Decision-Target Reversal under Informative Label Missingness

Introduction

Partially classified samples arise when feature measurements are observed for all training units but class labels are available for only a subset. This setting is common when definitive classification requires expert assessment, laboratory testing, follow-up, or another procedure that is more difficult or costly than obtaining the features themselves. In likelihood-based semi-supervised classification, observations without class labels can still contribute through the marginal feature distribution, but the unavailable labels ordinarily represent a loss of information about the discriminant structure.

This usual efficiency ordering can change when label availability is informative. Following the missing-data framework of Rubin [1], Ahfock and McLachlan [2] considered Gaussian discrimination in which the probability that a class label is missing depends on classification uncertainty. Because the missingness mechanism shares parameters with the classifier, the observed pattern of label availability can itself carry information about the discriminant model. The information supplied by the missingness indicators may then compensate, partly or more than fully, for the information lost through unavailable class labels. Consequently, an informatively partially classified sample can have smaller asymptotic classification risk than the corresponding ordinary completely classified sample.

Throughout this paper, complete classification (CC) refers to the ordinary fully labelled experiment based on $(Y, Z)$, without treating a mechanism-generated missingness indicator as an additional observation. This distinction is important. If both all class labels and the missingness indicator were observed, giving the augmented experiment $(Y, Z, M)$, the partially classified observation would be a coarsening of that experiment and could not dominate it. The favourable missingness phenomenon concerns comparison with the ordinary CC benchmark.

Subsequent work has clarified the information structure underlying this phenomenon and extended it beyond the original Gaussian setting. Information decompositions separate the conditional class-label information lost through missingness from the information contributed by the observed missingness indicators [3]. Related analyses have established favourable-missingness effects in other parametric classification models [4], [5]. More generally, Setoudehtazangi and McLachlan [6] developed a nuisance-adjusted information decomposition together with the local second-order geometry of multiclass zero–one classification risk. This analysis builds on the classical excess-risk approach to discriminant efficiency, in which the expected increase in classification error due to estimation of the discriminant rule admits an $O(n^{-1})$ expansion [7]. In the informative-missingness setting, favourable performance does not require improvement in every parameter direction: classification efficiency depends on how the resulting estimation uncertainty projects onto directions that perturb the active Bayes decision boundary.

These results compare statistical experiments under a fixed classification objective. At deployment, different errors may carry different losses, and changing those losses changes the Bayes action even when the data-generating distribution remains fixed [8]. Dependence of classifier comparisons on misclassification costs is well established. Adams and Hand [9] studied comparisons under uncertain costs, and Drummond and Holte [10] developed cost curves for comparing classifiers across operating conditions. Under correct posterior specification, a fixed fitted model can also be deployed under different costs by changing its posterior threshold [11].

Cost-sensitive learning with incompletely labelled samples has been studied through semi-supervised support vector machines and EM-based procedures [12], [13]. Our contribution concerns the first-order comparison of complete classification and classifier-linked informative partial classification: we derive explicit deployment-cost crossings and prove that their required covariance geometry is generated by a finite missingness mechanism at a fixed marginal missing-label proportion.

We study precisely this question. The data-generating model, the informative missingness mechanism, and the two statistical experiments are held fixed. Only the deployment loss is varied. We show that the resulting classification-risk ordering need not be invariant: the same informative partially classified experiment can be favourable for one deployment target and unfavourable for another. We refer to this phenomenon as decision-target reversal.

