 $\pm$ .0056	.0601 $\pm$ .0090	.0291 $\pm$ .0045	1.0448 $\pm$ .0168
NIG C0B1O1	.7313 $\pm$ .0118	.6215 $\pm$ .0034	.1408 $\pm$ .0061	.0595 $\pm$ .0107	.0348 $\pm$ .0042	1.0669 $\pm$ .0191
NIG C1B0O0	1.9900 $\pm$ .1392	.5933 $\pm$ .0065	.1434 $\pm$ .0049	.1497 $\pm$ .0302	.0747 $\pm$ .0046	1.3445 $\pm$ .0588

Appendix

Table 15: Matched component and head study on the fixed-boundary GP benchmark (five paired anchors; mean

\pm

sample SD). C, B, and O switch Condition, Bound (ObsRef), and Order (Rank) on or off. LogVar and Spearman compare the four-context score with the 124-mask oracle response; NLL uses the Gaussian variance

U_{\mathrm{data}}+U_{\mathrm{mean}}

for every head. Bold marks the best mean; width is not ranked.

Variant	MSE	NLL	AUROC	AURC	CovGap	Width	$L_{\mathrm{mean}}$
	$\downarrow$	$\downarrow$	$\uparrow$	$\downarrow$	$\downarrow$		$\downarrow$
NIG C0B0O0	.5137 $\pm$ .0326	$-.0565\pm$ .0714	.8715 $\pm$ .0074	.0543 $\pm$ .0030	.0694 $\pm$ .0062	2.4342 $\pm$ .1679	.1608 $\pm$ .0765
NIG C0B0O1	.5183 $\pm$ .0366	$-.0538\pm$ .0416	.8792 $\pm$ .0079	.0553 $\pm$ .0043	.0803 $\pm$ .0050	2.1985 $\pm$ .2129	.2314 $\pm$ .1294
NIG C0B1O0	.5115 $\pm$ .0294	$-.1705\pm$ .0419	.8748 $\pm$ .0114	.0534 $\pm$ .0030	.0149 $\pm$ .0090	1.4983 $\pm$ .0932	.2773 $\pm$ .1708
NIG C0B1O1	.5143 $\pm$ .0242	$-.1550\pm$ .0664	.8758 $\pm$ .0064	.0542 $\pm$ .0025	.0191 $\pm$ .0080	1.4646 $\pm$ .0627	.1402 $\pm$ .0827
NIG C1B0O0	.5265 $\pm$ .0181	$-.0571\pm$ .0445	.8761 $\pm$ .0088	.0557 $\pm$ .0026	.0707 $\pm$ .0069	2.5693 $\pm$ .3079	.1911 $\pm$ .1393

Appendix

Table 16: Matched component and head study on the pulse benchmark (ten seeds; mean

\pm

sample SD). This cohort is not mixed with Tables 6 – 9 . NLL uses the Gaussian variance

U_{\mathrm{data}}+U_{\mathrm{mean}}

for every head. Bold marks the best mean; width is not ranked.

Endpoint	Control	Difference [95% interval]	Wins	Holm $p$
GP (LogVar)	no Condition	$+$ 0.00356 [ $-$ 0.00845, $+$ 0.01383]	1/5	1.000
GP (LogVar)	no Bound	$-$ 1.21951 [ $-$ 1.32962, $-$ 1.11433]	5/5	0.375
GP (LogVar)	no Order	$+$ 0.02515 [ $+$ 0.01646, $+$ 0.03384]	0/5	0.375
GP (LogVar)	explicit head	$+$ 0.00994 [ $+$ 0.00080, $+$ 0.01802]	1/5	0.750
GP (LogVar)	fixed $\alpha=2$	$+$ 0.00411 [ $-$ 0.00470, $+$ 0.01234]	1/5	1.000
GP (LogVar)	direct mean variance	$+$ 0.00379 [ $-$ 0.00625, $+$ 0.01383]	2/5	1.000

Appendix

Table 17: Declared primary comparisons in the component study: full MORSE minus each control (negative favors MORSE), paired 95% intervals from 20,000 whole-anchor or whole-seed draws, wins, and Holm-adjusted exact sign-flip

p

within each domain. The five GP anchors are descriptive.

