面向多专家区间目标的不确定性感知学习

# Uncertainty-Aware Learning from Multi-Expert Interval Targets
Source: [https://arxiv.org/html/2610.00102](https://arxiv.org/html/2610.00102)
Samira Alkaee TaleghanAffiliation:University of Colorado Denver Denver, Colorado, USAsamira\.alkaeetaleghan@ucdenver\.eduAffiliation:Younghyun KooAffiliation:National Snow and Ice Data Center \(NSIDC\) CIRES, University of Colorado Boulder Boulder, Colorado, USAYounghyun\.Koo@colorado\.eduAffiliation:Andrew P\. BarrettAffiliation:National Snow and Ice Data Center \(NSIDC\) CIRES, University of Colorado Boulder Boulder, Colorado, USAandrew\.barrett@colorado\.eduAffiliation:Farnoush Banaei\-KashaniAffiliation:University of Colorado Denver Denver, Colorado, USAfarnoush\.banaei\-kashani@ucdenver\.eduAffiliation:

###### Abstract

Many machine learning \(ML\) applications rely on expert labels, and qualified experts may provide different but plausible interpretations of the same observation\. Such variation across expert labels may reflect genuine disagreement or ambiguity rather than annotation error\. When individual experts additionally report intervals rather than exact values, the supervision contains two distinct sources of label uncertainty: within\-label imprecision and between\-expert variation\. Existing methods treat these forms separately: multi\-expert approaches collapse labels to a consensus, interval\-target methods often yield a single prediction, and predictive\-uncertainty methods rarely validate their uncertainty estimates against observed expert disagreement\. To address this problem, we propose an approach that preserves individual expert intervals, separates within\-label imprecision from between\-expert variation, and validates the corresponding predictive uncertainty components\. First, heterogeneous label vocabularies are harmonized into a common probabilistic label space, separating encoding differences from expert judgement\. Second, individual label intervals are retained and modeled with a mixture of Beta distributions trained using a proper Cramér\-distance objective, preserving distinct expert\-reported labels\. Third, we decompose predictive uncertainty into within\-component, between\-component, and model uncertainty, and evaluate whether these components correspond to within\-label uncertainty, between\-label uncertainty, and model error, respectively\. Because this correspondence is not guaranteed, we introduce decomposition matching, which aligns the predictive components to their intended label\-side sources\. On sea\-ice concentration \(404 scenes labeled by multiple ice analysts\), the model reduces MAE by 31% over hard labels and outperforms aggregation, interval\-distribution, and interval\-regression baselines\. Uncertainty components do not automatically align with their sources: decomposition matching raises the correlation with between\-expert variation to 0\.500 while reducing correlation with model error, with essentially unchanged point accuracy\.

###### Index Terms:

Uncertainty\-aware learning, Interval targets, Multi\-expert learning

## IIntroduction

Many machine\-learning applications rely on expert labels, where the target itself is a matter of interpretation rather than a directly observed quantity\. Consequently, qualified experts may provide different but plausible labels for the same instance, introducing uncertainty in the supervision itself rather than only in the predictive model\. Radiologists may delineate the same lesion differently\[[1](https://arxiv.org/html/2610.00102#bib.bib12),[2](https://arxiv.org/html/2610.00102#bib.bib13)\], annotators may provide different judgments on the same item\[[3](https://arxiv.org/html/2610.00102#bib.bib14),[4](https://arxiv.org/html/2610.00102#bib.bib15)\], and institutional analysis pipelines may produce different labels for the same observation\. Such differences do not necessarily indicate annotation error\. Plank\[[5](https://arxiv.org/html/2610.00102#bib.bib35)\]described human label variation as arising from genuine disagreement, subjectivity, annotator uncertainty, or multiple plausible interpretations\. Preserving this variation can therefore be important: it identifies cases where the available evidence supports more than one interpretation, where predictions may warrant additional review, and where another expert assessment may be particularly informative\.

The problem becomes richer when individual labels are themselves imprecise\. An expert may report an interval of plausible values rather than a precise point, reflecting limited precision or ambiguity in the judgement\. With multiple experts, this creates two distinct forms of uncertainty in the labels: uncertainty within an individual label, represented by the range of values that expert considers plausible, and uncertainty between labels, arising when different experts report different intervals for the same instance\. A learned predictor introduces a third source of uncertainty: uncertainty in the model itself\. These sources reflect different phenomena and warrant separate treatment\. A learning framework for this setting should distinguish within\-label uncertainty, between\-label uncertainty, and model \(epistemic\) uncertainty rather than collapsing them into a single target or a single uncertainty measure\. Existing approaches address parts of this problem separately\. Multi\-expert methods typically infer a consensus or latent ground truth\[[2](https://arxiv.org/html/2610.00102#bib.bib13),[3](https://arxiv.org/html/2610.00102#bib.bib14)\], model expert reliability\[[4](https://arxiv.org/html/2610.00102#bib.bib15),[6](https://arxiv.org/html/2610.00102#bib.bib16)\], or simply select one trusted label\[[7](https://arxiv.org/html/2610.00102#bib.bib1)\]\. Other work preserves label variability through label distributions\[[8](https://arxiv.org/html/2610.00102#bib.bib10),[9](https://arxiv.org/html/2610.00102#bib.bib11)\]or probabilistic structured prediction\[[1](https://arxiv.org/html/2610.00102#bib.bib12)\], while predictive\-uncertainty methods quantify model uncertainty using ensembles, Bayesian approximations, or evidential prediction\[[10](https://arxiv.org/html/2610.00102#bib.bib3),[11](https://arxiv.org/html/2610.00102#bib.bib4),[12](https://arxiv.org/html/2610.00102#bib.bib5),[13](https://arxiv.org/html/2610.00102#bib.bib6)\]\. Recent interval\-supervision methods preserve bounded targets during training, but largely focus on deterministic regression\[[14](https://arxiv.org/html/2610.00102#bib.bib18),[15](https://arxiv.org/html/2610.00102#bib.bib19)\]\. However, these approaches do not jointly distinguish and validate label imprecision, between\-expert variation, and model uncertainty using observed multi\-expert interval labels\.

In this work, we propose a framework for learning from heterogeneous, interval\-valued expert labels\. We construct parallel label\-side and predictive variance decompositions and examine whether the predictive components correspond to the observed sources of label uncertainty\. First, we harmonize the labels into distributions over a common label space, separating encoding differences from differences in judgement\. Second, we retain the individual label intervals and learn a mixture of Beta distributions\[[16](https://arxiv.org/html/2610.00102#bib.bib9)\]using a proper Cramér\-distance objective\[[17](https://arxiv.org/html/2610.00102#bib.bib38)\], allowing distinct expert interpretations to be represented\. Third, we evaluate whether uncertainty within and between mixture components, together with model uncertainty estimated using a deep ensemble of independently trained models, correspond to label imprecision, between\-expert variation, and model error, respectively, and introduce decomposition matching to explicitly align the predictive components with their intended label\-side sources\.

We study this problem through sea\-ice concentration \(SIC\) estimation, an operational setting in which supervision is both interval and provided by multiple experts\. SIC describes the fraction of an area covered by sea ice and is important for climate monitoring, numerical modeling, maritime navigation, and decision\-making in polar regions\[[18](https://arxiv.org/html/2610.00102#bib.bib24),[19](https://arxiv.org/html/2610.00102#bib.bib22)\]\. Operational SIC information is commonly provided through ice charts produced by national ice services, such as the U\.S\. National Ice Center \(NIC\), whose trained analysts interpret satellite observations and manually delineate regions of relatively homogeneous ice conditions\. This interpretation is not always unique: SIC can be difficult to assess under intermediate concentrations, marginal ice conditions, poorly resolved floes, and melt\-season imagery\[[20](https://arxiv.org/html/2610.00102#bib.bib31)\], so qualified analysts may produce different but plausible interpretations of the same observation\. Each chart polygon is described using the standardized World Meteorological Organization \(WMO\) Egg Code and SIGRID3 data format\[[21](https://arxiv.org/html/2610.00102#bib.bib25),[22](https://arxiv.org/html/2610.00102#bib.bib23)\], and these chart\-derived labels are widely used to train SIC prediction models\[[23](https://arxiv.org/html/2610.00102#bib.bib20),[24](https://arxiv.org/html/2610.00102#bib.bib21)\]\. In this work, we treat each ice service as a distinct source of expert labels and use independently produced charts to characterize expert label variation\. The resulting supervision exposes three distinct sources of uncertainty: an individual SIC label may span a range of plausible values, producing within\-label imprecision; different ice services may report different intervals for the same observation, producing between\-expert variation; and the learned predictor introduces model uncertainty\. We study these sources using 404 satellite scenes independently charted by up to three operational ice services\.