The possibility of target-dependent ordering is consistent with the classical theory for comparison of statistical experiments [14], [15], [16]. Blackwell dominance gives a uniform risk ordering over decision problems; in its absence, a pair of experiments need not retain the same ordering under different losses. Our aim is not to restate this general decision-theoretic fact, but to identify the classification-specific mechanism that produces reversal under informative label missingness, to characterize explicitly the deployment losses at which the ordering changes, and to determine when the covariance geometry required for such a reversal can arise from the missingness mechanism itself. The mechanism is transparent in binary Gaussian discrimination. Let

denote the difference between the asymptotic covariance matrices of the classifier parameters under informative partial and complete classification. Informative missingness can reduce estimation uncertainty in some parameter directions while increasing it in others, so $D$ may be indefinite. For a binary cost matrix $C$, let $t_C$ denote the corresponding Bayes decision boundary. We show that the leading difference in deployment risk is governed by

Thus the statistical experiment determines the fixed covariance geometry $D$, whereas the deployment loss determines which part of that geometry is relevant to classification by moving the Bayes boundary. Changing the loss can therefore change the sign of the risk comparison without changing either training experiment.

In one dimension, this representation yields explicit critical cost ratios at which the ordering reverses. We also show that the required indefinite covariance geometry is not an artificial assumption: for a finite uncertainty-dependent missingness mechanism with a fixed marginal missing-label proportion, it occurs over a nonempty open region of the parameter space. The phenomenon is therefore structural rather than confined to a specially chosen numerical example.

The risk comparison is not specific to informative label missingness: it applies to any two regular estimators of the same model. Informative missingness provides the concrete likelihood construction for which we establish the covariance geometry required for reversal. The same principle extends to multiclass finite-action classification. For a loss matrix $C$, let $H_C$ denote the local curvature of the corresponding population Bayes risk. Under regularity conditions, an estimator with asymptotic covariance $V$ satisfies

For informative partial versus complete classification, the leading comparison is therefore

This yields the decision-target frontier

The general formulation also reveals a distinction from the one-dimensional binary case: indefiniteness of $D$ alone does not guarantee reversal, because the family of curvature matrices generated by admissible losses need not probe both favourable and unfavourable directions of $D$.

The paper therefore characterizes how deployment costs alter the first-order comparison of complete and informative partial classification. We derive the local cost-sensitive excess-risk expansion for binary Gaussian discrimination and explicit reversal thresholds in one dimension; establish that the covariance geometry required for reversal is generated by finite informative-missingness mechanisms at a fixed marginal missing-label proportion; and extend the analysis to multiclass finite-action problems through loss-specific active decision-boundary curvature. Monte Carlo experiments illustrate the predicted reversal at finite sample sizes, including under unequal class probabilities. A descriptive gastrointestinal-lesion application examines cost sensitivity when label availability is determined by expert agreement. The application treats the relative error costs as sensitivity parameters, rather than as estimates of clinical utility.

The remainder of the paper is organized as follows. Section 2 develops the loss-specific local risk geometry for binary classification and characterizes decision-target reversal. Section 3 derives the efficient information under informative label missingness and establishes the existence of the required covariance geometry. Section 4 gives the multiclass finite-action extension. Section 5 presents the population and finite-sample numerical results, Section 6 gives the gastrointestinal-lesion application, and Section 7 concludes.

Decision-target-dependent classification risk

Let $Z \in\{1,2\}$ denote the class label and $Y \in\mathbb{R}^{p}$ the feature vector. We begin with the canonical homoscedastic Gaussian discrimination model

where $\Delta>0$ and

Writing

the posterior log-odds at the true parameter value are

Suppose that the consequences of the two types of misclassification are not necessarily equal. Let $C_{12}>0$ denote the loss incurred by assigning an observation from class 2 to class 1, and let $C_{21}>0$ denote the loss incurred by assigning an observation from class 1 to class 2. Correct classifications have zero loss. The Bayes rule assigns an observation to class 1 whenever

or, equivalently,

Its decision boundary is therefore

Thus changing the relative misclassification losses moves the deployment boundary even though the underlying statistical model is unchanged.

Consider a local perturbation of the discriminant,

where $\mathbf{h}=(h_0,\ldots,h_p)^\mathsf{T}$ is close to zero. Let $R_C(\mathbf{h})$ denote the expected deployment loss of the classifier induced by (5), evaluated under the true distribution, and let $R_C^*=R_C(0)$ denote the Bayes risk.

