When Explanations Compete: Policy-Aware Selection Under Uncertainty

Helena Löfström; Tuwe Löfström; Johan Hallberg Szabadvary

cs.AIOct 7, 2024

When Explanations Compete: Policy-Aware Selection Under Uncertainty

Authors: Helena Löfström, Tuwe Löfström, Johan Hallberg Szabadvary

Organizations: Jönköping AI Lab, Department of Computing, Jönköping University, Jönköping, Sweden

Abstract

Uncertainty-aware explanation methods often produce several alternatives for the same prediction. Selecting among them requires a policy for balancing prediction confidence, uncertainty, and application constraints. This paper presents a framework for applying such policies to a fixed set of generated explanations. Candidates are characterised by uncertainty change, prediction direction, and, when available, interval position relative to a decision boundary. The framework combines these properties with eligibility rules, optional bidirectional Pareto screening, and policy-aware ranking. A fictitious prostate-cancer example illustrates how different explanatory purposes lead to different selections from the same candidate set. We instantiate the framework with Calibrated Explanations for classification, thresholded regression, and plain regression. Across 41 benchmark datasets, mean candidate counts range from 11.57 to 21.75 for single-feature explanations and from $29.48$ to $69.53$ when conjunctions are included. Equal-weight and confidence-only policies yield an average selection-disagreement rate of $28.7\%$ while favouring the same confidence direction. A supporting $δ$ -CLUE experiment demonstrates use with a second generator. By making the selection policy explicit, the framework allows applications to compare and prioritise explanations according to their intended use.

Figures & tables

Approach	Primary selection stage	Relation to the present framework
Explanation Sets [ 5 ]	Representation and restriction	Unifies counterfactual and semifactual explanations and supports restrictions. Not organised around candidate-level uncertainty trade-offs.
CLUE family [ 1 , 9 , 10 ]	During generation	Searches for uncertainty-aware alternatives. Multiple returned candidates can still require an external policy.
Stępka et al. [ 21 ]	After generation	Closest selection precedent. Applies multi-criteria filtering and ideal-point selection to counterfactuals from an ensemble of explainers.
Proposed framework	After generation	Separates uncertainty change, threshold ambiguity, confidence direction, eligibility, and policy across compatible alternative-explanation records.

Table 1 : Positioning against closely related approaches.

Panel of Figure 1 — Figure 1 : The framework operates after generation. It preserves candidate records and makes application-specific selection choices explicit.

Property	Condition	Interpretation
Ensured	$U_{j}<U_{0}$ . Optionally $U_{j}\leq U_{0}-\epsilon_{U}$ with $\epsilon_{U}>0$	Strict reduction in the specified uncertainty signal.
Uncertainty-non-increasing	$U_{j}\leq U_{0}$	Includes equal uncertainty, used by the CE diagnostic reported later.
Uncertainty-increasing	$U_{j}>U_{0}$	Candidate carries greater uncertainty than the factual reference.
Potential	$\ell_{j}\leq\tau\leq h_{j}$	Candidate interval spans or touches the supplied boundary.
Counter-oriented	$(C_{j}-\tau)(C_{0}-\tau)<0$	Point prediction crosses to the opposite side of the boundary.
Semi-oriented	$(C_{j}-\tau)(C_{0}-\tau)>0$ and $\|C_{j}-\tau\|<\|C_{0}-\tau\|$	Remains on the factual side but moves toward the boundary.

Table 2 : Operational candidate properties. “Ensured” concerns the selected uncertainty signal; it is not a guarantee of correctness, calibration after selection, feasibility, or actionability.

	Hypothetical alternative	$C_{j}$	Interval	$U_{j}$	Properties and possible eligibility
$a_{1}$	Refined prostate-volume measurement yields PSA density $<0.10$	0.25	$[0.20,0.30]$	0.10	Ensured and counter-oriented: interval wholly below $\tau$ .
$a_{2}$	Patient’s age is 65 years	0.30	$[0.15,0.45]$	0.30	Ensured, potential, and counter-oriented: excluded if age changes are prohibited.
$a_{3}$	MRI re-evaluation yields PI-RADS 5	0.85	$[0.80,0.90]$	0.10	Ensured and super-oriented: interval wholly above $\tau$ .
$a_{4}$	PI-RADS 5 together with PSA density $>0.20$	0.90	$[0.80,1.00]$	0.20	Ensured and super-oriented: higher confidence but greater uncertainty than $a_{3}$ .

Table 3 : Fictitious prostate-cancer alternatives. The factual record is

C_{0}=0.40

, interval

[0.20,0.60]

U_{0}=0.40

, and

\tau=0.35

. All predictive quantities and alternatives are illustrative.

Panel of Figure 2 — Figure 2 : The worked example separates direction, uncertainty, and eligibility. Candidates $a_{1}$ and $a_{3}$ have equal uncertainty but opposite confidence directions; $a_{3}$ and $a_{4}$ move in the same direction but trade confidence against uncertainty.

Panel of Figure 3 — Figure 3 : Exact score paths for the four-candidate example when higher confidence is preferred. The upper envelope selects $a_{3}$ for $0<w<2/3$ and $a_{4}$ for $w>2/3$ ; $a_{1}$ and $a_{3}$ tie at $w=0$ .

