{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:J3DWE6KNPKWIRJVVNUEILVM7OM","short_pith_number":"pith:J3DWE6KN","schema_version":"1.0","canonical_sha256":"4ec762794d7aac88a6b56d0885d59f7318d4517c94b31701a902cacffc503142","source":{"kind":"arxiv","id":"2405.13149","version":1},"attestation_state":"computed","paper":{"title":"Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.NA","math.PR","stat.CO"],"primary_cat":"stat.ML","authors_text":"Andrew M Stuart, Bamdad Hosseini, Houman Owhadi, Yifan Chen","submitted_at":"2024-05-21T18:38:14Z","abstract_excerpt":"The article presents a systematic study of the problem of conditioning a Gaussian random variable $\\xi$ on nonlinear observations of the form $F \\circ \\phi(\\xi)$ where $\\phi: \\mathcal{X} \\to \\mathbb{R}^N$ is a bounded linear operator and $F$ is nonlinear. Such problems arise in the context of Bayesian inference and recent machine learning-inspired PDE solvers. We give a representer theorem for the conditioned random variable $\\xi \\mid F\\circ \\phi(\\xi)$, stating that it decomposes as the sum of an infinite-dimensional Gaussian (which is identified analytically) as well as a finite-dimensional n"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2405.13149","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2024-05-21T18:38:14Z","cross_cats_sorted":["cs.LG","cs.NA","math.NA","math.PR","stat.CO"],"title_canon_sha256":"d8b857fd0512a058cb0006a7cce3add82fdd53f72a0b383f1e87da1d115e5e83","abstract_canon_sha256":"b59dd9964c0055c8ae62933a3dfd4d1e84a05e6fc64a9da8715c03cca0ddf844"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:21:58.599838Z","signature_b64":"ZT4ZwvAQ/p8wvV2hNwP64r6Z1TPGZYsbnCAI/tCZEbAibK500ql8x66ThG6/F1Yn5z3f0NE4pygI3IhWcnOwBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ec762794d7aac88a6b56d0885d59f7318d4517c94b31701a902cacffc503142","last_reissued_at":"2026-07-05T08:21:58.599344Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:21:58.599344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.NA","math.PR","stat.CO"],"primary_cat":"stat.ML","authors_text":"Andrew M Stuart, Bamdad Hosseini, Houman Owhadi, Yifan Chen","submitted_at":"2024-05-21T18:38:14Z","abstract_excerpt":"The article presents a systematic study of the problem of conditioning a Gaussian random variable $\\xi$ on nonlinear observations of the form $F \\circ \\phi(\\xi)$ where $\\phi: \\mathcal{X} \\to \\mathbb{R}^N$ is a bounded linear operator and $F$ is nonlinear. Such problems arise in the context of Bayesian inference and recent machine learning-inspired PDE solvers. We give a representer theorem for the conditioned random variable $\\xi \\mid F\\circ \\phi(\\xi)$, stating that it decomposes as the sum of an infinite-dimensional Gaussian (which is identified analytically) as well as a finite-dimensional n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.13149","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2405.13149/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2405.13149","created_at":"2026-07-05T08:21:58.599403+00:00"},{"alias_kind":"arxiv_version","alias_value":"2405.13149v1","created_at":"2026-07-05T08:21:58.599403+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.13149","created_at":"2026-07-05T08:21:58.599403+00:00"},{"alias_kind":"pith_short_12","alias_value":"J3DWE6KNPKWI","created_at":"2026-07-05T08:21:58.599403+00:00"},{"alias_kind":"pith_short_16","alias_value":"J3DWE6KNPKWIRJVV","created_at":"2026-07-05T08:21:58.599403+00:00"},{"alias_kind":"pith_short_8","alias_value":"J3DWE6KN","created_at":"2026-07-05T08:21:58.599403+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/J3DWE6KNPKWIRJVVNUEILVM7OM","json":"https://pith.science/pith/J3DWE6KNPKWIRJVVNUEILVM7OM.json","graph_json":"https://pith.science/api/pith-number/J3DWE6KNPKWIRJVVNUEILVM7OM/graph.json","events_json":"https://pith.science/api/pith-number/J3DWE6KNPKWIRJVVNUEILVM7OM/events.json","paper":"https://pith.science/paper/J3DWE6KN"},"agent_actions":{"view_html":"https://pith.science/pith/J3DWE6KNPKWIRJVVNUEILVM7OM","download_json":"https://pith.science/pith/J3DWE6KNPKWIRJVVNUEILVM7OM.json","view_paper":"https://pith.science/paper/J3DWE6KN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2405.13149&json=true","fetch_graph":"https://pith.science/api/pith-number/J3DWE6KNPKWIRJVVNUEILVM7OM/graph.json","fetch_events":"https://pith.science/api/pith-number/J3DWE6KNPKWIRJVVNUEILVM7OM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J3DWE6KNPKWIRJVVNUEILVM7OM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J3DWE6KNPKWIRJVVNUEILVM7OM/action/storage_attestation","attest_author":"https://pith.science/pith/J3DWE6KNPKWIRJVVNUEILVM7OM/action/author_attestation","sign_citation":"https://pith.science/pith/J3DWE6KNPKWIRJVVNUEILVM7OM/action/citation_signature","submit_replication":"https://pith.science/pith/J3DWE6KNPKWIRJVVNUEILVM7OM/action/replication_record"}},"created_at":"2026-07-05T08:21:58.599403+00:00","updated_at":"2026-07-05T08:21:58.599403+00:00"}