The following result gives the local cost-sensitive analogue of the classical excess-error expansion for normal discrimination.

The proof is given in Supplementary Section ??.

To connect the local expansion with estimation uncertainty, parameterize the fitted discriminant by

Thus the perturbation in (5) is $\mathbf{h}=\boldsymbol{\beta}-\boldsymbol{\beta}^*$. The constant $\lambda$ centers the true intercept for the analysis; the fitted intercept $\lambda+\widehat{\beta}_0$ is estimated, and the true class probabilities are not supplied as known quantities. Suppose that an estimator $\widehat{\boldsymbol{\beta}}$ in these discriminant coordinates satisfies

together with the moment conditions required to average the local quadratic expansion. Proposition 1 then yields

where $V=(v_{jk})$.

Remark 2 (Connection with the classical excess-error expansion). For equal misclassification costs, $C_{12}=C_{21}$, we have $r=0$ and hence

Consequently, the covariance contribution in (8) reduces to

which is the covariance combination appearing in the classical excess-error expansion of Efron [7] and subsequently used by Ahfock and McLachlan [2] for the ordinary zero-one classification comparison. Thus the equal-cost case is a single point in the loss-indexed family considered here. Varying the deployment loss moves the Bayes decision boundary and thereby changes the covariance projection relevant to classification.

Consider two statistical experiments with asymptotic covariance matrices $V_1$ and $V_2$, and write

Since $m_C/(2\Delta)>0$, their first-order deployment-risk ordering is determined by

The statistical experiment is fixed in (9); only the deployment loss, through $t_C$, changes. Consequently, an efficiency ordering obtained under one decision target need not persist under another.

For $p=1$, this dependence can be characterized exactly.

When $D_D > 0$, the first-order deployment-risk ordering changes at the two boundary locations

The corresponding positive cost ratios are

Writing

gives

When $D_D=0$, the quadratic is tangent to zero and there is no genuine sign reversal, whereas $D_D<0$ yields no real critical boundary location.

If $d_{11}=0$ and $d_{01}\ne0$, there is a single critical boundary location

and hence a single positive critical cost ratio

If $d_{11}=d_{01}=0$, the first-order comparison is independent of the decision target.

In particular, if $D$ is indefinite, then

and there exist positive cost ratios for which the two experiments have opposite first-order deployment-risk orderings.

The proof is given in Supplementary Section ??.

Theorem 3 separates two questions that are identical under a fixed loss but distinct when the deployment objective varies. The covariance difference $D$ describes how the statistical experiment changes estimation uncertainty, whereas the cost ratio determines which directions of that uncertainty perturb the relevant decision boundary. An indefinite $D$ therefore does not assign an intrinsic “better” experiment: the answer depends on the decision target.

Informative label missingness

Let $M$ denote the label-missingness indicator, with $M=1$ when the class label is unavailable. We assume

and allow the probability of a missing label to depend on the classifier through

where $\beta$ denotes the parameters governing classification and $\xi$ contains parameters specific to the missingness mechanism. Thus the missingness indicator may itself carry information about $\beta$.

We compare two experiments. Under complete classification,

is observed. Under partial classification,

is observed. These are distinct statistical experiments: in $\mathcal{E}_{\mathrm{PC}}$, the missingness indicator is an observed, parameter-linked random variable, whereas complete classification refers to direct availability of the class label.

For the information identities below, $\beta$ indexes the full distribution of $(Y,Z)$, and $\xi$ is the only additional nuisance parameter. Additional unknown location or covariance parameters require the full generative information calculation and subsequent nuisance adjustment or projection into discriminant coordinates.