We make three main contributions: \(i\) we formulate learning from multiple interval labels while explicitly separating within\-label uncertainty from between\-expert variation, and harmonize heterogeneous label vocabularies into a common probabilistic label space; \(ii\) we construct a multi\-expert interval target and train a Mixture\-of\-Betas predictor using a proper Cramér objective; and \(iii\) we decompose predictive uncertainty into within\-component, between\-component, and model uncertainty, validate each against within\-label uncertainty, between\-expert variation, and model error, respectively, and show that between\-component uncertainty reliably recovers between\-expert variation only under explicit decomposition matching\.

The remainder of the paper is organized as follows\. Section II reviews related work, Section III presents the methodology, Section IV describes the experimental setup and reports the results, including sensitivity to key design choices, and Section V concludes\.

## IIRelated Work

### II\-ALearning from Multiple Experts

##### Consensus and Expert Reliability

Learning from multiple experts traditionally formulated as a problem of recovering a latent ground\-truth label from noisy or inconsistent observations\. Classical approaches estimate expert reliability together with a consensus label\[[3](https://arxiv.org/html/2610.00102#bib.bib14),[2](https://arxiv.org/html/2610.00102#bib.bib13),[4](https://arxiv.org/html/2610.00102#bib.bib15)\], while later methods incorporate expert\-specific confusion or bias directly into neural models\[[6](https://arxiv.org/html/2610.00102#bib.bib16)\]\. These approaches are effective when disagreement primarily reflects differences in expert reliability, but their objective is generally to resolve disagreement in order to recover a single underlying target\.

##### Learning from Expert Disagreement

A complementary line of work treats disagreement as informative rather than as annotation noise\. Methods preserve multiple annotations or model individual annotators to capture variation in expert judgement\[[25](https://arxiv.org/html/2610.00102#bib.bib34),[26](https://arxiv.org/html/2610.00102#bib.bib36)\], while Plank\[[5](https://arxiv.org/html/2610.00102#bib.bib35)\]argued that human label variation may reflect genuine ambiguity or differences in interpretation\. Other approaches represent this variation through label distributions\[[8](https://arxiv.org/html/2610.00102#bib.bib10),[9](https://arxiv.org/html/2610.00102#bib.bib11)\]or multiple plausible structured outputs\[[1](https://arxiv.org/html/2610.00102#bib.bib12)\]\. These methods generally consider point\-valued annotations, with label distributions used to represent variation across individual labels\. Our setting instead considers multiple interval expert labels, allowing us to represent both imprecision within each expert label and variation across expert labels\. We preserve these sources in the target distribution, evaluate whether the corresponding predictive uncertainty components recover them, and use decomposition matching when the between\-component uncertainty does not emerge reliably\.

### II\-BLearning from Interval Targets

Interval supervision is approached from several perspectives\. Earlier work on interval data prediction learns interval outputs from interval observations, including center–range regression, constrained interval regression, kernel methods, and neural approaches\[[27](https://arxiv.org/html/2610.00102#bib.bib26),[28](https://arxiv.org/html/2610.00102#bib.bib27),[29](https://arxiv.org/html/2610.00102#bib.bib28)\]\. Related statistical work on interval\-censored regression similarly learns from outcomes known only to lie within bounds\[[30](https://arxiv.org/html/2610.00102#bib.bib29)\]\. More recently, regression with interval targets \(RIT\) considers the setting in which an interval contains an unknown point\-valued target\.\[[14](https://arxiv.org/html/2610.00102#bib.bib18)\]formulate RIT as weakly supervised regression and constrain predictions using the observed interval\.\[[15](https://arxiv.org/html/2610.00102#bib.bib19)\]further develop interval\-compatible and worst\-case objectives for this setting\. These methods generally consider a single interval target\. Our setting instead retains multiple expert\-reported intervals for the same observation, allowing within\-label imprecision to be distinguished from between\-expert variation and represented separately in the predictive uncertainty\.

### II\-CProbabilistic Prediction and Uncertainty

Predictive\-uncertainty methods quantify uncertainty arising from the data, the model, or both\. Approximate Bayesian inference through Monte Carlo dropout\[[11](https://arxiv.org/html/2610.00102#bib.bib4)\], deep ensembles\[[10](https://arxiv.org/html/2610.00102#bib.bib3)\], and evidential approaches\[[12](https://arxiv.org/html/2610.00102#bib.bib5),[13](https://arxiv.org/html/2610.00102#bib.bib6)\]provide widely used mechanisms for estimating predictive or epistemic uncertainty\. These methods identify uncertain predictions, but the uncertainty reflects model behavior rather than observed between\-expert variation\. High predictive variance alone cannot distinguish label imprecision, between\-expert variation, or missing model information\. Representing multiple expert interval labels requires modeling both bounded uncertainty and multiple plausible outcomes\. Beta distributions are well suited to bounded targets on\[0,1\]\[0,1\]\[[31](https://arxiv.org/html/2610.00102#bib.bib37)\], but a single Beta cannot represent separated modes\. Mixture models\[[16](https://arxiv.org/html/2610.00102#bib.bib9)\]address this limitation\. Our Mixture\-of\-Betas combines these ideas and tests whether its uncertainty components correspond to within\-label imprecision, between\-expert variation, and model uncertainty\.

### II\-DUncertainty in SIC Estimation

Operational ice charts provide expert\-derived estimates of SIC from satellite observations\. Because SIC is assigned through analyst interpretation, estimates for the same conditions can vary across analysts and operational ice services\. Previous studies quantified variability in these assessments both among individual ice analysts\[[32](https://arxiv.org/html/2610.00102#bib.bib30),[20](https://arxiv.org/html/2610.00102#bib.bib31)\]and across operational ice services\[[33](https://arxiv.org/html/2610.00102#bib.bib32)\]\. These studies show that differences in SIC estimates vary with concentration, location, and season, with larger differences at intermediate SIC values\. For SIC estimation, uncertainty mainly quantified from the predictive model\. Heteroscedastic Bayesian neural networks used to separate aleatoric uncertainty, represented as input\-dependent predictive variance, from epistemic uncertainty arising from uncertainty in the model parameters\[[34](https://arxiv.org/html/2610.00102#bib.bib33)\]\. Other work derives SIC and its predictive uncertainty from calibrated ensemble probabilities over discrete concentration levels\[[35](https://arxiv.org/html/2610.00102#bib.bib17)\]\. In these approaches, aleatoric or predictive uncertainty is inferred from the model output rather than measured from independent expert labels\. In contrast, our work explicitly models uncertainty in the SIC supervision itself\. We preserve interval labels from multiple experts, distinguish within\-label imprecision from between\-expert variation, and validate whether the learned predictive uncertainty corresponds to these observed sources of label uncertainty\.

## IIIMethodology

The methodology consists of three stages: harmonizing heterogeneous label codes into a common label space, constructing a probabilistic target that preserves within\-label imprecision and between\-expert variation, and decomposing predictive uncertainty into components that can be compared with these observed label uncertainties and model error\.

