{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:IYERAWICXZ5ACIHRSNCA35PCSY","short_pith_number":"pith:IYERAWIC","canonical_record":{"source":{"id":"1502.00124","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-31T15:46:39Z","cross_cats_sorted":[],"title_canon_sha256":"419a4af221962ca87e2543ac39af42e716cba86b70dbfcf151fe0b42f298b938","abstract_canon_sha256":"e5501bbf46494aede07d7daff78742ee0ec17c4d2581a2009c7e81648737a20a"},"schema_version":"1.0"},"canonical_sha256":"4609105902be7a0120f193440df5e2963a20905a6a9676f095d27628d3594713","source":{"kind":"arxiv","id":"1502.00124","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.00124","created_at":"2026-05-18T00:06:44Z"},{"alias_kind":"arxiv_version","alias_value":"1502.00124v2","created_at":"2026-05-18T00:06:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.00124","created_at":"2026-05-18T00:06:44Z"},{"alias_kind":"pith_short_12","alias_value":"IYERAWICXZ5A","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"IYERAWICXZ5ACIHR","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"IYERAWIC","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:IYERAWICXZ5ACIHRSNCA35PCSY","target":"record","payload":{"canonical_record":{"source":{"id":"1502.00124","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-31T15:46:39Z","cross_cats_sorted":[],"title_canon_sha256":"419a4af221962ca87e2543ac39af42e716cba86b70dbfcf151fe0b42f298b938","abstract_canon_sha256":"e5501bbf46494aede07d7daff78742ee0ec17c4d2581a2009c7e81648737a20a"},"schema_version":"1.0"},"canonical_sha256":"4609105902be7a0120f193440df5e2963a20905a6a9676f095d27628d3594713","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:44.183529Z","signature_b64":"fe+Kx6MK18ATX8A309FeOGhcCaru2f5WjdWRztA6gEudPYfgegF4YheiwsJlSsM4teUBcMr8U+OtrhfAdf/mBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4609105902be7a0120f193440df5e2963a20905a6a9676f095d27628d3594713","last_reissued_at":"2026-05-18T00:06:44.183195Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:44.183195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.00124","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:06:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y1TaLPqY2EsA8ClbuPNzWEKa4woDV/QgOKIbjBhNbGNMw3erEeEtoEyWpJKDcFiXXu0HFbxJBjKoLJ4xlxrxBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T07:21:23.122408Z"},"content_sha256":"e0ae973feb77964717e656997ab8470eb94267d134ea7e935ba1bed949a9e48b","schema_version":"1.0","event_id":"sha256:e0ae973feb77964717e656997ab8470eb94267d134ea7e935ba1bed949a9e48b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:IYERAWICXZ5ACIHRSNCA35PCSY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Law of total probability and Bayes' theorem in Riesz spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Liang Hong","submitted_at":"2015-01-31T15:46:39Z","abstract_excerpt":"This note generalizes the notion of conditional probability to Riesz spaces using the order-theoretic approach. With the aid of this concept, we establish the law of total probability and Bayes' theorem in Riesz spaces; we also prove an inclusion-exclusion formula in Riesz spaces. Several examples are provided to show that the law of total probability, Bayes' theorem and inclusion-exclusion formula in probability theory are special cases of our results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.00124","kind":"arxiv","version":2},"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":"2026-05-18T00:06:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H4gXVrJ/acKBYfrj5+LW28lnyTtM24DZ4EjsnuCPL9DnGqXcTzzgw664SXp7g1o6nG+t0onpBQgWi4Gr4k51CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T07:21:23.123037Z"},"content_sha256":"7a33bc294215c589b6b48ce364ead37f5e30cfc83292433b2354dcbf2407c96f","schema_version":"1.0","event_id":"sha256:7a33bc294215c589b6b48ce364ead37f5e30cfc83292433b2354dcbf2407c96f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IYERAWICXZ5ACIHRSNCA35PCSY/bundle.json","state_url":"https://pith.science/pith/IYERAWICXZ5ACIHRSNCA35PCSY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IYERAWICXZ5ACIHRSNCA35PCSY/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":"2026-05-30T07:21:23Z","links":{"resolver":"https://pith.science/pith/IYERAWICXZ5ACIHRSNCA35PCSY","bundle":"https://pith.science/pith/IYERAWICXZ5ACIHRSNCA35PCSY/bundle.json","state":"https://pith.science/pith/IYERAWICXZ5ACIHRSNCA35PCSY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IYERAWICXZ5ACIHRSNCA35PCSY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:IYERAWICXZ5ACIHRSNCA35PCSY","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":"e5501bbf46494aede07d7daff78742ee0ec17c4d2581a2009c7e81648737a20a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-31T15:46:39Z","title_canon_sha256":"419a4af221962ca87e2543ac39af42e716cba86b70dbfcf151fe0b42f298b938"},"schema_version":"1.0","source":{"id":"1502.00124","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.00124","created_at":"2026-05-18T00:06:44Z"},{"alias_kind":"arxiv_version","alias_value":"1502.00124v2","created_at":"2026-05-18T00:06:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.00124","created_at":"2026-05-18T00:06:44Z"},{"alias_kind":"pith_short_12","alias_value":"IYERAWICXZ5A","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"IYERAWICXZ5ACIHR","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"IYERAWIC","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:7a33bc294215c589b6b48ce364ead37f5e30cfc83292433b2354dcbf2407c96f","target":"graph","created_at":"2026-05-18T00:06:44Z","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":"This note generalizes the notion of conditional probability to Riesz spaces using the order-theoretic approach. With the aid of this concept, we establish the law of total probability and Bayes' theorem in Riesz spaces; we also prove an inclusion-exclusion formula in Riesz spaces. Several examples are provided to show that the law of total probability, Bayes' theorem and inclusion-exclusion formula in probability theory are special cases of our results.","authors_text":"Liang Hong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-31T15:46:39Z","title":"Law of total probability and Bayes' theorem in Riesz spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.00124","kind":"arxiv","version":2},"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:e0ae973feb77964717e656997ab8470eb94267d134ea7e935ba1bed949a9e48b","target":"record","created_at":"2026-05-18T00:06:44Z","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":"e5501bbf46494aede07d7daff78742ee0ec17c4d2581a2009c7e81648737a20a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-01-31T15:46:39Z","title_canon_sha256":"419a4af221962ca87e2543ac39af42e716cba86b70dbfcf151fe0b42f298b938"},"schema_version":"1.0","source":{"id":"1502.00124","kind":"arxiv","version":2}},"canonical_sha256":"4609105902be7a0120f193440df5e2963a20905a6a9676f095d27628d3594713","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4609105902be7a0120f193440df5e2963a20905a6a9676f095d27628d3594713","first_computed_at":"2026-05-18T00:06:44.183195Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:44.183195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fe+Kx6MK18ATX8A309FeOGhcCaru2f5WjdWRztA6gEudPYfgegF4YheiwsJlSsM4teUBcMr8U+OtrhfAdf/mBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:44.183529Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.00124","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e0ae973feb77964717e656997ab8470eb94267d134ea7e935ba1bed949a9e48b","sha256:7a33bc294215c589b6b48ce364ead37f5e30cfc83292433b2354dcbf2407c96f"],"state_sha256":"6c528f5913cd156f0a803369305126b34b6f5f9807277256442ce50e25bb855a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Adt6wcnTMMJv/+z8foT7YAaXLNj9d8U2MNeChDP4dD56Pk34atDaHZuTQUpb6/xoqVIR8O6dQ2ilItIu6bAQAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T07:21:23.126614Z","bundle_sha256":"9b92dac2900415921ee3de913325f798132baae3476cba41b008dcab5677daab"}}