Let $I_Y$ denote the Fisher information for $\beta$ in the marginal distribution of $Y$, and let

denote the conditional information in the class label. The information under complete classification is

Under partial classification, an unavailable label removes its conditional label information, but the missingness event contributes information whenever $q$ depends on $\beta$. After eliminating the nuisance parameter $\xi$, the efficient information is

where

The two terms on the right-hand side of (13) have different roles. The first is the information lost through unavailable class labels; the second is the classifier information retained in the pattern of label availability after nuisance variation in the missingness mechanism has been removed.

More generally, if the generative parameter is $\theta$ and the discriminant coefficients are $\beta=\psi(\theta)$, apply (13)–(14) in the full $\theta$ coordinates. If $K_{\mathrm{PC}}$ is the resulting information after eliminating $\xi$, then

These covariance matrices follow from nuisance elimination and the delta method; see Setoudet h zangi and McLachlan [6].

To expose the resulting geometry, consider the two-parameter Gaussian likelihood

where $b\in\mathbb{R}$ and $\delta>0$ are estimated. The midpoint of the two means and the within-class variance are known and fixed at zero and one, respectively. These are restrictions on the fitted likelihood, not consequences of standardizing the true distribution. The fitted posterior log-odds are $b+\delta y$.

The symmetric configuration has true values $b = 0$ and $\delta= \Delta$, so that $\pi_1 = \pi_2 = 1/2$ and $\eta_0(y) = \Delta y$. Equal class probabilities are not imposed during fitting. At this configuration $(\beta_0, \beta_1) = (b, \delta)$. For an unequal-prior configuration, the centered coordinates in Section 2 are $(\beta_0, \beta_1) = (b - \lambda, \delta)$. Both parameters of the mechanism $q(y; b, \delta, \xi) = \operatorname{expit}\{\xi_0 + \xi_1(b + \delta y)^2\}$ are also estimated freely.

We use the uncertainty-dependent mechanism

For $\xi_1 < 0$, labels are more likely to be unavailable near the Bayes decision boundary, where the posterior evidence for the two classes is weakest.

Let

In the discriminant parameterization introduced in Section 2,

the conditional label information is

The next result shows that informative missingness affects the two discriminant directions differently.

The proof is given in Supplementary Section ??.

Since the expectation subtracted in (22) is strictly positive whenever labels are missing with positive probability,

Thus partial classification necessarily loses information in the slope direction in this symmetric model. The intercept direction is different. It gains information relative to complete classification precisely when

When (23) holds,

has one negative and one positive eigenvalue. The partially classified experiment then improves estimation in one discriminant direction while degrading it in another. By Theorem 3, neither experiment has a uniform first-order advantage over all deployment cost ratios.

Existence of a target-sensitive regime

Condition (23) identifies the geometry required for decision-target reversal, but does not by itself establish that such a configuration can arise at a fixed amount of missingness. We now show that it does.

Write

and, for each $a$, choose $\xi_0 = \xi_0(a)$ so that

where

The calibration in (24) is used only to compare data-generating mechanisms having the same marginal proportion of missing labels. The parameters $\xi_0$ and $\xi_1$ remain unrestricted nuisance parameters in the statistical model.

Let

and define

The proof is given in Supplementary Section ??.

Theorem 5 shows that target dependence is not merely a possibility created by an arbitrary indefinite covariance matrix. It arises from the informative-missingness model itself while the marginal proportion of missing labels is held fixed. In the symmetric model, the missingness indicator can contribute sufficient information about the intercept direction to offset the information lost through unavailable labels, while no corresponding efficient missingness information remains in the slope direction.

The result is established through a limiting argument as $a$ becomes large, but the conclusion concerns finite values of $a$. Moreover, whenever the inequalities in (26) are strict at a finite parameter value, continuity of the information matrices implies that they persist in an ordinary neighborhood of that configuration. If the fixed-$\gamma$ calibration is imposed, the implicit-function theorem shows that the same inequalities persist in a relative neighborhood within the calibrated family. Thus the target-sensitive regime is not confined to an isolated parameter configuration.