$N$	$R_{\mathrm{obs}}(\theta)$ vs $R^{\ast}_{c}$ LogVar $\downarrow$	$U_{\mathrm{mean}}$ vs $R^{\ast}_{c}$ LogVar $\downarrow$	Mean $R_{\mathrm{obs}}(\theta)$	Mean $V_{\mathrm{ref}}$
512	1.987 $\pm$ .187	4.477	.009	.0427
2048	1.562 $\pm$ .092	3.327	.022	.0445
8192	1.281 $\pm$ .091	2.345	.038	.0361
32768	.917 $\pm$ .087	1.687	.040	.0200

Appendix

Table 18: Pulse training-size audit on the 32,594 resolvable common targets (ten paired anchors per

N

). The first column is the 128-mask response of each MORSE predictor,

R_{\mathrm{obs}}(\theta)

, against the corrected proxy reference

R^{\ast}_{c}

(mean

\pm

sample standard deviation); the second is MORSE’s learned

U_{\mathrm{mean}}

against the same reference. The mean of

R^{\ast}_{c}

is .03395 at every

N

. The last column is the finite-training-set reference on the same targets.

Figure 8: Finite-training-set audit with ten paired anchors and an independent 50-model reference bank per $N$ . (A) Mean $U_{\mathrm{mean}}$ and mean $V_{\mathrm{ref}}$ . (B) Token-level log-variance MAE to $V_{\mathrm{ref}}$ ; lower is better. Curves show mean $\pm$ standard deviation; pale markers show individual anchors.

$N$	DER	RN-DER	Gaussian-MV	MORSE	DER	RN-DER	Gaussian-MV	MORSE
	A. Mean $U_{\mathrm{mean}}$				A. LogVar MAE to $V_{\mathrm{ref}}$ $\downarrow$
512	26.21 $\pm$ 7.65	28.90 $\pm$ 5.43	3.72 $\pm$ 1.61	.084 $\pm$ .022	6.371 $\pm$ .153	7.166 $\pm$ .139	3.517 $\pm$ .245	1.708 $\pm$ .061
1024	411.22 $\pm$ 227.15	307.81 $\pm$ 236.56	1.542 $\pm$ .351	.0479 $\pm$ .0065	7.719 $\pm$ .529	6.315 $\pm$ 1.161	3.893 $\pm$ .137	1.564 $\pm$ .079
2048	.897 $\pm$ .252	1.293 $\pm$ .340	1.297 $\pm$ .286	.0249 $\pm$ .0069	3.850 $\pm$ .085	4.186 $\pm$ .111	4.641 $\pm$ .119	1.546 $\pm$ .040
4096	.815 $\pm$ .180	.991 $\pm$ .202	1.585 $\pm$ .302	.0159 $\pm$ .0032	4.069 $\pm$ .122	4.434 $\pm$ .061	5.759 $\pm$ .226	1.505 $\pm$ .038
8192	5.50 $\pm$ 1.39	4.21 $\pm$ .77	1.592 $\pm$ .202	.0140 $\pm$ .0033	4.452 $\pm$ .073	4.788 $\pm$ .051	6.421 $\pm$ .118	1.223 $\pm$ .030

Appendix

Table 19: Finite-training-set audit across seven training sizes. Panel A: ten paired second-order anchors per size (mean

\pm

sample standard deviation); mean

U_{\mathrm{mean}}

is descriptive. LogVar MAE is the common-target average of

|\log(U_{\mathrm{mean}}+\epsilon)-\log(V_{\mathrm{ref}}+\epsilon)|

. Panel B: empirical reference variance; each row uses an independent bank of 50 first-order models on the same fixed test and mask bank.

