{"paper":{"title":"Bundles of Probability Schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Bundles record quotients of sample spaces, algebras of random variables, and conditional schemes simultaneously.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Wai Yan Pong","submitted_at":"2026-05-05T15:55:48Z","abstract_excerpt":"We study finite probability theory through a category of finite probability schemes and probability-preserving maps, called \\emph{bundles}. A bundle simultaneously records a quotient of a sample space, an algebra of random variables, and the family of conditional schemes over the quotient. The two natural linear functors associated with a bundle give a compact construction of conditional expectation and explain its projection properties. Within this framework we recover the laws of total expectation, variance, and covariance, the weak law of large numbers, and the variance decomposition behind"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"A bundle simultaneously records a quotient of a sample space, an algebra of random variables, and the family of conditional schemes over the quotient. The two natural linear functors associated with a bundle give a compact construction of conditional expectation and explain its projection properties. From this point of view we recover the laws of total expectation, total variance, total covariance, the weak law of large numbers, and the variance decomposition behind simple linear regression. We also introduce fiber products of bundles and show that they encode conditional independence, sequential random experiments, and discrete-time Markov chains.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the category of finite probability schemes and probability-preserving maps can be defined so that bundles accurately capture quotients, random variables, and all conditional schemes without information loss or the need for extra structures beyond the stated linear functors.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Bundles of probability schemes give a categorical construction of conditional expectation that recovers laws of total expectation, variance, and covariance while using fiber products to encode conditional independence and Markov chains.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Bundles record quotients of sample spaces, algebras of random variables, and conditional schemes simultaneously.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"efb1ad58a8e1806416f395359c24aba45fe3044d35b89c9c4ad213b832de27cc"},"source":{"id":"2605.03902","kind":"arxiv","version":2},"verdict":{"id":"bfa06347-f0a7-4478-947a-1ab1b7049a38","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T14:04:04.913643Z","strongest_claim":"A bundle simultaneously records a quotient of a sample space, an algebra of random variables, and the family of conditional schemes over the quotient. The two natural linear functors associated with a bundle give a compact construction of conditional expectation and explain its projection properties. From this point of view we recover the laws of total expectation, total variance, total covariance, the weak law of large numbers, and the variance decomposition behind simple linear regression. We also introduce fiber products of bundles and show that they encode conditional independence, sequential random experiments, and discrete-time Markov chains.","one_line_summary":"Bundles of probability schemes give a categorical construction of conditional expectation that recovers laws of total expectation, variance, and covariance while using fiber products to encode conditional independence and Markov chains.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the category of finite probability schemes and probability-preserving maps can be defined so that bundles accurately capture quotients, random variables, and all conditional schemes without information loss or the need for extra structures beyond the stated linear functors.","pith_extraction_headline":"Bundles record quotients of sample spaces, algebras of random variables, and conditional schemes simultaneously."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.03902/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-20T00:01:21.621772Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:57:41.553862Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"fbbb387dd391fc5dc087daf09d85142889c03937774641b0b55f11a8f20d5cdc"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}