Extension to multiclass finite-action losses

The one-dimensional binary Gaussian model provides explicit decision-target reversal thresholds. We now consider multiclass problems with finite action losses. Building on the zero–one active-boundary expansion of Setoudehtazangi and McLachlan [6], we derive the loss-specific curvature matrix and characterize reversal through the family of curvature matrices attainable by varying the deployment loss.

Let $Z \in\{1,\ldots,K\}$ and let $\tau_k(y;\theta)$ denote the posterior probability of class $k$. Suppose that the available actions are $a \in\{1,\ldots,A\}$ and that $C=(C_{ak})$ is a finite loss matrix. The conditional risk of action $a$ is

and the Bayes action minimizes (27).

The decision problem is invariant to multiplication of $C$ by a common positive constant and to addition of the same class-dependent vector to every action row, since these transformations preserve the ordering of the conditional action risks. Accordingly, the decision-target frontier below is understood on the resulting equivalence classes of loss matrices.

For two actions $a$ and $b$, define the contrast

At the true parameter $\theta_0$, let $\mathcal{F}_{ab}^{C}$ denote the active decision face on which

Assume that the feature distribution has fixed support $\mathcal{S}\subseteq\mathbb{R}^{p}$ and a continuous, locally bounded density, and that the contrasts $g_{ab}$ are jointly twice continuously differentiable in $(y,\theta)$ near $\theta_0$. The minimizing action is unique almost everywhere. On compact subsets of the interior of $\mathcal{S}$, the active tie sets have a finite regular stratification: pairwise faces are smooth with nonzero spatial contrast gradient, and higher-order active intersections are transverse and have codimension at least two. Assume that compact sets $K_L$ exhaust $\mathcal{S}$, with boundaries transverse to these strata. The local chart and codimension conditions are detailed in Supplementary Section ??.

Define

Assume the surface-integrability condition

Let $a_h(y)$ minimize $\rho_a(y;\theta_0+h)$ under a fixed tie convention, and set

In addition, assume the uniform quadratic tail condition

Proposition 6. Under the preceding regularity conditions, as $\lVert h\rVert\to0$,

where

If, in addition,

the sequence

is uniformly integrable, and the remainder in (32), evaluated at $h=\widehat{\theta}-\theta_0$, is uniformly integrable after multiplication by $n$, then

The proof is given in Supplementary Section ??.

For the complete- and partially classified experiments, write

Their first-order deployment-risk difference is then governed by

With the convention in (35),

means that partial classification has smaller first-order deployment risk, whereas $\Delta_C>0$ favors complete classification.

The set

defines the decision-target frontier. In contrast with the one-dimensional binary Gaussian case, this frontier is generally not available in closed form because changing $C$ moves the active decision faces and may also change which actions are locally optimal. Decision-target reversal occurs whenever the attainable family $\{H_C\}$ intersects both sides of the hyperplane

More generally, for a class $\mathcal{C}$ of admissible deployment losses, decision-target reversal occurs precisely when

Thus indefiniteness of $D$ alone is not sufficient in the general multiclass problem: the relevant question is whether the attainable family of loss-specific curvature matrices $\{H_C:C\in\mathcal{C}\}$ probes both favourable and unfavourable directions of the covariance difference.

Finite-sample validation of decision-target reversal

We next examine whether the decision-target reversal predicted by the asymptotic theory is visible at practically relevant sample sizes, and whether the local risk expansion explains the finite-sample ordering of the two experiments. The purpose of the experiment is to compare deployment risks and examine how closely the local quadratic expansion tracks the estimated finite-sample risk curves. Differences from the population curve can reflect finite-sample estimation effects as well as Monte Carlo variability.

