Decision-Target Reversal under Informative Label Missingness

Farinborz Setoudehtazangi; Geoffrey J. McLachlan

stat.MLOct 18, 2026

Decision-Target Reversal under Informative Label Missingness

Authors: Farinborz Setoudehtazangi, Geoffrey J. McLachlan

Organizations: Dipartimento di Scienze Statistiche, Università di Padova, Padova, Italy · School of Mathematics and Physics, The University of Queensland, Brisbane, Australia

Abstract

Informative label missingness can improve classification efficiency because the pattern of label availability may itself contain information about the discriminant structure. We show that this advantage need not persist across deployment objectives: with the data-generating model and statistical experiments held fixed, informative partial classification can be favourable under one loss and unfavourable under another. We call this phenomenon decision-target reversal. For binary Gaussian discrimination, we derive a local cost-sensitive excess-risk expansion and, in one dimension, obtain explicit critical cost ratios at which the first-order risk ordering reverses. We show that the required covariance geometry arises under finite uncertainty-dependent missingness mechanisms with a fixed marginal missing-label proportion. We then extend the analysis to multiclass finite-action problems through loss-specific active-boundary curvature, yielding a general decision-target frontier. Monte Carlo experiments illustrate the predicted reversals, and a descriptive gastrointestinal-lesion application with agreement-defined label availability illustrates sensitivity to deployment costs. The results show that favourable informative missingness is a property of the statistical experiment relative to the deployment decision, rather than of the experiment alone.

Figures & tables

Figure 1: Finite-sample decision-target reversal for

\pi_{1}=\pi_{2}=1/2

\Delta=1.5

\Pr(M=1)=0.30

, and

\xi_{1}=-0.25

, based on 5000 paired replications at

n=2000

, with

C_{21}=1

. The solid curve estimates

n\widehat{\Delta R}_{n}(c)

; the dashed curve is the quadratic comparison using the empirical covariance difference

\widehat{\bm{D}}_{n}

; and the dotted curve is the population first-order approximation. The blue ribbon gives pointwise 95% Monte Carlo intervals calculated from paired risk differences; it is not a simultaneous band. The grey region and vertical lines indicate the population favourable interval, bounded by

c_{\rm low}=0.214374

and

c_{\rm high}=4.664740

. Negative values favour partial classification and positive values favour complete classification.

	Monte Carlo risk		Empirical $\widehat{\bm{D}}_{n}$		Population theory
$n$	$c_{\rm low}$	$c_{\rm high}$	$c_{\rm low}$	$c_{\rm high}$	$c_{\rm low}$	$c_{\rm high}$
500	0.2928	4.2418	0.2874	4.6125	0.2144	4.6647
1000	0.3043	3.3520	0.2993	3.3785	0.2144	4.6647
2000	0.2310	5.2777	0.2269	5.2774	0.2144	4.6647

Table I: Critical cost ratios obtained from the Monte Carlo deployment-risk curve, from the first-order approximation using the empirical covariance difference

\widehat{\bm{D}}_{n}

, and from the population theory. The simulation uses

5000

replications for each sample size. The Monte Carlo crossings are obtained by linear interpolation of the mean risk difference in

\log c

on the evaluated cost grid.

Figure 2: Gastrointestinal-lesion application under a shared-covariance Gaussian specification. Panel (a) shows the first two standardized principal components of the four previously selected features, with histological class indicated by colour and label availability by symbol; the projection is used only for visualization. Panel (b) shows entropy from the held-out FULL posterior predictions. Panel (c) shows the empirical leave-one-out risk difference

\widehat{R}_{\mathrm{FULL}}(c)-\widehat{R}_{\mathrm{CC}}(c)

. Negative values favour FULL and positive values favour CC. The inset shows the narrow reversals and tie interval at lower cost ratios. For the held-out predictions, preprocessing, initialization and fitting use training observations only, and all 76 paired fits pass the numerical checks. The principal-component projection in panel (a) uses all lesions for visualization only. Cost ratios are sensitivity parameters rather than clinical recommendations.

Learning with Missing Data

Learning with missing data develops methods to estimate, predict, and make decisions when variables, labels, modalities, or time points are unobserved.

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Aug 31, 2026J/K move · Enter open · S save

Informative Label Missingness in Multiclass Classification Information Geometry and Excess Risk

Fariborz Setoudehtazang, Geoffrey J. McLachlan

Dipartimento di Scienze Statistiche, Universit`a di Padova, Padova, Italy · School of Mathematics and Physics, The University of Queensland, Australia

Information Geometry Learning with Missing Data Multiclass Classification Classification Statistical Learning Theory

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Informative label missingness can change the usual efficiency ordering between completely and partially labelled classifiers because the pattern of missing labels may itself carry information about the classification model. We develop a general likelihood-based theory for this phenomenon in parametric multiclass classification. An efficient-information decomposition separates information lost through unavailable class memberships from information contributed by the missing-label mechanism. We then derive a quadratic expansion of plug-in excess risk over the active pairwise faces of the multiclass Bayes boundary, showing that classification efficiency depends on how information gains and losses align with directions that perturb the decision boundary. This yields a classification-weighted generalized-eigenvalue criterion under which informative partial classification may have smaller asymptotic classification risk without globally dominating complete classification in Fisher information. Near missing completely at random, with the marginal missing-label proportion fixed, redistribution of missing labels changes lost class-label information at first order, whereas efficient information from the missingness pattern appears only at second order. Three-class quadratic discriminant calculations, finite-sample experiments, and a semi-synthetic multiclass application illustrate the resulting regime-dependent behaviour.

Abstract

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