{
  "abstract": "We ask how much of a large random system must be observed before a global average can be inferred with controlled uncertainty. For the random transposition shuffle, a conjugate Bayesian estimator of the average displacement, computed from $m$ uniformly sampled cards, has relative uncertainty $1/\\sqrt{2m}$, independent of the deck size $n$. The law of total variance exposes a second error, the floor $R_n$, which arises because the observed configuration is a single draw from equilibrium: no relative tolerance below $κ\\sqrt{R_n}$ can be certified, even by a census. For the shuffle we derive $R_n=(2n^2+7)/\\bigl(5(n-1)^2(n+1)\\bigr)$ and show that the posterior remains calibrated. At the critical point of the two-dimensional Ising model the floor of the energy vanishes, whereas that of the magnetisation stays near $0.072$; its mean-field limit is $\\sqrt{2}-1$. For correlated fields read by sensor networks, the usual $95\\%$ interval covers the ensemble mean in $7\\%$ of cases. Bounding the floor from one realisation, as in single-chain Monte Carlo diagnostics, requires an assumption on the decay of correlations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35973",
  "author_total": 1,
  "authors": [
    {
      "name": "Mihika Singhania"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0t8z7fjeh3bpt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35973",
  "fulltext_url": null,
  "id": "0t8z7fjeh3bpt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35973",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7224096588",
  "pdf_url": "https://arxiv.org/pdf/2610.35973",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0t8z7fjeh3bpt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:22Z",
      "native_id": "2610.35973",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35973"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Bayesian Inference in Partially Observed Critical Systems",
  "title_text": "Bayesian Inference in Partially Observed Critical Systems",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/0t8z7fjeh3bpt",
  "venue": "arXiv",
  "versions": []
}