We use the symmetric Gaussian configuration

with missing-label probability

The mechanism strength is fixed at

and the intercept is calibrated to

which gives the marginal missing-label proportion

For this configuration, the population covariance difference derived in § 3 is

Thus the partially classified experiment is asymptotically more efficient in the intercept direction and less efficient in the slope direction. The corresponding population critical cost ratios are

Consequently, the first-order theory predicts that partial classification is favourable for deployment targets between these critical values and unfavourable for sufficiently asymmetric costs outside this interval.

For each

we generated 5000 independent samples. For every sample we fitted both the complete-classification estimator and the partially classified maximum-likelihood estimator, including estimation of the missingness parameters. The same generated sample was used for the two estimators within each replication. For deployment-risk evaluation, we normalized $C_{21} = 1$ and set $C_{12} = c$. Risk was evaluated exactly under the data-generating distribution over a grid of positive cost ratios. All 15000 paired fits were successful. Across all replications and sample sizes, the realized mean missing-label proportion was 0.29994, close to its target value of 0.30.

For a given cost ratio $c$, let

denote the Monte Carlo estimate of the difference in deployment risk. Negative values therefore favour partial classification. We compare this quantity with the first-order approximation

first using the population covariance difference $D$, and then using the empirical covariance difference

This is a diagnostic comparison based on the same paired replications as the Monte Carlo risk curve. Agreement between these summaries does not independently establish crossing precision or separate finite-sample effects from Monte Carlo variability. We also computed empirical second-moment differences about the true parameter. Replacing the centered covariance differences by these second-moment differences changes the matrices by approximately 0.08%–0.11% in relative Frobenius norm across the four simulation configurations. Empirical estimation bias therefore contributes little to the reported quadratic comparisons.

We also compared covariance-based approximations with risk curves computed from disjoint batches of simulation replications. The symmetric equal-cost advantage persists in both risk-evaluation batches, whereas the unequal-prior advantage remains unresolved. The split comparison and its Monte Carlo limitations are reported in Supplementary Section ??.

Figure 1 shows the resulting scaled risk difference

for $n = 2000$, together with the two first-order approximations. The estimated finite-sample curve exhibits the sign pattern predicted by the theory: partial classification is favourable over an intermediate range of deployment costs, whereas complete classification is preferable in the outer portions of the displayed range. The curve is closely tracked by the quadratic approximation using $\widehat{D}_n$. This agreement is consistent with the covariance–risk relation underlying the asymptotic calculation.

The agreement can be quantified through the zero crossings of the risk difference. Table 1 reports the empirical crossings obtained directly from the Monte Carlo risk curve, the crossings implied by $\widehat{D}_n$, and the population critical values. At every sample size, the crossings of the observed risk curve are closely reproduced by those obtained from the empirical covariance geometry. For example, at $n = 2000$, the Monte Carlo crossings are 0.2310 and 5.2777, while the corresponding first-order values based on $\widehat{D}_{2000}$ are 0.2269 and 5.2774. This agreement is a diagnostic of the quadratic risk representation, since both estimates use the same replications. It does not establish independent predictive accuracy or precise crossing locations.

$n$	Monte Carlo risk $c_{\mathrm{low}}$	Monte Carlo risk $c_{\mathrm{high}}$	Empirical $\widehat{D}_n$ $c_{\mathrm{low}}$	Empirical $\widehat{D}_n$ $c_{\mathrm{high}}$	Population theory $c_{\mathrm{low}}$	Population theory $c_{\mathrm{high}}$
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

The same conclusion is obtained from a global comparison of the risk curves. The root mean squared discrepancy between the Monte Carlo risk difference and the population first-order approximation is $6.12 \times10^{-6}$, $9.03 \times10^{-6}$, and $1.52 \times10^{-6}$ for $n = 500$, 1000, and 2000, respectively. When the population covariance difference is replaced by $\widehat{D}_n$, the corresponding discrepancies are $2.70 \times10^{-6}$, $2.65 \times10^{-7}$, and $1.19 \times10^{-7}$. The substantially closer agreement with the empirical-covariance approximation supports the local quadratic description of the reported risk curves. The discrepancy from the population curve may include both finite-sample effects and Monte Carlo variability in the estimated covariance difference.