Method	Context LogVar $\downarrow$	Context $\rho$ $\uparrow$	$V_{\mathrm{ref}}$ LogVar $\downarrow$	$V_{\mathrm{ref}}$ $\rho$ $\uparrow$
DER	$6.140\pm 0.096$	$0.458\pm 0.008$	$3.847\pm 0.089$	$0.806\pm 0.012$
RN-DER	$6.541\pm 0.143$	$0.464\pm 0.009$	$4.201\pm 0.136$	$0.861\pm 0.006$
Gaussian-MV	$7.040\pm 0.104$	$\mathbf{0.464\pm 0.016}$	$4.678\pm 0.108$	$0.831\pm 0.010$
MORSE	$3.332\pm 0.037$	$0.428\pm 0.015$	$1.545\pm 0.051$	$0.855\pm 0.011$
MORSE-IS	$3.526\pm 0.052$	$0.447\pm 0.010$	$1.485\pm 0.073$	$\mathbf{0.867\pm 0.005}$
Deep ensemble (5)	$\mathbf{2.555\pm 0.210}$	$0.429\pm 0.057$	$\mathbf{0.722\pm 0.100}$	$0.751\pm 0.109$

Appendix

Table 20: Pulse proxy audit on five paired anchors and 32,594 resolvable targets (mean

\pm

sample SD). Context variance uses the selected four-model large-architecture proxy, not an exact Bayes reference. The independent training-set reference contains 50 first-order models. Both ensemble controls outperform MORSE in LogVar MAE to both targets in this benchmark.

Method	Center RMSE $\downarrow$	Context LogVar $\downarrow$	Context $\rho$ $\uparrow$	$V_{\mathrm{ref}}$ LogVar $\downarrow$	$V_{\mathrm{ref}}$ $\rho$ $\uparrow$
DER	$0.135\pm 0.004$	$2.150\pm 0.105$	$0.526\pm 0.006$	$3.432\pm 0.105$	$0.486\pm 0.009$
RN-DER	$\mathbf{0.129\pm 0.004}$	$2.414\pm 0.071$	$0.525\pm 0.006$	$3.704\pm 0.070$	$0.481\pm 0.006$
Gaussian-MV	$0.139\pm 0.008$	$4.336\pm 0.111$	$0.506\pm 0.017$	$5.632\pm 0.111$	$0.455\pm 0.021$
Deep ensemble (5)	$0.132\pm 0.002$	$2.166\pm 0.140$	$0.303\pm 0.046$	$0.915\pm 0.091$	$0.680\pm 0.052$
Bootstrap ensemble (5)	$0.129\pm 0.002$	$1.607\pm 0.092$	$0.303\pm 0.055$	$\mathbf{0.548\pm 0.027}$	$0.697\pm 0.035$
MC dropout (50)	$0.133\pm 0.002$	$0.807\pm 0.013$	$0.467\pm 0.026$	$1.185\pm 0.107$	$\mathbf{0.797\pm 0.017}$

Appendix

Table 21: GP two-reference audit on five paired anchors (mean

\pm

sample SD). Center is RMSE of first-order conditional means against analytic Bayes means. The same four evaluation contexts are used for every learned score and for the 20-model independent

V_{\mathrm{ref}}

bank; model-axis variances are computed before averaging over contexts. Raw rows use 4 or 128 masks and estimate context variation directly. The two reference quantities measure different sources of variation. The no-attention MC dropout row is retrained without attention-probability dropout (Appendix E.3 ).

