{
  "abstract": "Uncertainty in the solution of an inverse problem and in the tasks performed on it is quantified by posterior expectations, each an average of an integrand over $M$ posterior samples. While designed quadratures improve on the $O(M^{-1/2})$ error of Monte-Carlo estimation, they solve an optimization problem, often against the posterior density, for every new observation, which can be computationally costly. To address this limitation, we introduce the quadrature field, a set-equivariant network that maps an observation and its $M$ posterior samples to an $M$-node signed-weight quadrature in one forward pass. Trained once on a family of posteriors to minimize the worst-case integration error over a class of functions, it serves any observation, any $M$ and any integrand in that class with no further optimization. We show that, with high probability and up to a computable slack, the resulting quadrature is never worse than the Monte-Carlo estimate built from the same samples. We validate the quadrature field on closed-form and on learned posteriors, one constrained by a partial differential equation, where it improves on the Monte-Carlo estimate in median at every node count, often by orders of magnitude.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.15871",
  "author_total": 1,
  "authors": [
    {
      "name": "Ali Siahkoohi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dcyaf7d7gpz0t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.15871",
  "fulltext_url": null,
  "id": "dcyaf7d7gpz0t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.15871",
  "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.CO"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Gaussian Processes and Bayesian Inference"
    ]
  ],
  "openalex": "W7164885744",
  "pdf_url": "https://arxiv.org/pdf/2606.15871",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dcyaf7d7gpz0t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:03Z",
      "native_id": "2606.15871",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.15871"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.CO",
      "subfield": "statistics"
    }
  ],
  "title": "Amortized quadrature for posterior expectations in inverse problems",
  "title_text": "Amortized quadrature for posterior expectations in inverse problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:03Z",
  "url": "https://research.draconitic.com/paper/dcyaf7d7gpz0t",
  "venue": "arXiv",
  "versions": []
}