### III\-AProblem Formulation and Label Harmonization

Letx∈𝒳x\\in\\mathcal\{X\}denote an instance, here a pixel together with its input features, and let the ordered label space be𝒴=\[0,1\]\\mathcal\{Y\}=\[0,1\]\. Each instance is labeled by a subset𝒜⁡\(x\)⊆\{1,…,S\}\\mathcal\{A\}\(x\)\\subseteq\\\{1,\\dots,S\\\}of a pool ofSSexperts, withA⁡\(x\)=\|𝒜⁡\(x\)\|≥1A\(x\)=\|\\mathcal\{A\}\(x\)\|\\geq 1varying across instances\. Expertaareports a label encoded as codecac\_\{a\}from a vocabulary𝒱a\\mathcal\{V\}\_\{a\}, which may differ across experts\. A decoding map

ιa:𝒱a→\{\[L,U\]:0≤L≤U≤1\}\\iota\_\{a\}:\\mathcal\{V\}\_\{a\}\\rightarrow\\\{\[L,U\]:0\\leq L\\leq U\\leq 1\\\}\(1\)maps each code to an interval\[La,Ua\]=ιa​\(ca\)\[L\_\{a\},U\_\{a\}\]=\\iota\_\{a\}\(c\_\{a\}\), whereLa=UaL\_\{a\}=U\_\{a\}denotes an exact\-valued label\. The observed label set forxxis therefore

𝒟⁡\(x\)=\(\[La,Ua\]\)a∈𝒜⁡\(x\),\\mathcal\{D\}\(x\)=\\bigl\(\[L\_\{a\},U\_\{a\}\]\\bigr\)\_\{a\\in\\mathcal\{A\}\(x\)\},\(2\)and the training set is\{\(xi,𝒟⁡\(xi\)\)\}i=1N\\\{\(x\_\{i\},\\mathcal\{D\}\(x\_\{i\}\)\)\\\}\_\{i=1\}^\{N\}\. This label set contains two forms of label uncertainty: within\-label uncertainty, represented by the range of plausible values within each label reported by an expert, and between\-label uncertainty, induced by differences among the labels reported by different experts for the same instance\.

Because experts use different coding vocabularies and quantization conventions, the raw codes are not directly comparable\. We therefore treat each label as uniform on its interval and discretize it onto a common grid: partition\[0,1\]\[0,1\]intoJJbinsℬj=\[ℓj,uj\)\\mathcal\{B\}\_\{j\}=\[\\ell\_\{j\},u\_\{j\}\)of width1/J1/Jwith centerszjz\_\{j\}, the final bin closed\. ForLa<UaL\_\{a\}<U\_\{a\}, the unnormalized mass inℬj\\mathcal\{B\}\_\{j\}is the overlap length

p~a​\(j\)=max⁡\(0,min⁡\(Ua,uj\)−max⁡\(La,ℓj\)\),\\tilde\{p\}\_\{a\}\(j\)=\\max\\bigl\(0,\\;\\min\(U\_\{a\},u\_\{j\}\)\-\\max\(L\_\{a\},\\ell\_\{j\}\)\\bigr\),\(3\)normalized by∑j′p~a​\(j′\)=Ua−La\\sum\_\{j^\{\\prime\}\}\\tilde\{p\}\_\{a\}\(j^\{\\prime\}\)=U\_\{a\}\-L\_\{a\}; for an exact labelLa=UaL\_\{a\}=U\_\{a\}all mass goes to the bin containing it\. This yields a harmonized distributionpap\_\{a\}for each reported label\. The harmonized distributionspap\_\{a\}provide a common representation for comparing labels expressed under heterogeneous coding conventions, whereas the original intervals\[La,Ua\]\[L\_\{a\},U\_\{a\}\]are retained for distributional learning and for quantifying within\-label uncertainty and between\-label uncertainty\. In particular, between\-label uncertainty arises from variation among the labels reported by different experts for the same instance\. To quantify the magnitude of this between\-expert variation, we use the mean pairwise Wasserstein\-1 distance\. For brevity, letA=A⁡\(x\)A=A\(x\):

dsrc​\(x\)=2A⁡\(A−1\)​∑a<bW1​\(pa,pb\),A≥2,d\_\{\\mathrm\{src\}\}\(x\)=\\frac\{2\}\{A\(A\-1\)\}\\sum\_\{a<b\}W\_\{1\}\(p\_\{a\},p\_\{b\}\),\\qquad A\\geq 2,\(4\)whereW1W\_\{1\}is computed on the ordered label grid using the bin locations as the ground metric\. WhenA⁡\(x\)=1A\(x\)=1, between\-expert variation cannot be assessed, anddsrcd\_\{\\mathrm\{src\}\}is therefore undefined rather than zero\. We additionally define two forms of uncertainty directly from the reported intervals\. Letma=\(La\+Ua\)/2m\_\{a\}=\(L\_\{a\}\+U\_\{a\}\)/2, andm¯=1A​∑a∈𝒜⁡\(x\)ma\\bar\{m\}=\\frac\{1\}\{A\}\\sum\_\{a\\in\\mathcal\{A\}\(x\)\}m\_\{a\}\. Treating each label as uniform on its interval, the equal\-weight mixture1A​∑a∈𝒜⁡\(x\)Unif⁡\[La,Ua\]\\tfrac\{1\}\{A\}\\sum\_\{a\\in\\mathcal\{A\}\(x\)\}\\mathrm\{Unif\}\[L\_\{a\},U\_\{a\}\]has variancetw2​\(x\)\+tb2​\(x\)t\_\{w\}^\{2\}\(x\)\+t\_\{b\}^\{2\}\(x\)by the law of total variance, where

tw2​\(x\)\\displaystyle t\_\{w\}^\{2\}\(x\)=1A∑a∈𝒜⁡\(x\)\(Ua−La\)212,A≥1,\\displaystyle=\\frac\{1\}\{A\}\\sum\_\{a\\in\\mathcal\{A\}\(x\)\}\\frac\{\(U\_\{a\}\-L\_\{a\}\)^\{2\}\}\{12\},\\qquad A\\geq 1,\(5\)tb2​\(x\)\\displaystyle t\_\{b\}^\{2\}\(x\)=1A∑a∈𝒜⁡\(x\)\(ma−m¯\)2,A≥2\.\\displaystyle=\\frac\{1\}\{A\}\\sum\_\{a\\in\\mathcal\{A\}\(x\)\}\(m\_\{a\}\-\\bar\{m\}\)^\{2\},\\qquad A\\geq 2\.Here,tw2t\_\{w\}^\{2\}measures within\-label uncertainty, defined as the average within\-label variance induced by the widths of the reported intervals, whiletb2t\_\{b\}^\{2\}measures between\-label uncertainty through the variance of their interval centers\. The1/A1/Anormalization makestb2t\_\{b\}^\{2\}the between\-label component of the variance of the equal\-weight label mixture\. These quantities characterize uncertainty observed from the expert labels\. Model uncertainty, which arises from variability in the learned predictor, is estimated separately using a deep ensemble\.

