{
  "abstract": "Approximate message passing on factor graphs underlies two dominant families of probabilistic inference algorithms: expectation propagation (EP) and variational message passing (VMP). Both methods approximate the marginal at each factor edge, forcing an iterative round-robin schedule, risking negative-precision messages, and, for VMP, collapsing to point estimates at Dirac-delta factors. We introduce Direct Message Approximation (DMA), which approximates factor-to-variable messages directly rather than the marginal. For normalisable factors, we define a consistency condition (requiring exactness when all other incoming messages are Dirac deltas) to guide message construction. We prove a master theorem (proper messages, any graph) bounding marginal KL from message KL, with three structural corollaries: Dirac-input consistency, no EP-style inner-loop iteration, and no negative-precision messages. Further, we prove a complementary $O(1/r^2)$ guarantee for the inherently improper backward message of the product factor, whose closed-form treatment has resisted prior work. As a concrete instantiation, we derive explicit DMA messages for the product and leaky-ReLU factors and assemble a Bayesian neural network (BNN) inference algorithm with one forward/backward sweep per training example and no gradient learning-rate hyperparameter, validating that the structural guarantees translate to predictive uncertainty that widens in data-sparse regions, including under model mismatch.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29466",
  "author_total": 14,
  "authors": [
    {
      "name": "Ralf Herbrich"
    },
    {
      "name": "Rainer Schlosser"
    },
    {
      "name": "Jan Lemcke"
    },
    {
      "name": "Johann Ukrow"
    },
    {
      "name": "Anna Kazachkova"
    },
    {
      "name": "Nicolas Alder"
    },
    {
      "name": "Leonhard Hennicke"
    },
    {
      "name": "Theo Bardey"
    },
    {
      "name": "Nico Grimm"
    },
    {
      "name": "Luca Kleinschmidt"
    },
    {
      "name": "Philipp Kolbe"
    },
    {
      "name": "Cezary Kujath"
    },
    {
      "name": "Johanna Schlimme"
    },
    {
      "name": "Karl Matti Schütz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2asnw1ee8e2rp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29466",
  "fulltext_url": null,
  "id": "2asnw1ee8e2rp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29466",
  "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.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Gaussian Processes and Bayesian Inference"
    ]
  ],
  "openalex": "W7214376503",
  "pdf_url": "https://arxiv.org/pdf/2609.29466",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2asnw1ee8e2rp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:21Z",
      "native_id": "2609.29466",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29466"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Direct Message Approximation (DMA): A Consistency-Based Framework for Tractable Approximate Inference on Factor Graphs",
  "title_text": "Direct Message Approximation (DMA): A Consistency-Based Framework for Tractable Approximate Inference on Factor Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:21Z",
  "url": "https://research.draconitic.com/paper/2asnw1ee8e2rp",
  "venue": "arXiv",
  "versions": []
}