Silvano Zappalà | Searcharxiv

arXiv2026

Bayesian Deck-of-cards-based Ordinal Regression with Sequential Preference Elicitation

The Deck-of-cards-based Ordinal Regression (DOR) infers a value function from a ranking of reference alternatives in which the Decision Maker (DM) inserts blank cards between consecutive levels to express preference intensity. DOR, and its stochastic extension (SMAA-DOR), treat these answers as hard constraints defining a set of compatible value functions. We propose B-DOR, a probabilistic reformulation of DOR in which each pair of adjacent levels yields an ordinal observation, the declared direction and the number of cards, modelled through a cumulative-link likelihood that relates the number of blank cards to the latent value difference between alternatives. Two Bayesian inference algorithms are proposed: BAYES-DOR samples the whole posterior distribution by Hamiltonian Monte Carlo; FTRL-DOR tracks the maximum a posteriori estimate by constrained convex optimization. Moreover, through a multi-step elicitation process, elicitation can be spread over several short sessions reducing the cognitive burden on the DM. Both algorithms enjoy logarithmic regret bounds for prediction that hold for any sequence of DM responses and that guide the choice of the prior hyperparameters. A Monte Carlo study over 768 configurations shows that accuracy grows with the number of sessions, that blank cards add significant information over preference directions alone, that both algorithms maintain good performance under inconsistent answers, and that both outperform DOR and SMAA-DOR. An illustrative application to Italian regional healthcare performance demonstrates the practical applicability of the approach for building composite indicators.

stat.ML↗

arXiv2026

An Explainable and Interpretable Composite Indicator Based on Decision Rules

Composite indicators are widely used to score or classify units evaluated on multiple criteria. Their construction typically involves aggregating criteria evaluations, a common practice in Multiple Criteria Decision Aiding (MCDA). Beyond producing a final score or classification, however, ensuring explainability, interpretability, and transparency is crucial. This paper proposes a novel framework for constructing explainable and interpretable composite indicators using if-then decision rules. We explore four scenarios: (i) decision rules explaining classifications derived from the sum of ordinal indicator codes; (ii) interpretation of an opaque numerical composite indicator used to classify units into quantiles; (iii) construction of a composite indicator from decision-maker preference information, given as classifications of reference units; and (iv) explanation of classifications generated by an existing MCDA method. To induce the rules from scored or classified units, we apply the Dominance-based Rough Set Approach. The resulting rules relate class assignments or scores to threshold conditions on indicator values in a clear and intelligible way, clarifying the underlying rationale and supporting the assessment of new units. Our main methodological contribution is the introduction of a decision-rule-based framework for constructing composite indicators. Moreover, the framework extends naturally to continuous composite indicators by treating each distinct score as an ordered class. This is enabled by a new algorithm that efficiently induces all minimal rules in a single run. Although this may yield many rules, explainability is preserved by showing only those satisfied by the unit of interest. Finally, the methodology can handle datasets with missing values, enhancing its practical applicability.

cs.LG↗