{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6P2AL27ZY3WLMWLCYTKKCR562R","short_pith_number":"pith:6P2AL27Z","canonical_record":{"source":{"id":"1501.07878","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-30T18:45:46Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"55071d6ec47bb65d6cd6f69f62bc9e6d1ec1998d1f31b6ec7cc0bac2b73ecf6f","abstract_canon_sha256":"d56fcb6ffae0934a1d60cc7be21c6d3b294f8d71a7f0f4e3b2b254bcd544a3a0"},"schema_version":"1.0"},"canonical_sha256":"f3f405ebf9c6ecb65962c4d4a147bed44ec411585277946b535618770fb94141","source":{"kind":"arxiv","id":"1501.07878","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.07878","created_at":"2026-05-18T02:28:13Z"},{"alias_kind":"arxiv_version","alias_value":"1501.07878v1","created_at":"2026-05-18T02:28:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.07878","created_at":"2026-05-18T02:28:13Z"},{"alias_kind":"pith_short_12","alias_value":"6P2AL27ZY3WL","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6P2AL27ZY3WLMWLC","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6P2AL27Z","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6P2AL27ZY3WLMWLCYTKKCR562R","target":"record","payload":{"canonical_record":{"source":{"id":"1501.07878","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-30T18:45:46Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"55071d6ec47bb65d6cd6f69f62bc9e6d1ec1998d1f31b6ec7cc0bac2b73ecf6f","abstract_canon_sha256":"d56fcb6ffae0934a1d60cc7be21c6d3b294f8d71a7f0f4e3b2b254bcd544a3a0"},"schema_version":"1.0"},"canonical_sha256":"f3f405ebf9c6ecb65962c4d4a147bed44ec411585277946b535618770fb94141","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:13.861553Z","signature_b64":"D/cpRRYxscYEECOni8z3BToHjuxQ9OU+vm4XqZtSf/ds1v/xtwGOt7EwvI+zMgq0fXXjjsbSn1GRs80mrMQQBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3f405ebf9c6ecb65962c4d4a147bed44ec411585277946b535618770fb94141","last_reissued_at":"2026-05-18T02:28:13.860891Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:13.860891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.07878","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:28:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sIEBJQmme/6pUsHl3nyfT/SIckaTjAh9afRGl898E8ywMxy5sKKhkLmk+ZU0IFCqfQJUWy0F6K42rhbA1mElBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T18:20:20.933651Z"},"content_sha256":"c1735ac5cc3abbdaf55e7eb0f4f975c197a4e14ed76db4a8efaf60cd19827ab8","schema_version":"1.0","event_id":"sha256:c1735ac5cc3abbdaf55e7eb0f4f975c197a4e14ed76db4a8efaf60cd19827ab8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6P2AL27ZY3WLMWLCYTKKCR562R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Graphical Markov models for infinitely many variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Bala Rajaratnam, David Montague","submitted_at":"2015-01-30T18:45:46Z","abstract_excerpt":"Representing the conditional independences present in a multivariate random vector via graphs has found widespread use in applications, and such representations are popularly known as graphical models or Markov random fields. These models have many useful properties, but their fundamental attractive feature is their ability to reflect conditional independences between blocks of variables through graph separation, a consequence of the equivalence of the pairwise, local and global Markov properties demonstrated by Pearl and Paz (1985). Modern day applications often necessitate working with eithe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07878","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:28:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VBUoH8UkH9Tgk+/mOscJoY9udzadQ1XZDueHUaD7V3/VevoJ+2nT4hWeAOrmPiUvAZMpg+gWIkw2XQju/1XZAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T18:20:20.933994Z"},"content_sha256":"f0441e5a472940d817bf08c5b5f26b541e4fd160789a41adf3764830868a8789","schema_version":"1.0","event_id":"sha256:f0441e5a472940d817bf08c5b5f26b541e4fd160789a41adf3764830868a8789"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6P2AL27ZY3WLMWLCYTKKCR562R/bundle.json","state_url":"https://pith.science