Method	Variance ratio $\to 1$	Coverage $\to.95$	CovGap 95 $\downarrow$	Width ratio $\to 1$
DER	$8.357\pm 0.891$	$0.992\pm 0.001$	$0.042\pm 0.001$	$2.911\pm 0.157$
RN-DER	$10.859\pm 0.799$	$0.995\pm 0.001$	$0.045\pm 0.001$	$3.306\pm 0.121$
Gaussian-MV	$79.267\pm 8.858$	$1.000\pm 0.000$	$0.050\pm 0.000$	$9.250\pm 0.457$
Deep ensemble (5)	$0.115\pm 0.021$	$0.395\pm 0.033$	$0.555\pm 0.033$	$0.373\pm 0.028$
Bootstrap ensemble (5)	$0.223\pm 0.026$	$0.523\pm 0.023$	$0.427\pm 0.023$	$0.501\pm 0.023$
MC dropout (50)	$0.809\pm 0.083$	$0.776\pm 0.024$	$0.174\pm 0.024$	$0.929\pm 0.047$

Appendix

Table 22: GP context-distribution diagnostics (five-anchor mean

\pm

sample SD). A Gaussian moment interval is centered at the four-context average prediction with variance equal to the reported score. Coverage is measured against the 128 Bayes conditional means, not noisy outcomes. Width is relative to their empirical central 95% interval. Variance ratio is the within-anchor median score/reference ratio. These are operational intervals, not native NIG posteriors.

(A) Contextual reference
$\ell$	Noise	MORSE	MC dropout	Bootstrap	$\Delta$ [95% interval]	Wins
.025	Step	$0.749\pm 0.012$	$1.176\pm 0.034$	$2.210\pm 0.028$	$-0.427$ $[-0.461,-0.395]$	5/5
.025	Window-homosc.	$0.742\pm 0.014$	$1.196\pm 0.037$	$2.163\pm 0.149$	$-0.454$ $[-0.477,-0.431]$	5/5
.025	Smooth	$0.749\pm 0.009$	$1.175\pm 0.035$	$2.207\pm 0.042$	$-0.426$ $[-0.464,-0.399]$	5/5
.05	Step (default)	$0.779\pm 0.022$	$0.809\pm 0.013$	$1.606\pm 0.092$	$-0.029$ $[-0.039,-0.019]$	5/5
.05	Window-homosc.	$0.657\pm 0.005$	$0.772\pm 0.026$	$1.713\pm 0.147$	$-0.114$ $[-0.137,-0.094]$	5/5

Appendix

Table 23: GP configuration scan (five paired anchors per setting; LogVar MAE, mean

\pm

sample SD; lower is better;

\ell

is the length scale). (A) Contextual reference;

\Delta

is MORSE minus MC dropout with a descriptive 95% interval from 20,000 paired anchor bootstrap draws, and wins count anchors on which MORSE is lower. (B) Training-set reference

V_{\mathrm{ref}}

. In every row, MORSE has the lowest mean contextual error and the bootstrap ensemble the lowest mean

V_{\mathrm{ref}}

error among the seven estimators; all seven are listed in the supplement ( results/gp_config_scan/ ), and Table 24 adds MC dropout without attention-probability dropout. The default row re-scores the checkpoints of Table 1 on an independent oracle bank.

(A) Contextual reference
$\ell$	Noise	All	No attn.	MORSE	$\Delta$ [95% interval]
.025	Step	$1.176\pm 0.034$	$1.914\pm 0.102$	$0.749\pm 0.012$	$+0.738$ $[+0.625,+0.827]$
.025	Window-homosc.	$1.196\pm 0.037$	$1.883\pm 0.109$	$0.742\pm 0.014$	$+0.687$ $[+0.608,+0.777]$
.025	Smooth	$1.175\pm 0.035$	$1.909\pm 0.134$	$0.749\pm 0.009$	$+0.735$ $[+0.593,+0.841]$
.05	Step (default)	$0.809\pm 0.013$	$1.012\pm 0.064$	$0.779\pm 0.022$	$+0.203$ $[+0.151,+0.240]$
.05	Window-homosc.	$0.772\pm 0.026$	$1.052\pm 0.073$	$0.657\pm 0.005$	$+0.281$ $[+0.208,+0.348]$

Appendix

Table 24: MC dropout with all dropout sites active (All) and with attention-probability dropout disabled (No attn.) in the nine GP settings (five paired anchors; LogVar MAE, mean

\pm

sample SD; lower is better;

\ell

is the length scale).