As a simple interior target, consider equal misclassification costs, $c = 1$. The population theory predicts that partial classification is favourable at this target. The Monte Carlo risk differences are

for $n = 500$, 1000, and 2000, respectively. Their corresponding 95% Monte Carlo intervals are

all lying below zero. The first-order population values are, respectively,

Thus the favourable ordering at equal costs is resolved at the pointwise 95% Monte Carlo level for every sample size considered.

The symmetric simulation resolves opposite risk orderings at separated deployment targets. Crossing locations nevertheless have appreciable Monte Carlo uncertainty. At $n = 2000$, we resampled the 5000 paired replication IDs 2000 times, recomputed the mean risk curve, and interpolated its crossings in $\log c$. All resamples produced two crossings within the evaluated range. Their 2.5th and 97.5th percentile ranges are $(0.155, 0.331)$ and $(3.635, 8.036)$, respectively. Both contain the corresponding population critical values. The observed crossing displacements should therefore not be attributed solely to finite-sample effects.

As a robustness check, we considered unequal class probabilities, $\pi_1 = 0.30$ and $\pi_2 = 0.70$. The population covariance difference has a substantial off-diagonal component and remains indefinite, with eigenvalues $-0.3085$ and 0.4123. This establishes that reversal is attainable beyond the equal-prior configuration. The finite-sample mean curve has two estimated sign changes, but its favourable portion is not resolved from zero at the present Monte Carlo precision. The population calculation therefore supplies the robustness argument; the finite-sample curve remains illustrative. Full results are reported in Supplementary Section ??.

Gastrointestinal-lesion application

We illustrate decision-target dependence using the gastrointestinal-lesion data of Mesejo and Pizarro [17] and the partially classified construction of Lyu et al. [18]. The data comprise 76 lesions, with histology providing the reference class: 55 require resection and 21 do not. We retain the four previously selected image-derived features from the published analysis. The comparison is conditional on this historical representation and does not evaluate the original feature-selection procedure.

Agreement among seven endoscopists determines label availability. When all seven agree, the histological class label is available for fitting; otherwise it is masked. This gives 35 labelled and 41 unlabelled lesions. Expert agreement determines only label availability. Histology supplies every available training label and every evaluation label, including cases in which a unanimous expert decision disagrees with histology.

We compare complete classification (CC) with informative partial classification (FULL) using two Gaussian classes with unrestricted means and a shared covariance matrix. This shared-covariance specification replaces the component-specific covariance specification of the published analysis. Under FULL, label availability is modelled jointly through a logistic function of log posterior classification entropy, with both missingness parameters estimated.

Evaluation uses leave-one-out fitting. In each fold, feature standardization and all starting values are computed from the 75 training lesions. CC uses all training histological labels, whereas FULL uses only the agreement-defined available labels. The evaluated lesion contributes neither to preprocessing nor to initialization or fitting. CC is fitted by its exact Gaussian maximum-likelihood solution; FULL is fitted from multiple training-only starts. All 76 paired fits satisfy the numerical checks described in Supplementary Section ??. These checks establish reproducible numerical local optima, rather than a guarantee of global maximization.

Panel (a) of Figure 2 shows the first two standardized principal components, explaining 44.8% and 26.2% of feature variation. This projection is used only for visualization. Panel (b) compares entropy calculated from the held-out FULL posterior predictions. Mean entropy is 0.0267 for lesions with available labels and 0.2007 for lesions with masked labels; the corresponding medians are 0.0024 and 0.1132. These descriptive differences do not establish $M \perp Z \mid Y$ or validate the parametric missingness model, which remain working assumptions.

