{
  "abstract": "For a one-dimensional process in a bounded interval, splitting probabilities quantify the likelihood of exiting said interval for the first time via either boundary. For two-dimensional Markov processes {X(t), Y(t)} t∈T , a joint analogue of the splitting probabilities can be defined, which captures the likelihood that the variable X(t), having been initialized at x 0 ∈ L, exits L for the first time via either of the interval boundaries and that the variable Y(t), initialized at y 0 , is y exit at the time of exit. Drawing on Bayes' theorem, we introduce the related notion of conditional splitting probabilities and formalize a simple scheme that leverages them to partially infer the assumedly hidden state Y(t) from detecting the particle exiting at either interval boundary. We demonstrate the viability of this boundary-based inference scheme for two classes of processes: processes where X(t) is Brownian motion and Y(t) is a decoupled internal state, and unidirectionally coupled processes where X(t) is drift-diffusive and depends on Y(t), while Y(t) evolves independently. For the first class we obtain generic expressions in terms of the eigensystem of the Fokker-Planck operator for the Y dynamics, while for the second we carry out explicit derivations for three paradigmatic cases (run-and-tumble motion, diffusion in an intermittent piecewise-linear potential and diffusion with stochastic resetting). Drawing on Bayes' theorem, we subsequently introduce the related notion of conditional splitting probabilities, defined as the posterior likelihoods of the internal state Y \\emph{given} that the observable degree of freedom X has undergone a specific exit event. After computing these conditional splitting probabilities, we propose a simple scheme that leverages them to partially infer the assumedly hidden state Y(t) from point-wise detection events.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2508.07386",
  "author_total": 7,
  "authors": [
    {
      "name": "Emir Sezik"
    },
    {
      "name": "Jacob Knight"
    },
    {
      "name": "Henry Alston"
    },
    {
      "name": "Connor Roberts"
    },
    {
      "name": "Thibault Bertrand"
    },
    {
      "name": "Gunnar Pruessner"
    },
    {
      "name": "Luca Cocconi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cjkmmym2zqx12.bib",
  "cited_by": 0,
  "doi": "10.1098/rspa.2025.0716",
  "fulltext_url": null,
  "id": "cjkmmym2zqx12",
  "issns": [
    "1364-5021",
    "1471-2946"
  ],
  "landing_url": "https://hal.science/hal-05758232v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
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      "2613|Probability and Statistical Research"
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  "openalex": "W4416242150",
  "pdf_url": "https://hal.science/hal-05758232/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-06-10",
  "publisher": "Royal Society",
  "ris": "https://research.draconitic.com/paper/cjkmmym2zqx12.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:37Z",
      "native_id": "hal-05758232",
      "source": "hal",
      "url": "https://hal.science/hal-05758232v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-mp",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-pr",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "hal:2.phys.phys.phys-data-an",
      "subfield": "computational"
    },
    {
      "field": "physics",
      "origin": "hal:1.phys.cond",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "hal:2.phys.cond.cm-sm",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Conditional splitting probabilities for hidden-state inference in drift-diffusive processes",
  "title_text": "Conditional splitting probabilities for hidden-state inference in drift-diffusive processes",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:17:37Z",
  "url": "https://research.draconitic.com/paper/cjkmmym2zqx12",
  "venue": "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",
  "versions": []
}