\Delta

is No attn. minus All with a descriptive 95% interval from 20,000 paired anchor bootstrap draws. (A) Contextual reference, with MORSE. (B) Training-set reference

V_{\mathrm{ref}}

, with the bootstrap ensemble.

MORSE versus SimMTM
Labels	Metric	Benefit	Paired 95% CI	Wins
5%	AUROC	.0169	[.0121, .0208]	5/5
	AUPRC	.0277	[.0251, .0308]	5/5
	Macro-F1	.0147	[.0090, .0223]	5/5
	Brier	.0026	[ $-$ .0043, .0096]	2/5
	AURC	.0009	[ $-$ .0078, .0109]	2/5

Appendix

Table 25: Paired PTB-XL differences over five matched seeds. Positive values favor MORSE; intervals are paired bootstrap 95% intervals.

5% labeled records
Method	AUROC $\uparrow$	AUPRC $\uparrow$	Macro-F1 $\uparrow$	Brier $\downarrow$	ECE $\downarrow$	AURC $\downarrow$
Supervised	.808 $\pm$ .005	.587 $\pm$ .010	.578 $\pm$ .004	.166 $\pm$ .003	.067 $\pm$ .016	.120 $\pm$ .003
DAE	.809 $\pm$ .011	.588 $\pm$ .019	.569 $\pm$ .017	.170 $\pm$ .010	.111 $\pm$ .036	.110 $\pm$ .010
MAE	.835 $\pm$ .002	.628 $\pm$ .012	.610 $\pm$ .005	.152 $\pm$ .006	.046 $\pm$ .021	.105 $\pm$ .005
PatchTST	.809 $\pm$ .009	.587 $\pm$ .021	.575 $\pm$ .007	.165 $\pm$ .010	.026 $\pm$ .013	.128 $\pm$ .014
SimMTM	.835 $\pm$ .005	.634 $\pm$ .006	.607 $\pm$ .009	.151 $\pm$ .005	.049 $\pm$ .019	.098 $\pm$ .008

Appendix

Table 26: PTB-XL low-label fine-tuning over five matched seeds (mean

\pm

sample standard deviation). All methods use the official patient-separated folds, identical seed-specific labeled subsets, and validation Macro-AUROC selection. Bold marks the best mean within each label budget.

Method	Accuracy $\uparrow$	Macro-F1 $\uparrow$	Balanced $\uparrow$	Directional F1 $\uparrow$	Brier $\downarrow$	ECE $\downarrow$	AURC $\downarrow$
Supervised	.7154 $\pm$ .0029	.6362 $\pm$ .0178	.6479 $\pm$ .0109	.2951 $\pm$ .0552	.3751 $\pm$ .0009	.0243 $\pm$ .0074	.1435 $\pm$ .0010
DAE	.7240 $\pm$ .0027	.6565 $\pm$ .0074	.6622 $\pm$ .0052	.3392 $\pm$ .0215	.3539 $\pm$ .0019	.0211 $\pm$ .0007	.1309 $\pm$ .0016
Masked AE	.7304 $\pm$ .0027	.6645 $\pm$ .0049	.6693 $\pm$ .0032	.3539 $\pm$ .0161	.3459 $\pm$ .0026	.0171 $\pm$ .0034	.1284 $\pm$ .0030
PatchTST-style	.7363 $\pm$ .0015	.6742 $\pm$ .0150	.6783 $\pm$ .0102	.3697 $\pm$ .0503	.3346 $\pm$ .0059	.0244 $\pm$ .0064	.1193 $\pm$ .0049
SimMTM-style	.7338 $\pm$ .0018	.6563 $\pm$ .0049	.6655 $\pm$ .0030	.3152 $\pm$ .0175	.3414 $\pm$ .0024	.0248 $\pm$ .0056	.1244 $\pm$ .0020
MORSE	.7407 $\pm$ .0018	.6882 $\pm$ .0087	.6878 $\pm$ .0066	.4084 $\pm$ .0339	.3273 $\pm$ .0036	.0319 $\pm$ .0038	.1081 $\pm$ .0034