Let class 1 denote resection and class 2 no resection. Set $C_{21}=1$ and $c=C_{12}/C_{21}$, where $C_{12}$ is the loss of recommending resection for a true no-resection lesion and $C_{21}$ is the loss of failing to recommend resection for a true resection lesion. For each fitted method, the held-out decision assigns a lesion to class 1 when

We hold the fitted posterior predictions fixed and vary only $c$. At equal costs,

Thus FULL makes three fewer observed classification errors. Panel (c) displays

over $\log c \in[-3,3]$. The risk difference takes both signs: it is positive at $c=0.1$, negative at $c=1$, and positive at $c=5$. FULL has lower empirical risk throughout the interval $0.283236 < c < 2.462888$, while additional narrow reversals and a tie interval occur at lower cost ratios. Supplementary Section ?? reports the complete ordering.

The application illustrates the cost sensitivity of the fitted decision rules. It does not establish a population predictive advantage or identify the covariance mechanism derived for the restricted Gaussian model. The cost ratios are sensitivity parameters, rather than recommended clinical costs.

Discussion

The paper gives explicit first-order deployment-cost crossings for the one-dimensional Gaussian discrimination problem and proves that finite classifier-linked missingness mechanisms can generate the covariance geometry required for reversal at fixed marginal missingness. The existence result concerns a likelihood in which the class-probability logit and separation are estimated, while the class-mean midpoint and within-class variance are known. Estimating additional generative parameters requires their information and nuisance adjustment; canonical standardization alone does not remove them.

For general finite-action losses, the leading comparison is $\operatorname{tr}(H_C D)$, where $D=V_{\mathrm{PC}}-V_{\mathrm{CC}}$ describes estimation uncertainty and $H_C$ describes the active decision geometry. Indefiniteness is sufficient for reversal in the one-dimensional binary Gaussian case, but need not suffice in higher dimensions or multiclass problems. The attainable loss-specific curvature matrices must expose both signs of the covariance difference.

The reported simulation curves display changes in estimated risk ordering as the deployment loss varies. Their close agreement with an empirical quadratic comparison is consistent with the local risk mechanism, including under unequal priors. Differences between empirical and population crossings can reflect finite-sample effects and Monte Carlo variability; agreement between two summaries of the same replications does not separate these contributions. The gastrointestinal application illustrates cost sensitivity using agreement-defined availability of histological labels. The shared-covariance analysis uses training-only preprocessing and initialization, and all 76 paired fits pass the numerical checks. FULL has lower observed risk at equal costs, but the empirical ordering reverses as the deployment cost ratio varies. This exploratory comparison is conditional on the historical feature representation; it neither establishes a population predictive advantage nor identifies the covariance mechanism derived for the restricted Gaussian model. The relative costs remain sensitivity parameters rather than clinical utility estimates.

The theoretical comparison is local and asymptotic, and concerns the leading risk of regular estimators. It does not establish finite-sample dominance. Correct specification of the generative and classifier-linked missingness models is essential to the information calculation. The multiclass expansion assumes regular active faces, transverse higher-order ties, integrable boundary curvature, and uniform quadratic control of regret in the tails. Positive-measure ties, nontransverse intersections, or uncontrolled tail decision changes require a different analysis.

Extensions include comparison over a class of uncertain deployment losses and studying the frontier under classifier or mechanism misspecification. Designing label acquisition for a specified deployment loss is a related question. Work on classification-risk-optimal label acquisition [19] concerns changing the training experiment; the present comparison instead holds that experiment fixed while varying the deployment loss.

Statements and declarations

Data and code availability

The gastrointestinal-lesion dataset is publicly available from the UCI Machine Learning Repository at https://doi.org/10.24432/C5V02D. The analysis inputs, comprising the four previously selected features, histological reference classes and agreement-defined label mask, together with the code for the simulations, real-data analysis and manuscript figures, are available at https://github.com/fariborz-setoudeh/decision-target-reversal.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References