{
  "abstract": "Pearl's causal hierarchy shows that observational, interventional, and counterfactual queries are qualitatively distinct. We ask a quantitative version of this question: how many additional bits are needed to specify higher-rung causal answers once lower-rung answers are known? We formalize this via query-class description length, the Kolmogorov complexity of the answers a structural causal model (SCM) gives to a class of queries, with system size given. Our main construction gives acyclic SCMs on $n$ binary variables whose common observational distribution has constant description length, while single-variable intervention answers can require $Θ(n^2)$ additional bits. For SCMs with independent noises, fixed finite gate-schema and noise libraries, and at most $d$ parents per variable, a degree-sensitive upper bound gives an observational-interventional gap of at most $O(nd\\log(en/d)+n\\log n)$. This makes the quadratic construction order-optimal in the dense regime and a rooted-tree construction order-optimal for bounded indegree. The quadratic separation persists under $\\varepsilon$-accurate total-variation descriptions for every fixed $\\varepsilon<1/4$. At the next rung, even knowing every hard-intervention distribution can leave a $Θ(n)$ counterfactual description gap; a masked-parity construction extends this to $Θ(n^2)$. These separation rates hold in the worst case and with high probability under uniform hidden parameters. A general ambiguity-to-bits theorem and Shannon analogue, together with explicit encodings, show that the gaps in our constructions equal the logarithm of residual higher-rung ambiguity up to lower-order terms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.02177",
  "author_total": 1,
  "authors": [
    {
      "name": "Seyed Morteza Emadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6j4rrhjt64z63.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.02177",
  "fulltext_url": null,
  "id": "6j4rrhjt64z63",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.02177",
  "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.AI"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Bayesian Modeling and Causal Inference"
    ]
  ],
  "openalex": "W7160248446",
  "pdf_url": "https://arxiv.org/pdf/2605.02177",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6j4rrhjt64z63.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2605.02177",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.02177"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "The Causal Description Gap: Information-Theoretic Separations Across Pearl's Hierarchy",
  "title_text": "The Causal Description Gap: Information-Theoretic Separations Across Pearl's Hierarchy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:54Z",
  "url": "https://research.draconitic.com/paper/6j4rrhjt64z63",
  "venue": "arXiv",
  "versions": []
}