{
  "abstract": "We study Gibbs averaging of finite Markov chains with a prescribed stationary distribution $π$. Given a transition matrix $P$ and the Gibbs kernel $G$ induced by a state-space partition, we prove that $GPG$ is an information projection for every admissible Bregman generator $ϕ$, with divergences defined using transition probabilities relative to $π$. A Pythagorean identity characterizes this projection and recovers the relative entropy and weighted Frobenius cases. A dual characterization identifies $GPG$ as a kernel closest to stationarity among those with the same projection chain as $P$; for relative entropy, it maximizes the entropy rate. The identities also give monotonicity under partition coarsening. For every reversible ergodic baseline $P$, Gibbs averaging cannot increase the squared weighted Frobenius distance to stationarity at any positive integer time, and the inequality is strict unless $P=GPG$. The corresponding mixing time therefore cannot increase. To choose the partition, we identify the divergence of $G$ from stationarity as the exact worst-case divergence of $GPG$ over all $π$-stationary baselines $P$. This objective depends only on the block masses. We give an exact algorithm for concave block costs, including those arising from relative entropy and power generators $ϕ(u)=u^p$ with $1<p<2$, and a greedy approximation algorithm with a provable guarantee for $p>2$. For power generators $ϕ(u)=u^p$ with $p>1$, the objective is proportional to Tsallis entropy of order $2-p$: its minimization favors imbalanced block masses when $1<p<2$ and balanced block masses when $p>2$, while all $k$-block partitions have the same objective value when $p=2$. A numerical experiment compares the resulting projected chains with a local Metropolis sampler on a multi-modal $π$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39239",
  "author_total": 2,
  "authors": [
    {
      "name": "Michael C. H. Choi"
    },
    {
      "name": "Youjia Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8g99abydrcqs4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39239",
  "fulltext_url": null,
  "id": "8g99abydrcqs4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39239",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "stat.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39239",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8g99abydrcqs4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39239",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39239"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.CO",
      "subfield": "statistics"
    }
  ],
  "title": "Bregman projections and partition design for accelerating finite Markov chains",
  "title_text": "Bregman projections and partition design for accelerating finite Markov chains",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:00Z",
  "url": "https://research.draconitic.com/paper/8g99abydrcqs4",
  "venue": "arXiv",
  "versions": []
}