### III\-BDistributional Learning and Uncertainty Decomposition

The quantities in \([5](https://arxiv.org/html/2610.00102#S3.E5)\) describe within\-label and between\-label uncertainty, but these distinctions must also be preserved in the training target\. We therefore construct the target from the reported intervals by representing each as a probability distribution\. For expertaa, the reported interval\[La,Ua\]\[L\_\{a\},U\_\{a\}\]is treated as a uniform distribution, assigning equal probability density to all values within the reported range\. LetQa​\(y\)Q\_\{a\}\(y\)denote its cumulative distribution function \(CDF\)\. Then

Qa​\(y\)=\{clip⁡\(y−LaUa−La,0,1\),Ua\>La,𝕀\[y≥La\],Ua=La\.Q\_\{a\}\(y\)=\\begin\{cases\}\\mathrm\{clip\}\\\!\\left\(\\dfrac\{y\-L\_\{a\}\}\{U\_\{a\}\-L\_\{a\}\},0,1\\right\),&U\_\{a\}\>L\_\{a\},\\\\\[4\.0pt\] \\mathbb\{I\}\[y\\geq L\_\{a\}\],&U\_\{a\}=L\_\{a\}\.\\end\{cases\}\(6\)WhenUa\>LaU\_\{a\}\>L\_\{a\}, the CDF increases from00to11across the reported interval; whenUa=LaU\_\{a\}=L\_\{a\}, the label is exact and the distribution reduces to a point mass atLaL\_\{a\}\. For an instance withAAavailable expert\-reported labels, the target CDF gives each label equal weight:Q⁡\(y∣x\)=A−1​∑a∈𝒜⁡\(x\)Qa​\(y\)Q\(y\\mid x\)=A^\{\-1\}\\sum\_\{a\\in\\mathcal\{A\}\(x\)\}Q\_\{a\}\(y\)\. For interval labels, the corresponding density isq⁡\(y∣x\)=A−1​∑a∈𝒜⁡\(x\)Uniform⁡\(y,La,Ua\)q\(y\\mid x\)=A^\{\-1\}\\sum\_\{a\\in\\mathcal\{A\}\(x\)\}\\mathrm\{Uniform\}\(y;L\_\{a\},U\_\{a\}\)\. This construction retains the width of each reported interval and the differences among the intervals reported by different experts\. Overlapping intervals reinforce common support, while separated intervals remain separated rather than being merged into a single interval\.

The constructed target may contain both broad intervals and separated regions of probability when different experts report conflicting intervals\. The predictive model therefore needs a distribution that can represent this structure while respecting the bounded label space𝒴=\[0,1\]\\mathcal\{Y\}=\[0,1\]\. A Beta distribution is a continuous distribution on\[0,1\]\[0,1\], parameterized by two positive shape parametersα\\alphaandβ\\beta, making it suitable for modeling bounded label space\. A single Beta distribution, however, cannot represent multiple separated high\-probability regions without also assigning probability to the values between them\. We therefore model the predictive distribution as a mixture of Beta distributions\[[16](https://arxiv.org/html/2610.00102#bib.bib9)\],

pθ​\(y∣x\)=∑k=1Kπk​\(x\)​Beta​\(y,αk​\(x\),βk​\(x\)\),p\_\{\\theta\}\(y\\mid x\)=\\sum\_\{k=1\}^\{K\}\\pi\_\{k\}\(x\)\\,\\mathrm\{Beta\}\\bigl\(y;\\alpha\_\{k\}\(x\),\\beta\_\{k\}\(x\)\\bigr\),\(7\)whereπk​\(x\)≥0\\pi\_\{k\}\(x\)\\geq 0and∑kπk​\(x\)=1\\sum\_\{k\}\\pi\_\{k\}\(x\)=1, andαk​\(x\),βk​\(x\)\>0\\alpha\_\{k\}\(x\),\\beta\_\{k\}\(x\)\>0determine the shape of each component\. The mixture allows different components to represent different regions favored by the reported labels while maintaining a continuous predictive distribution over\[0,1\]\[0,1\]\.

We train the predictive distribution to match the full target constructed from the label set𝒟⁡\(x\)\\mathcal\{D\}\(x\)rather than a point summary\. LetFθ​\(y∣x\)F\_\{\\theta\}\(y\\mid x\)denote the predictive CDF, and we minimize the Cramér distance betweenFθF\_\{\\theta\}and the target CDF:

DCr​\(Fθ,Q\)=∫01\[Fθ​\(y∣x\)−Q⁡\(y∣x\)\]2​𝑑y\.D\_\{\\mathrm\{Cr\}\}\(F\_\{\\theta\},Q\)=\\int\_\{0\}^\{1\}\\left\[F\_\{\\theta\}\(y\\mid x\)\-Q\(y\\mid x\)\\right\]^\{2\}dy\.\(8\)This objective is closely related to the Continuous Ranked Probability Score \(CRPS\), a proper scoring rule for probabilistic predictions\. Specifically, the expected CRPS forY∼QY\\sim Qdiffers fromDCr​\(Fθ,Q\)D\_\{\\mathrm\{Cr\}\}\(F\_\{\\theta\},Q\)only by a term that depends onQQ\. Therefore, minimizing \([8](https://arxiv.org/html/2610.00102#S3.E8)\) is equivalent to minimizing expected CRPS with respect to the target distribution\[[36](https://arxiv.org/html/2610.00102#bib.bib8)\]\. This encourages the model to reproduce the full target distribution rather than only its mean\.

Beyond matching the target distribution, we also examine how uncertainty is represented within the predictive mixture\. Because the target uncertainty was separated into within\-label and between\-label terms in \([5](https://arxiv.org/html/2610.00102#S3.E5)\), we derive an analogous decomposition for the predictive distribution\. For mixture componentkk, let

μk=αkαk\+βk,σk2=αk​βk\(αk\+βk\)2​\(αk\+βk\+1\),\\mu\_\{k\}=\\frac\{\\alpha\_\{k\}\}\{\\alpha\_\{k\}\+\\beta\_\{k\}\},\\qquad\\sigma\_\{k\}^\{2\}=\\frac\{\\alpha\_\{k\}\\beta\_\{k\}\}\{\(\\alpha\_\{k\}\+\\beta\_\{k\}\)^\{2\}\(\\alpha\_\{k\}\+\\beta\_\{k\}\+1\)\},and letμ=∑kπk​μk\\mu=\\sum\_\{k\}\\pi\_\{k\}\\mu\_\{k\}\. By the law of total variance,

Var⁡\(Y∣x\)=∑kπk​σk2⏟sw2​\(x\)\+∑kπk​\(μk−μ\)2⏟sb2​\(x\)\.\\mathrm\{Var\}\(Y\\mid x\)=\\underbrace\{\\sum\_\{k\}\\pi\_\{k\}\\sigma\_\{k\}^\{2\}\}\_\{s\_\{w\}^\{2\}\(x\)\}\+\\underbrace\{\\sum\_\{k\}\\pi\_\{k\}\(\\mu\_\{k\}\-\\mu\)^\{2\}\}\_\{s\_\{b\}^\{2\}\(x\)\}\.\(9\)The first term,sw2s\_\{w\}^\{2\}, measures variation within predictive components, whereassb2s\_\{b\}^\{2\}measures variation between component means\. This decomposition has the same within/between structure as the quantities derived from the reported labels in \([5](https://arxiv.org/html/2610.00102#S3.E5)\):tw2t\_\{w\}^\{2\}andsw2s\_\{w\}^\{2\}describe within\-label and within\-component uncertainty, respectively, whiletb2t\_\{b\}^\{2\}andsb2s\_\{b\}^\{2\}describe between\-label and between\-component uncertainty\. We use a variance\-based rather than entropy\-based decomposition so thatsw2s\_\{w\}^\{2\}andsb2s\_\{b\}^\{2\}are directly comparable to the variancestw2t\_\{w\}^\{2\}andtb2t\_\{b\}^\{2\}\. However, the Cramér loss constrains only the overall predictive distribution, so this correspondence is not guaranteed\.

