{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T13:05:58Z","timestamp":1775480758701,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,8]],"date-time":"2025-07-08T00:00:00Z","timestamp":1751932800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000925","name":"John Templeton Foundation","doi-asserted-by":"publisher","award":["62312"],"award-info":[{"award-number":["62312"]}],"id":[{"id":"10.13039\/100000925","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The usual inputs for a causal identification task are a graph representing qualitative causal hypotheses and a joint probability distribution for some of the causal model\u2019s variables when they are observed rather than intervened on. Alternatively, the available probabilities sometimes come from a combination of passive observations and controlled experiments. It also makes sense, however, to consider causal identification with data collected via schemes more generic than (perfect) passive observation or perfect controlled experiments. For example, observation procedures may be noisy, may disturb the variables, or may yield only coarse-grained specification of the variables\u2019 values. In this work, we investigate identification of causal quantities when the probabilities available for inference are the probabilities of outcomes of these more generic schemes. Using process theories (aka symmetric monoidal categories), we formulate graphical causal models as second-order processes that respond to such data collection instruments. We pose the causal identification problem relative to arbitrary sets of available instruments. Perfect passive observation instruments\u2014those that produce the usual observational probabilities used in causal inference\u2014satisfy an abstract process-theoretic property called marginal informational completeness. This property also holds for other (sets of) instruments. The main finding is that in the case of Markovian models, as long as the available instruments satisfy this property, the probabilities they produce suffice for identification of interventional quantities, just as those produced by perfect passive observations do. This finding sharpens the distinction between the Markovianity of a causal model and that of a probability distribution, suggesting a more extensive line of investigation of causal inference within a process-theoretic framework.<\/jats:p>","DOI":"10.3390\/e27070732","type":"journal-article","created":{"date-parts":[[2025,7,8]],"date-time":"2025-07-08T07:31:57Z","timestamp":1751959917000},"page":"732","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Compositional Causal Identification from Imperfect or Disturbing Observations"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6320-5861","authenticated-orcid":false,"given":"Isaac","family":"Friend","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6090-9684","authenticated-orcid":false,"given":"Aleks","family":"Kissinger","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5235-4255","authenticated-orcid":false,"given":"Robert W.","family":"Spekkens","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6960-3796","authenticated-orcid":false,"given":"Elie","family":"Wolfe","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,8]]},"reference":[{"key":"ref_1","unstructured":"Pearl, J. 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