[
    {
        "id": "osp-22089",
        "type": "article-journal",
        "title": "A Hierarchy of Entropy-Shapley Games for Multivariate Predictive Uncertainty",
        "author": [
            {
                "family": "Koenen",
                "given": "Niklas"
            },
            {
                "family": "Battistin",
                "given": "Claudia"
            },
            {
                "family": "Abeele",
                "given": "Jeriek Van den"
            },
            {
                "family": "Jullum",
                "given": "Martin"
            }
        ],
        "URL": "https://omanscience.com/en/articles/a-hierarchy-of-entropy-shapley-games-for-multivariate-predictive-uncertainty",
        "language": "en",
        "issued": {
            "date-parts": [
                [
                    2026
                ]
            ]
        },
        "abstract": "Modern probabilistic machine learning models increasingly produce multivariate outputs with complex dependence structure, from multi-step time-series forecasts to sample path predictions. Understanding which input features drive the predictive uncertainty is important for risk-aware decisions, model diagnostics, and deciding whether the uncertainty should be mitigated or hedged against. This attribution problem requires a choice of how dependencies between output components are treated. Existing approaches reduce the output to a scalar through aggregation or projection before attribution, thereby obscuring whether features affect marginal uncertainty, dependence structure, or both, while component-wise analyses can miss dependence effects entirely. We close this gap by introducing a hierarchy of three entropy-based Shapley games that make this output-side choice explicit for any ordered multivariate outcome, ranging from per-component marginal entropy to fully joint entropy. The hierarchy isolates a cross-component attribution term that captures how each feature shifts the dependence between output components, a quantity invisible to component-wise methods. We establish a chain-rule decomposition of the joint attribution and characterize the cross-component term through conditional total correlation, providing both closed-form and sample-based estimators. Finally, we demonstrate how the framework captures differences in learned joint structure across probabilistic models from distributional regression to a zero-shot time series foundation model."
    }
]