Because this correspondence is not guaranteed by the Cramér objective, we first evaluate whether it emerges in the unconstrained mixture\. We then introduce decomposition matching, which adds auxiliary loss terms that match the within\- and between\-component uncertainties to their label\-derived counterparts,tw2t\_\{w\}^\{2\}andtb2t\_\{b\}^\{2\}\. Letℐ\\mathcal\{I\}denote the set of valid training pixels andℐ2=\{x∈ℐ:A⁡\(x\)≥2\}\\mathcal\{I\}\_\{2\}=\\\{x\\in\\mathcal\{I\}:A\(x\)\\geq 2\\\}denote those with at least two expert\-reported labels\. The resulting objective is

ℒdecomp=⟨DCr⟩ℐ\+λw​⟨\|sw2\+ϵ−tw2\+ϵ\|⟩ℐ\+λb​⟨\|sb2\+ϵ−tb2\+ϵ\|⟩ℐ2,\\begin\{split\}\\mathcal\{L\}\_\{\\mathrm\{decomp\}\}=\{\}&\\left\\langle D\_\{\\mathrm\{Cr\}\}\\right\\rangle\_\{\\mathcal\{I\}\}\+\\lambda\_\{w\}\\left\\langle\\left\|\\sqrt\{s\_\{w\}^\{2\}\+\\epsilon\}\-\\sqrt\{t\_\{w\}^\{2\}\+\\epsilon\}\\right\|\\right\\rangle\_\{\\mathcal\{I\}\}\\\\ &\+\\lambda\_\{b\}\\left\\langle\\left\|\\sqrt\{s\_\{b\}^\{2\}\+\\epsilon\}\-\\sqrt\{t\_\{b\}^\{2\}\+\\epsilon\}\\right\|\\right\\rangle\_\{\\mathcal\{I\}\_\{2\}\},\\end\{split\}\(10\)whereϵ=10−6\\epsilon=10^\{\-6\}\. The within\-component term is aligned with within\-label uncertainty wherever a valid interval is available, whereas the between\-component term is aligned with between\-label uncertainty only when at least two labels are available\. With the Cramér objective alone, any correspondence arises naturally; under \([10](https://arxiv.org/html/2610.00102#S3.E10)\), it is explicitly enforced during training\.

Finally, we distinguish uncertainty represented within the predictive distribution from uncertainty associated with the learned model\. We trainRRindependently initialized and trained models as a deep ensemble\[[10](https://arxiv.org/html/2610.00102#bib.bib3)\]\. For ensemble memberrr, letπr​k\\pi\_\{rk\},μr​k\\mu\_\{rk\}, andσr​k2\\sigma\_\{rk\}^\{2\}denote the weight, mean, and variance of mixture componentkk, respectively\. The predictive mean of ensemble memberrrand the ensemble mean areμr=∑kπr​k​μr​k\\mu\_\{r\}=\\sum\_\{k\}\\pi\_\{rk\}\\mu\_\{rk\}andμ¯=1R​∑r=1Rμr\\bar\{\\mu\}=\\frac\{1\}\{R\}\\sum\_\{r=1\}^\{R\}\\mu\_\{r\}\. Applying the law of total variance over both mixture components and ensemble members gives

Var⁡\(Y∣x\)=\\displaystyle\\mathrm\{Var\}\(Y\\mid x\)=\{\}𝔼r​\[∑kπr​k​σr​k2\]⏟uw2\\displaystyle\\underbrace\{\\mathbb\{E\}\_\{r\}\\left\[\\sum\_\{k\}\\pi\_\{rk\}\\sigma\_\{rk\}^\{2\}\\right\]\}\_\{u\_\{w\}^\{2\}\}\(11\)\+𝔼r​\[∑kπr​k​\(μr​k−μr\)2\]⏟ub2\\displaystyle\+\\underbrace\{\\mathbb\{E\}\_\{r\}\\left\[\\sum\_\{k\}\\pi\_\{rk\}\(\\mu\_\{rk\}\-\\mu\_\{r\}\)^\{2\}\\right\]\}\_\{u\_\{b\}^\{2\}\}\+1R​∑r=1R\(μr−μ¯\)2⏟um2\.\\displaystyle\+\\underbrace\{\\frac\{1\}\{R\}\\sum\_\{r=1\}^\{R\}\(\\mu\_\{r\}\-\\bar\{\\mu\}\)^\{2\}\}\_\{u\_\{m\}^\{2\}\}\.where𝔼r​\[⋅\]\\mathbb\{E\}\_\{r\}\[\\cdot\]denotes the average over ensemble members\. The three terms describe within\-component, between\-component, and model \(epistemic\) variability, respectively\. The model termum2u\_\{m\}^\{2\}measures variation in the predictive mean across independently trained ensemble members\. On held\-out data, we evaluate their interpretation:uw2u\_\{w\}^\{2\}is compared against the interval\-derived within\-label uncertaintytw2t\_\{w\}^\{2\},ub2u\_\{b\}^\{2\}is compared against the between\-label uncertaintytb2t\_\{b\}^\{2\}and the complementary measure of between\-expert variationdsrcd\_\{\\mathrm\{src\}\}, andum2u\_\{m\}^\{2\}is compared against prediction error\.

## IVExperimental Evaluation

This section describes the multi\-expert SIC dataset, experimental setup, comparison baselines, and evaluation results\.

![Refer to caption](https://arxiv.org/html/2610.00102v1/figures/scene_charts.png)Fig\. 1:One Sentinel\-1 EW scene \(2025\-12\-26, Davis Strait\)\. \(a\) HH backscatter and \(b\) HV backscatter with polygon boundaries from CIS, DMI, and NIC\. \(c\)–\(d\) AMSR2 18\.7\-GHz H\- and 36\.5\-GHz V\-polarized brightness temperatures\. \(e\)–\(g\) SIC charts from CIS, DMI, and NIC, shown at the midpoint of each reported interval\. \(h\) Mean pairwiseW1W\_\{1\}distance among the available agency labels, concentrated near the marginal ice zone\.### IV\-ADataset

We construct a multi\-expert sea\-ice concentration \(SIC\) dataset by matching the same satellite observation with independently produced operational ice charts\. Trained expert ice analysts construct each ice chart by delineating polygons that represent regions of approximately homogeneous ice conditions and assigning a sea\-ice concentration \(SIC\) label to each polygon\. SIC denotes the fraction of the surface covered by sea ice, from 0 to 100%, and is reported using standardized World Meteorological Organization \(WMO\) codes that encode concentration as an interval rather than a single exact value\[[37](https://arxiv.org/html/2610.00102#bib.bib2)\]\. The labels are provided by four operational sources: the U\.S\. National Ice Center \(NIC\), the Danish Meteorological Institute \(DMI\), the Canadian Ice Service \(CIS\), and the National Oceanic and Atmospheric Administration \(NOAA\)\. We treat each independently produced agency chart as one expert\-reported label set for the corresponding satellite observation\. The width of each SIC interval represents within\-label uncertainty, while differences among the intervals reported by different agencies for the same observation represent between\-label uncertainty\. Figure[1](https://arxiv.org/html/2610.00102#S4.F1)illustrates this setting for one Sentinel\-1 observation and the independently produced expert charts\.

The agencies use different SIC vocabularies: NIC and NOAA predominantly use SIGRID\-3 range codes, whereas DMI and CIS use decile\-based intervals\. We therefore harmonize the labels to a common 11\-bin concentration representation before comparing expert reports \(e\.g\., NIC 46, i\.e\. 40–60%, vs\. DMI 40 becomes aW1W\_\{1\}of 1\.5 tenths after harmonization rather than a categorical mismatch\)\. Table[I](https://arxiv.org/html/2610.00102#S4.T1)presents the resulting pairwise comparison across 404 distinct scenes, of which 364 are charted by two agencies and 40 by three agencies; one DMI–NIC scene\-pair with no shared valid pixels is excluded\. Both the frequency and magnitude of between\-expert variation differ substantially across agency pairs\. DMI–NIC, CIS–NIC, and NIC–NOAA show relatively large meanW1W\_\{1\}distances of 0\.80–0\.89 tenths, whereas CIS–NOAA shows substantially closer agreement\. Across the reported label intervals, approximately 83% of target variance is between experts and 17% is within individual labels, providing label\-side references for evaluating the model uncertainty decomposition\.