/pith/6P2AL27ZY3WLMWLCYTKKCR562R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6P2AL27ZY3WLMWLCYTKKCR562R/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T18:20:20Z","links":{"resolver":"https://pith.science/pith/6P2AL27ZY3WLMWLCYTKKCR562R","bundle":"https://pith.science/pith/6P2AL27ZY3WLMWLCYTKKCR562R/bundle.json","state":"https://pith.science/pith/6P2AL27ZY3WLMWLCYTKKCR562R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6P2AL27ZY3WLMWLCYTKKCR562R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6P2AL27ZY3WLMWLCYTKKCR562R","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d56fcb6ffae0934a1d60cc7be21c6d3b294f8d71a7f0f4e3b2b254bcd544a3a0","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-30T18:45:46Z","title_canon_sha256":"55071d6ec47bb65d6cd6f69f62bc9e6d1ec1998d1f31b6ec7cc0bac2b73ecf6f"},"schema_version":"1.0","source":{"id":"1501.07878","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.07878","created_at":"2026-05-18T02:28:13Z"},{"alias_kind":"arxiv_version","alias_value":"1501.07878v1","created_at":"2026-05-18T02:28:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.07878","created_at":"2026-05-18T02:28:13Z"},{"alias_kind":"pith_short_12","alias_value":"6P2AL27ZY3WL","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6P2AL27ZY3WLMWLC","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6P2AL27Z","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:f0441e5a472940d817bf08c5b5f26b541e4fd160789a41adf3764830868a8789","target":"graph","created_at":"2026-05-18T02:28:13Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Representing the conditional independences present in a multivariate random vector via graphs has found widespread use in applications, and such representations are popularly known as graphical models or Markov random fields. These models have many useful properties, but their fundamental attractive feature is their ability to reflect conditional independences between blocks of variables through graph separation, a consequence of the equivalence of the pairwise, local and global Markov properties demonstrated by Pearl and Paz (1985). Modern day applications often necessitate working with eithe","authors_text":"Bala Rajaratnam, David Montague","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-30T18:45:46Z","title":"Graphical Markov models for infinitely many variables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07878","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c1735ac5cc3abbdaf55e7eb0f4f975c197a4e14ed76db4a8efaf60cd19827ab8","target":"record","created_at":"2026-05-18T02:28:13Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d56fcb6ffae0934a1d60cc7be21c6d3b294f8d71a7f0f4e3b2b254bcd544a3a0","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-30T18:45:46Z","title_canon_sha256":"55071d6ec47bb65d6cd6f69f62bc9e6d1ec1998d1f31b6ec7cc0bac2b73ecf6f"},"schema_version":"1.0","source":{"id":"1501.07878","kind":"arxiv","version":1}},"canonical_sha256":"f3f405ebf9c6ecb65962c4d4a147bed44ec411585277946b535618770fb94141","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3f405ebf9c6ecb65962c4d4a147bed44ec411585277946b535618770fb94141","first_computed_at":"2026-05-18T02:28:13.860891Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:13.860891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"D/cpRRYxscYEECOni8z3BToHjuxQ9OU+vm4XqZtSf/ds1v/xtwGOt7EwvI+zMgq0fXXjjsbSn1GRs80mrMQQBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:13.861553Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.07878","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c1735ac5cc3abbdaf55e7eb0f4f975c197a4e14ed76db4a8efaf60cd19827ab8","sha256:f0441e5a472940d817bf08c5b5f26b541e4fd160789a41adf3764830868a8789"],"state_sha256":"88455514a4e922d728c10c971262452b9f757db0efa988b0a8846805885ee212"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lUA3pHGXMZyoho2FINbnCVcAgFOS2aJD5cWwYJnZxMn6TcUla9zu5UMUcrhhasZPikhYuMECKTUuF2RvpI0IAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T18:20:20.935921Z","bundle_sha256":"bc4aae809fb2f0532ca020b182db0d490de621d4a458d7400612eb2f36c372ce"}}