Provider	Candidate coordinates and qualification
Calibrated Explanations	Task-specific calibrated estimate and associated interval width: main evaluated instantiation.
CLUE / $\delta$ -CLUE	Decision-relevant prediction and the generator’s uncertainty signal, such as predictive entropy or variance: $\delta$ -CLUE is evaluated in a supporting transfer check.
QUCE	Prediction and a candidate-comparable summary of path uncertainty: path semantics must remain explicit.
Multi-objective or robust recourse	Compatible when records expose a decision-relevant prediction and an explicit uncertainty or sensitivity signal: ordinary validity and proximity objectives alone do not establish the mapping.

Table 4 : Candidate-provider mappings and qualifications. Proposed mappings are interface arguments, not additional transfer experiments.

Task	Prediction coordinate	Uncertainty and directional interpretation
Binary classification	Calibrated support for the decision-relevant class	Width of the associated Venn–Abers output; boundary-relative types when a decision threshold is defined.
Multiclass classification	Calibrated support for a specified target class	Width for that class; decreasing target support does not select a particular competing class.
Thresholded regression	Calibrated estimate for the supplied threshold event	Width of the event-specific bounds; threshold is part of the task definition.
Plain regression	Numeric prediction	Width of a conformal prediction interval; direction is relative to the factual value or an application reference.

Table 5 : Task-specific interpretation of the CE mapping.

Task and mode	Total	Non-increasing	Share (%)
Binary, single	18.12	6.48	35.8
Binary, conjunctive	69.26	22.77	32.9
Multiclass, single	21.75	9.97	45.8
Multiclass, conjunctive	69.53	31.59	45.4
Regression 25th threshold, single	11.82	5.59	47.3
Regression 25th threshold, conjunctive	29.90	13.66	45.7

Table 6 : Candidate multiplicity. Total is the mean number generated. Non-increasing candidates satisfy

U_{j}\leq U_{0}

. Conjunctive mode includes two-feature conjunctions.

Policy comparison	Disagree	Mean med. $\|\Delta C\|$	Mean med. $\|\Delta U\|$
Equal weight vs confidence only ( $0.5$ vs $1$ )	28.7%	0.13	0.46
Confidence decrease vs increase ( $-0.5$ vs $0.5$ )	85.5%	0.65	0.18
Equal weight vs uncertainty only ( $0.5$ vs $0$ )	51.9%	0.53	0.18
Uncertainty only vs confidence only ( $0$ vs $1$ )	66.4%	0.50	0.47

Table 7 : Policy disagreement averaged over dataset-setting-mode blocks. Coordinate differences are means of blockwise medians over changed selections, using the coordinates employed for ranking.

Panel of Figure 4 — Figure 4 : Policy sensitivity for a fixed thresholded California Housing candidate set. The panels show the generated set and the top ten records under three policies.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

DS	t	n	f	DS	t	n	f	DS	t	n	f
diabetes	b	768	8	german	b	955	27	kc1	b	1192	21
pc4	b	1343	37	ttt	b	958	27	cars	m	1728	6
cmc	m	1473	9	cool	m	768	8	heat	m	768	8
image	m	2310	19	steel	m	1941	27	vehicle	m	846	18
vowel	m	990	11	wave	m	5000	40	wineR	m	1599	11
wineW	m	4898	11	yeast	m	1484	8	abalone	r	4177	8

Appendix

Table 1 : Datasets used in the diagnostic evaluation. Columns are repeated three times to fit the table: DS = dataset, t = task code (b = binary, m = multiclass, r = regression), n = instances, f = features.

Panel of Figure 5 — Figure 5 : Feasible confidence–uncertainty regions for probability intervals. Solid lines delimit feasible records; dashed lines separate intervals wholly on one side of $\tau=1/2$ from potential intervals containing it. The feasible region above the dashed boundaries is potential. These are representation boundaries, not Pareto frontiers. The curved guides in Figure 4 use the second geometry.

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Formal explainability provides mathematically grounded justifications for individual predictions. However, abductive explanations often exceed human cognitive limits by involving too many features, while probabilistic relaxations have remained largely limited to categorical classification. We present a unified framework for probabilistic explainability based on sparse, anchored linear models, applicable to both binary classification and continuous regression. By mapping instances to the Boolean hypercube, our linear explanations strictly generalize subset-based approaches: they capture both the magnitude and direction of feature contributions while enforcing a prescribed sparsity budget

k

. We show that minimizing the relevance error for such explanations is \ClassNPPP-hard when the underlying model is a neural network, and we relate this intractable objective to a tractable surrogate---the fidelity error. For a parameterized family of local distributions, the relevance error of any

k

-sparse explanation is bounded by its fidelity error up to a multiplicative factor that remains small locally. We address the resulting empirical problem using two complementary approaches: a Mixed Integer Programming (MIP) formulation that yields provably optimal empirical solutions while maintaining polynomial sample complexity, and a polynomial-time Iterative Hard Thresholding (IHT) algorithm with provable approximation guarantees. Empirical evaluations show that, unlike state-of-the-art baselines such as LIME and MAPLE, our explanations satisfy both the anchoring and sparsity constraints by construction, while consistently achieving lower relevance error.

Abstract

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