The primary input is Sentinel\-1 Extra\-Wide \(EW\) Ground\-Range\-Detected Synthetic Aperture Radar \(SAR\) imagery, using HH and HV backscatter and local incidence angle at 40 m pixel spacing\. Sentinel\-1 C\-band SAR provides high\-resolution, all\-weather observations of sea\-ice surface and structural properties\. Scenes are thermal\-noise corrected, radiometrically calibrated, and reprojected to the NSIDC \(National Snow and Ice Data Center\) Sea\-Ice Polar Stereographic North grid \(EPSG:3413\)\. Auxiliary inputs include four AMSR2 passive\-microwave brightness\-temperature channels at 18\.7 and 36\.5 GHz in horizontal and vertical polarization\. Passive\-microwave brightness temperatures respond to differences in surface emissivity and physical temperature, providing contrast between open water, sea ice, and snow\-covered ice\. The polarization and frequency dependence also carries information about ice surface state, snow cover, and melt conditions\. Coarser auxiliary variables are aligned to the SAR grid and standardized channel\-wise\. Each Sentinel\-1 acquisition is paired with temporally valid, spatially overlapping operational ice charts\. We rasterize all charts to the common Sentinel\-1 grid, allowing pixelwise comparison without registration error\. Matching the agency archives yields 404 multi\-expert scenes from 2025, whose spatial and seasonal patterns in inter\-agency label differences are summarized in Fig\.[2](https://arxiv.org/html/2610.00102#S4.F2)\.

TABLE I:Inter\-expert comparison on shared pixels after label harmonization\. Scene counts denote scene–pair combinations rather than distinct scenes\. Harm\. mism\. denotes the fraction of pixels with differing harmonized labels, andW1W\_\{1\}measures the magnitude of label differences in tenths\.Fig\. 2:Inter\-agency SIC differences across 404 scenes\. Left: mean absolute SIC difference as a function of distance to the ice edge\. Right: monthly variation in mean absolute SIC difference for the DMI–NIC and NIC–NOAA agency pairs\.
### IV\-BExperimental Setup

All methods use the same U\-Net encoder–decoder architecture with four downsampling stages, channel widths32/64/128/25632/64/128/256, a 256\-channel bridge, bilinear upsampling, and skip connections\. The methods differ only in their output heads and training objectives\. Models are trained on random256×256256\\times 256crops with batch size 16 for 30 epochs of 400 iterations using AdamW with a learning rate of3×10−43\\times 10^\{\-4\}, weight decay of10−410^\{\-4\}, and cosine learning\-rate annealing\. The architecture, crop sampling, optimization schedule, and random seeds are fixed across methods, with no method\-specific hyperparameter tuning\. Model selection is based on the lowest validation MAE relative to the consensus midpoint\. Each experiment is repeated using three random seeds, with the resulting three models also serving as deep\-ensemble members for estimating model uncertainty\. Experiments are implemented in PyTorch 2\.6 with CUDA 12\.4 and run on NVIDIA RTX PRO 6000 GPUs\. The scenes are split into 282 training, 41 validation, and 81 test scenes\.

### IV\-CBaselines

We organize the baselines into four groups according to how they represent multiple expert labels and interval targets during training\. The first group consists of label\-aggregation baselines\. These methods use an 11\-bin SIC prediction and differ in how the expert labels are combined\. The Hard baseline uses the midpoint of one reference expert label, while Consensus uses the mean midpoint across the available experts\. The Soft baseline follows label\-distribution learning\[[8](https://arxiv.org/html/2610.00102#bib.bib10),[9](https://arxiv.org/html/2610.00102#bib.bib11)\]and averages the harmonized expert distributions asp¯​\(j\)=1A​∑a∈𝒜⁡\(x\)pa​\(j\)\\bar\{p\}\(j\)=\\frac\{1\}\{A\}\\sum\_\{a\\in\\mathcal\{A\}\(x\)\}p\_\{a\}\(j\)\. The second group contains interval\-distribution baselines\. These methods retain the expert\-reported intervals and represent the prediction with a single continuous Beta distribution\. They differ in how the expert intervals enter the loss: AvgNLL averages the interval negative log\-likelihood across experts, Cons\. uses the consensus range spanning the available expert intervals\. The third group contains deterministic interval\-regression baselines\. The limiting loss\(LIM\)\[[14](https://arxiv.org/html/2610.00102#bib.bib18)\]gives zero loss when the point prediction lies inside the reported interval and penalizes deviations outside it\. The Min–max method \(MM\)\[[15](https://arxiv.org/html/2610.00102#bib.bib19)\]minimizes the worst\-case squared error over values within the interval\. Unlike the distributional methods, these baselines produce a single point prediction\. Finally, we include a distribution\-matching control and twoK=2K=2variants\. TheK=1K=1model uses the same multi\-expert target and Cramér objective as theK=2K=2mixture but has no between\-component variation, isolating the effect of the mixture representation\. The baseK=2K=2model learns the decomposition from the interval target alone, while a secondK=2K=2variant adds decomposition matching withλ=1\\lambda=1\.

### IV\-DMetrics

Point accuracy is evaluated using mean absolute error \(MAE\) against the consensus midpoint,MAE=1N​∑i=1N\|y^i−y¯i\|\\mathrm\{MAE\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\|\\hat\{y\}\_\{i\}\-\\bar\{y\}\_\{i\}\|, wherey^i\\hat\{y\}\_\{i\}is the predictive mean andy¯i\\bar\{y\}\_\{i\}is the mean midpoint of the available expert labels\. MAE is reported in SIC tenths\. For probabilistic predictions, predictive entropy is defined asH=−∑jp\(j\)logp\(j\)H=\-\\sum\_\{j\}p\(j\)\\log p\(j\)on the common 11\-bin SIC representation\. Distributional accuracy is measured using the Cramér/expected\-CRPS score in \([8](https://arxiv.org/html/2610.00102#S3.E8)\), where lower values indicate closer agreement with the multi\-expert target\. Associations between predicted uncertainty and observed quantities are measured using Spearman rank correlation,ρs​\(u,z\)=Corr⁡\(rank⁡\(u\),rank⁡\(z\)\)\\rho\_\{s\}\(u,z\)=\\mathrm\{Corr\}\(\\mathrm\{rank\}\(u\),\\mathrm\{rank\}\(z\)\)\. We reportρ⁡\(u,W1\)\\rho\(u,W\_\{1\}\)to measure association with observed expert disagreement\. For the uncertainty decomposition, we additionally reportρ⁡\(sw,tw\)\\rho\(s\_\{w\},t\_\{w\}\)andρ⁡\(sb,tb\)\\rho\(s\_\{b\},t\_\{b\}\)to measure correspondence between the predicted and observed within\- and between\-label uncertainty, respectively\. For transfer experiments,ρtot=ρ⁡\(utot,W1\)\\rho\_\{\\mathrm\{tot\}\}=\\rho\(u\_\{\\mathrm\{tot\}\},W\_\{1\}\)andρb=ρ⁡\(ub,tb\)\\rho\_\{b\}=\\rho\(u\_\{b\},t\_\{b\}\)denote the corresponding total\- and between\-component associations\. Finally, selective prediction is evaluated using the excess area under the error\-retention curve \(E\-AURC\)\[[38](https://arxiv.org/html/2610.00102#bib.bib7)\]\. Let𝒮u​\(c\)\\mathcal\{S\}\_\{u\}\(c\)contain the fractionccof predictions with the lowest uncertainty\. The retained error isRu​\(c\)=1\|𝒮u​\(c\)\|​∑i∈𝒮u​\(c\)\|y^i−y¯i\|R\_\{u\}\(c\)=\\frac\{1\}\{\|\\mathcal\{S\}\_\{u\}\(c\)\|\}\\sum\_\{i\\in\\mathcal\{S\}\_\{u\}\(c\)\}\|\\hat\{y\}\_\{i\}\-\\bar\{y\}\_\{i\}\|, andE​\-​AURC=∫01\[Ru​\(c\)−Roracle​\(c\)\]​𝑑c\\mathrm\{E\\text\{\-\}AURC\}=\\int\_\{0\}^\{1\}\[R\_\{u\}\(c\)\-R\_\{\\mathrm\{oracle\}\}\(c\)\]\\,dc, where the oracle retains predictions in increasing order of their actual error\.

