{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35217",
  "author_total": 4,
  "authors": [
    {
      "name": "Niklas Koenen"
    },
    {
      "name": "Claudia Battistin"
    },
    {
      "name": "Jeriek Van den Abeele"
    },
    {
      "name": "Martin Jullum"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2hzm5mbtrxe20.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35217",
  "fulltext_url": null,
  "id": "2hzm5mbtrxe20",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35217",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Explainable Artificial Intelligence (XAI)"
    ]
  ],
  "openalex": "W7214736737",
  "pdf_url": "https://arxiv.org/pdf/2609.35217",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2hzm5mbtrxe20.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:26Z",
      "native_id": "2609.35217",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35217"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "A Hierarchy of Entropy-Shapley Games for Multivariate Predictive Uncertainty",
  "title_text": "A Hierarchy of Entropy-Shapley Games for Multivariate Predictive Uncertainty",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/2hzm5mbtrxe20",
  "venue": "arXiv",
  "versions": []
}