Appendix

Table 27: IMPTC in-distribution Balanced FEM3, 30% labels and partial fine-tuning over five seeds (mean

\pm

sample standard deviation).

Method	Accuracy $\uparrow$	Macro-F1 $\uparrow$	Balanced $\uparrow$	Directional F1 $\uparrow$	Brier $\downarrow$	ECE $\downarrow$	AURC $\downarrow$
Supervised	.6194 $\pm$ .0495	.4714 $\pm$ .0422	.5352 $\pm$ .0430	.0030 $\pm$ .0046	.4956 $\pm$ .0657	.0435 $\pm$ .0175	.2688 $\pm$ .0781
DAE	.7134 $\pm$ .0029	.6022 $\pm$ .0149	.6311 $\pm$ .0059	.1711 $\pm$ .0506	.3822 $\pm$ .0025	.0212 $\pm$ .0038	.1621 $\pm$ .0044
Masked AE	.7199 $\pm$ .0032	.6202 $\pm$ .0140	.6418 $\pm$ .0067	.2196 $\pm$ .0455	.3731 $\pm$ .0023	.0237 $\pm$ .0099	.1569 $\pm$ .0029
PatchTST-style	.7256 $\pm$ .0052	.6158 $\pm$ .0095	.6418 $\pm$ .0068	.1899 $\pm$ .0264	.3708 $\pm$ .0077	.0249 $\pm$ .0088	.1568 $\pm$ .0084
SimMTM-style	.7198 $\pm$ .0025	.6022 $\pm$ .0090	.6335 $\pm$ .0041	.1570 $\pm$ .0264	.3807 $\pm$ .0024	.0346 $\pm$ .0066	.1656 $\pm$ .0032
MORSE	.7225 $\pm$ .0084	.6129 $\pm$ .0122	.6425 $\pm$ .0078	.1891 $\pm$ .0203	.3761 $\pm$ .0131	.0411 $\pm$ .0224	.1595 $\pm$ .0126

Appendix

Table 28: IMPTC held-out environment 5 under the same protocol (five seeds, mean

\pm

sample standard deviation). The mixed ranking is an explicit boundary on the paper’s out-of-distribution claim.

Split	Baseline	Metric	Oriented $\Delta$ [95% CI]	Wins/5	$d_{z}$
In-dist.	Masked AE	Macro-F1	$.0237\ [.0154,.0313]$	5	2.30
In-dist.	Masked AE	Brier	$.0186\ [.0144,.0228]$	5	3.41
In-dist.	Masked AE	ECE	$-.0147\ [-.0179,-.0111]$	0	$-3.46$
In-dist.	Masked AE	AURC	$.0203\ [.0159,.0247]$	5	3.62
In-dist.	PatchTST-style	Macro-F1	$.0140\ [.0078,.0218]$	5	1.57
In-dist.	PatchTST-style	Brier	$.0073\ [.0005,.0140]$	3	.85

Appendix

Table 29: Matched-seed IMPTC comparisons with the two closest reconstruction baselines. Positive

\Delta

favors MORSE. Intervals are deterministic paired bootstrap 95% CIs (20,000 replicates);

d_{z}

is the paired standardized effect. With five pairs, the minimum two-sided exact sign-test

p

is .0625; intervals, wins, and effects are reported without significance claims.

Table 38

Experiment	Model/data seeds	Fixed auxiliary seeds
Fixed and parameterized one-dimensional tasks	10000–10049	fixed-generator settings in merge manifests
Pulse-window comparison/components/ $K$	1–10	paired masks and data within each seed
MORSE versus MORSE-RC	1–20	paired data, initialization, and masks
Strict five-anchor reference	anchors 900001–900005; reference 1–50	test 424242; normalization 777777; init/mask/eval/loader 12001–12004
Strict $N$ sensitivity	anchors 900001–900010; reference 1–50 at each $N$	same 424242, 777777, and 12001–12004
Pulse proxy reference	standard 700011–700018; large 700021–700024	128-mask bank 424243

Appendix

Table 32: Frozen seed axes for reported experiments. Ranges are inclusive.

Claim	Primary evidence	Scope
Marginal fitting does not identify the relative allocation	Proposition 1	Concerns evidence from the marginal, not uniqueness under arbitrary regularization
Cross-context mean variance is observable and has a population limit	Propositions 2 and 3	Policy-relative; not a parameter posterior
The learned score aligns with the contextual reference	GP audit, Tables 1 , 21 , 22 , and 23	Four-context score; scale error in nine GP settings, five anchors each, not ordering (MC dropout has the higher Spearman correlation in five); MC dropout close at length scale .10; undercoverage; benchmark-dependent (Table 20 )
The learned score costs fewer inference passes than direct repeated masking of comparable accuracy	GP amortization audit, Appendix E	Default setting; four passes versus 16 to 128; inference only, not training cost
Contextual and training-set variation are distinct	GP two-reference audit in nine settings (Table 23 ); Proposition 3	MORSE–bootstrap reversal in all nine GP settings; disabling attention dropout moves MC dropout toward $V_{\mathrm{ref}}$ (Table 24 ), by refitting rather than an inference-only switch; not observed on the pulse benchmark
The predictor’s contextual response converges with $N$	Pulse training-size audit, Table 18	Approximate proxy reference, not exact Bayes

Appendix

Table 33: Claim-to-evidence map for this paper.

Uncertainty Quantification

Uncertainty quantification estimates how reliable predictions and scientific inferences are, and how uncertainty propagates into decisions.

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Observability-Aligned Evidential Masked Pretraining for Reliable VRU Motion Prediction
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Observability-Aligned Evidential Masked Pretraining for Reliable VRU Motion Prediction

Shang Gao, Franz Götz-Hahn, Konrad Doll, Bernhard Sick

Intelligent Embedded Systems, University of Kassel, Germany · Aschaffenburg University of Applied Sciences, Germany

Uncertainty Quantification Masked Autoencoders Evidential Deep Learning Sequence Modeling Pedestrian Trajectory Prediction Pedestrian Intention Prediction

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Vulnerable road user (VRU) prediction must operate under partial observability and distribution shift, so reliable uncertainty matters alongside accuracy. Masked pretraining is attractive in this setting because it naturally exposes the model to missing content during self-supervision, but existing time-series masked autoencoding methods are typically deterministic and do not constrain how confidence should change with observability. We introduce eCoMAE, an evidential masked pretraining method for multivariate time series that explicitly aligns predictive evidence with what is observed. eCoMAE combines block-wise time-step masking with a Normal-Inverse-Gamma decoder and three observability-aware mechanisms: uncertainty-weighted mean consistency on positions visible in both a masked view and a reference pass, evidence suppression on masked positions, and representation consistency across stochastic masks. On IMPTC benchmark for VRU intention prediction under leave-one-environment-out weather shift, eCoMAE improves the strongest baseline from 0.7971 to 0.8068 in OOD accuracy and from 0.0718 to 0.0648 in OOD AURC. Additional evaluation on pedestrian trajectory forecasting across IMPTC, Waymo, nuScenes, and inD further yields the best fine-tuning MSE on all four datasets. These results show that observability-aligned evidential pretraining improves reliability under shift and remains effective beyond the primary benchmark.

Abstract

Uncertainty Quantification

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