### IV\-EMain Results

TABLE II:Point accuracy across supervision regimes and input sets: MAE \(tenths\) mean±\\pmSD over three seeds on test data\.Table[II](https://arxiv.org/html/2610.00102#S4.T2)compares how input information and learning strategy affect point accuracy\. The rows use SAR \(3 channels\) and SAR\+AMSR2 \(7 channels\), while the columns compare label aggregation, interval distribution methods, deterministic interval regression, and distribution matching withK=1K=1andK=2K=2components\. Smaller MAE indicates better accuracy\. Retaining richer label information generally improves performance\. Within label aggregation, soft labels outperform hard labels in both input settings, reducing MAE from2\.2212\.221to1\.8791\.879for SAR and from0\.8090\.809to0\.7690\.769for SAR\+AMSR2\. The interval\-based consensus distribution further improves on aggregation\-based supervision \(1\.8201\.820and0\.6410\.641\), indicating that preserving the interval structure of the annotations, rather than collapsing them to a single target, benefits point accuracy\. The deterministic LIM and MM baselines are weaker than distribution matching when AMSR2 is available \(0\.6940\.694and0\.7320\.732vs\.0\.5670\.567forK=2K=2and0\.5590\.559with decomposition\), and, unlike the mixture\-based models, they do not yield predictive distributions that support subsequent uncertainty analysis\. The benefit of additional mixture components depends on the input:K=2K=2outperformsK=1K=1with SAR\+AMSR2 \(0\.5670\.567vs\.0\.7090\.709\) but underperforms it with SAR alone \(1\.7141\.714vs\.1\.4841\.484\)\. The proposed decomposition resolves this: it recovers and surpasses theK=1K=1result for SAR \(1\.2301\.230, the lowest SAR\-only error\) and further reduces the SAR\+AMSR2 MAE to0\.5590\.559, the lowest in the table\. Decomposition supervision thus stabilizes the mixture when the input alone cannot disambiguate the components, which explains its larger gain in the more ambiguous SAR\-only setting\. Finally, adding AMSR2 consistently improves all methods, reducing MAE by approximately 52–69% relative to the SAR\-only models \(e\.g\.,2\.2212\.221to0\.8090\.809under hard supervision\), confirming that the passive microwave channels carry substantial complementary information independent of the learning strategy\.

Figure[3](https://arxiv.org/html/2610.00102#S4.F3)illustrates why a single component is insufficient when expert annotations are separated\. The target consists of two disjoint expert intervals with an unsupported region between them\. Minimizing the Cramér objective forces a single Beta to spread mass across this gap \(0\.610\.61\), whereas the two\-component mixture preserves both modes, reduces the gap mass to0\.200\.20, and lowers the CRPS by19×19\\times\. The CDF view \(bottom\) makes the mechanism explicit: only the mixture can reproduce the flat plateau of the target CDF over the unsupported region\. This failure mode is exactly the regime addressed by theK=2K\{=\}2model in Table[II](https://arxiv.org/html/2610.00102#S4.T2)\.

![Refer to caption](https://arxiv.org/html/2610.00102v1/figures/mixture_example.png)Fig\. 3:Why one component is insufficient for separated labels\. Both models use the same target and Cramér objective\. A single Beta fills the unsupported gap between the two agency intervals, whereas the mixture preserves the two modes and reduces CRPS by 19×\\times\.TABLE III:Predictive uncertainty alignment withW1W\_\{1\}and model error under the 7\-channel setting\.HHdenotes tenth\-bin predictive entropy anduupredictive width\.ρ⁡\(H,err\)\\rho\(H,\\mathrm\{err\}\)and E\-AURC useHHfor ranking in all methods\.Table[III](https://arxiv.org/html/2610.00102#S4.T3)evaluates what predictive uncertainty captures under the SAR\+AMSR2 input setting by comparing its correlation withW1W\_\{1\}, its correlation with model error, and its selective\-prediction performance \(E\-AURC\)\. Alignment withW1W\_\{1\}is strongest for the mixture\-based models, and decomposition strengthens it further\. The proposedK=2\+decompK\{=\}2\{\+\}\\mathrm\{decomp\}model achieves the highest correlations withW1W\_\{1\}for both entropy \(ρ⁡\(H,W1\)=0\.397\\rho\(H,W\_\{1\}\)=0\.397\) and predictive width \(ρ⁡\(u,W1\)=0\.402\\rho\(u,W\_\{1\}\)=0\.402\), doubling the width–W1W\_\{1\}correlation of the plainK=2K\{=\}2model \(0\.2000\.200\) and reducing its variance across seeds\. In contrast, hard and consensus categorical supervision yield uncertainty that is essentially uninformative aboutW1W\_\{1\}\(ρ⁡\(H,W1\)=0\.015\\rho\(H,W\_\{1\}\)=0\.015and0\.0750\.075, respectively\), which is expected because these targets collapse inter\-expert label variation during training\. The pattern reverses for error alignment\. The methods with the weakest correlations withW1W\_\{1\}—AvgNLL, hard, and consensus—show the strongest correlation between entropy and error \(0\.8310\.831,0\.7470\.747, and0\.7440\.744\) and the best selective\-prediction performance, with AvgNLL attaining the lowest E\-AURC \(0\.0820\.082\)\. The distribution\-matching models \(K=1K\{=\}1,K=2K\{=\}2,K=2K\{=\}2\+decomp\.\), whose uncertainty better reflects variation in the expert labels, correlate only weakly with error \(0\.1350\.135–0\.4430\.443\) and rank examples less effectively \(E\-AURC≥0\.204\\geq 0\.204\)\. Together, these results indicate that uncertainty aligned with expert\-label variation and uncertainty aligned with model error are complementary rather than interchangeable: objectives that preserve inter\-expert label variation yield uncertainty that better reflects label ambiguity, whereas objectives that collapse this variation can yield uncertainty that is more effective for ranking prediction errors\. The decomposed model is thus preferable when uncertainty is intended to represent ambiguity in the supervision, whereas AvgNLL is preferable when uncertainty is used to identify predictions likely to have large errors\.

TABLE IV:Validation of the three\-way uncertainty decomposition on test data \(SAR\+AMSR2\)\. Entries are Spearmanρ\\rho; bold indicates the intended reference\.umu\_\{m\}is computed across seeds\.Table[IV](https://arxiv.org/html/2610.00102#S4.T4)compares each predictive component with four references:twt\_\{w\}measures within\-label uncertainty,tbt\_\{b\}between\-expert variation,dsrcd\_\{\\rm src\}direct expert disagreement, anderr\\mathrm\{err\}prediction error\. For plainK=2K\{=\}2,uwu\_\{w\}aligns well withtwt\_\{w\}\(0\.667±0\.0130\.667\\pm 0\.013\), butubu\_\{b\}aligns weakly withtbt\_\{b\}anddsrcd\_\{\\rm src\}\(0\.1070\.107and0\.1490\.149\) and instead tracks error \(0\.6350\.635\), as the Cramér objective constrains only the overall CDF;umu\_\{m\}also tracks error \(0\.6890\.689\)\. With decomposition,ubu\_\{b\}becomes substantially better aligned withtbt\_\{b\}anddsrcd\_\{\\rm src\}\(0\.5000\.500and0\.3960\.396\) while its error correlation falls to0\.2330\.233, andumu\_\{m\}loses its error association \(0\.0980\.098\) as ensemble members converge\. Cross\-correlations remain, showing improved source alignment rather than complete disentanglement\.

TABLE V:Transfer performance under cross\-agency and temporal shifts\. A dash indicates that no between\-component uncertainty channel is available\.Table[V](https://arxiv.org/html/2610.00102#S4.T5)evaluates point accuracy and uncertainty alignment under cross\-agency and temporal shifts\. Here,ρtot=ρ⁡\(utot,W1\)\\rho\_\{\\rm tot\}=\\rho\(u\_\{\\rm tot\},W\_\{1\}\)measures alignment of total predictive uncertainty with expert disagreement, whileρb=ρ⁡\(ub,tb\)\\rho\_\{b\}=\\rho\(u\_\{b\},t\_\{b\}\)measures alignment of between\-component and between\-label uncertainty\. Under temporal transfer \(train on NIC charts alone from 2018–2024, test on the 2025 multi\-agency scenes\), bothK=2K\{=\}2variants outperform the categorical baselines \(0\.6260\.626and0\.5620\.562vs\.1\.4751\.475and1\.3461\.346\), with the decomposed model lowest and most stable \(0\.562±\.0160\.562\\,\{\\scriptstyle\\pm\.016\}\)\. The uncertainty channels also transfer: total uncertainty stays correlated with expert disagreement \(ρtot≈0\.40\\rho\_\{\\rm tot\}\\approx 0\.40\), and the between\-component channel aligns even more strongly with between\-label uncertainty \(ρb=0\.519\\rho\_\{b\}=0\.519and0\.5280\.528\), a signal unavailable to hard and soft baselines\. Cross\-agency transfer is harder: decomposition still lowers MAE \(0\.6840\.684vs\.0\.7270\.727, within seed variance\), but both alignment correlations drop \(ρtot≤0\.150\\rho\_\{\\rm tot\}\\leq 0\.150,ρb≤0\.197\\rho\_\{b\}\\leq 0\.197\)\. Point predictions transfer to unseen agency pairs, but the uncertainty channels reflect the annotation conventions of the training agencies\.

TABLE VI:Irreducible MAE and model error by distance to the ice edge \(SAR\+AMSR2\), evaluated against individual agency charts on pixels labeled by≥2\\geq 2agencies\. Ours \(K=2K\{=\}2\+ decomp\.\)Table[VI](https://arxiv.org/html/2610.00102#S4.T6)compares the irreducible MAE estimated from inter\-expert label variation \(floor\) with observed model error across distance\-to\-ice\-edge strata\. The floor is strongly stratified: it is highest immediately at the ice edge \(1\.3391\.339within00–1010km\) and decays monotonically with distance \(0\.1800\.180with no edge present\), mirroring the inter\-agency label\-difference pattern in Fig\.[2](https://arxiv.org/html/2610.00102#S4.F2)and confirming that irreducible error concentrates where annotators themselves diverge\. The proposed model attains lower MAE than hard supervision in every stratum and operates closer to the floor throughout, raising the Floor/MAE ratio from38%38\\%to56%56\\%pooled\. The gains are largest where the error is most reducible: far from the edge, the decomposed model recovers much of the headroom left by hard supervision, whereas near the edge both models already sit close to the floor \(59%59\\%and61%61\\%within00–1010km\), suggesting that the remaining error is largely attributable to label ambiguity rather than model deficiency\. These strata separate two error regimes: away from the edge, errors are largely reducible and the proposed supervision reduces them; at the edge, inter\-expert label variation imposes a larger irreducible error floor, leaving less room for improvement through additional modeling\.

TABLE VII:Synthetic identifiability study\. noise replaces the feature with pure noise\.Table[VII](https://arxiv.org/html/2610.00102#S4.T7)tests whether between\-component uncertainty recovers latent between\-expert variation\. Both variants remain at the ceiling across noise levels, while the noise control gives near\-zero correlation\. Decomposition leaves this correlation unchanged but shifts variance into the between\-component channel, whose share increases withσs\\sigma\_\{s\}\(0\.8320\.832–0\.9490\.949\) instead of decreasing as in plainK=2K\{=\}2\(0\.7830\.783–0\.6180\.618\)\. Thus, the variance share is meaningful only when interpreted alongsideρ⁡\(sb,s\)\\rho\(s\_\{b\},s\)\.

#### IV\-E1Sensitivity Analysis

TABLE VIII:Sensitivity to the number of mixture components under the SAR\+AMSR2 input setting\. ForK≥2K\\geq 2,ρb=ρ⁡\(sb,tb\)\\rho\_\{b\}=\\rho\(s\_\{b\},t\_\{b\}\)andρerr=ρ⁡\(sb,err\)\\rho\_\{\\rm err\}=\\rho\(s\_\{b\},\\mathrm\{err\}\)\.Table[VIII](https://arxiv.org/html/2610.00102#S4.T8)varies mixture size at seven channels\. MAE improves fromK=1K\{=\}1toK=2K\{=\}2\(0\.7090\.709to0\.5670\.567\) and then saturates, while CRPS is best atK=2K\{=\}2\(0\.02430\.0243\)\. For allK≥2K\\geq 2,Keff≈1\.1K\_\{\\rm eff\}\\approx 1\.1, indicating little use of additional components\. Althoughρb\\rho\_\{b\}increases withKK, its seed variance remains large, whileρerr\\rho\_\{\\rm err\}peaks atK=2K\{=\}2\(0\.6350\.635\)\. We therefore useK=2K\{=\}2throughout the paper, as it gives the best CRPS with comparable MAE to larger mixtures\.

TABLE IX:Sensitivity to decomposition weights for theK=2K=2mixture under the SAR\+AMSR2 input setting\. Here,ρb=ρ⁡\(sb,tb\)\\rho\_\{b\}=\\rho\(s\_\{b\},t\_\{b\}\)andρerr=ρ⁡\(sb,err\)\\rho\_\{\\rm err\}=\\rho\(s\_\{b\},\\mathrm\{err\}\)\.Table[IX](https://arxiv.org/html/2610.00102#S4.T9)varies the decomposition weights\. Onlyλb\\lambda\_\{b\}improves between\-label alignment \(ρb\\rho\_\{b\}0\.107→0\.4910\.107\\to 0\.491at\(0,1\)\(0,1\);λw\\lambda\_\{w\}alone leaves it near zero\), but supervisingubu\_\{b\}alone degrades within\-label alignment \(0\.667→0\.4540\.667\\to 0\.454\); supervising both restores it, with\(1,1\)\(1,1\)aligning both components \(0\.5000\.500,0\.6770\.677\) at MAE comparable to no supervision \(0\.5590\.559vs\.0\.5670\.567\)\. Two effects bound the useful range:ρerr\\rho\_\{\\rm err\}falls asρb\\rho\_\{b\}rises \(0\.635→−0\.1930\.635\\to\-0\.193\), and over\-weighting hurts accuracy \(0\.6630\.663at\(3,3\)\(3,3\)\)\.\(1,1\)\(1,1\)with only a0\.0040\.004MAE increase over the best setting\(0\.3,0\.3\)\(0\.3,0\.3\), well within seed variability; we therefore adopt\(1,1\)\(1,1\)\.

## VConclusion

We introduce a framework for learning from multiple expert interval labels while distinguishing within\-label uncertainty, between\-label uncertainty arising from variation across experts, and predictive model uncertainty\. By harmonizing heterogeneous label conventions and retaining the full interval supervision, the proposed Mixture\-of\-Betas model reduced MAE by45%45\\%with SAR alone and31%31\\%with SAR\+AMSR2 relative to hard supervision, outperforming single\-component and deterministic interval\-target baselines\. Our results show that predictive uncertainty components do not automatically align with their intended sources\. Decomposition matching substantially improved alignment between between\-component and between\-label uncertainty and increased the width–W1W\_\{1\}correlation without degrading point accuracy\. Uncertainty aligned with between\-expert variation and model error remained complementary rather than interchangeable\. Future work will extend the framework to spatially structured uncertainty, semi\-supervised decomposition when multi\-expert labels are available for only a subset of scenes, and expert\-conditioned modeling of agency\-specific disagreement\. A further direction is to jointly represent uncertainty associated with between\-expert variation and model error within a single predictive distribution\.

## Acknowledgment

This research was funded by the National Science Foundation \(NSF\) under grant number 2531101\.

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