{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:OQ4DUP2RTRT6XOTW2KPF767C53","short_pith_number":"pith:OQ4DUP2R","canonical_record":{"source":{"id":"2005.14199","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2020-05-28T18:00:00Z","cross_cats_sorted":["astro-ph.IM","physics.data-an"],"title_canon_sha256":"d7550cb1d9a0a4b4d2e0afae1593cc1805033eced7d333f3d3cc951d9e4c78d7","abstract_canon_sha256":"24bf78abaa56526774e3907e6df56164b564e811fc9900926ced3fad02098ca9"},"schema_version":"1.0"},"canonical_sha256":"74383a3f519c67ebba76d29e5ffbe2eeebd86110dff79e243282a0455aecadd3","source":{"kind":"arxiv","id":"2005.14199","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.14199","created_at":"2026-07-05T01:06:35Z"},{"alias_kind":"arxiv_version","alias_value":"2005.14199v1","created_at":"2026-07-05T01:06:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.14199","created_at":"2026-07-05T01:06:35Z"},{"alias_kind":"pith_short_12","alias_value":"OQ4DUP2RTRT6","created_at":"2026-07-05T01:06:35Z"},{"alias_kind":"pith_short_16","alias_value":"OQ4DUP2RTRT6XOTW","created_at":"2026-07-05T01:06:35Z"},{"alias_kind":"pith_short_8","alias_value":"OQ4DUP2R","created_at":"2026-07-05T01:06:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:OQ4DUP2RTRT6XOTW2KPF767C53","target":"record","payload":{"canonical_record":{"source":{"id":"2005.14199","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2020-05-28T18:00:00Z","cross_cats_sorted":["astro-ph.IM","physics.data-an"],"title_canon_sha256":"d7550cb1d9a0a4b4d2e0afae1593cc1805033eced7d333f3d3cc951d9e4c78d7","abstract_canon_sha256":"24bf78abaa56526774e3907e6df56164b564e811fc9900926ced3fad02098ca9"},"schema_version":"1.0"},"canonical_sha256":"74383a3f519c67ebba76d29e5ffbe2eeebd86110dff79e243282a0455aecadd3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:06:35.315148Z","signature_b64":"hO/b54my5vQcqP98IKeWc1L27UNQaHzsf8HgujNGbmJAUy1MGbdhhJ9JwwdO19mCXzJGu9Qdr9m6Iy7fxLo4Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74383a3f519c67ebba76d29e5ffbe2eeebd86110dff79e243282a0455aecadd3","last_reissued_at":"2026-07-05T01:06:35.314453Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:06:35.314453Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2005.14199","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":"2026-07-05T01:06:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JQ7D642f13kSDwOuMha3KloQOA8It5Qh/il8vPzuukTWWt6mLWik02H6t5FMrs9YOFfVFAouWPjiB/h4OSFJCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T07:32:17.260641Z"},"content_sha256":"c73fab4b7f8a781ca46afd1e4690f0f8274fc3a2a426994f5c277ad3e9d235af","schema_version":"1.0","event_id":"sha256:c73fab4b7f8a781ca46afd1e4690f0f8274fc3a2a426994f5c277ad3e9d235af"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:OQ4DUP2RTRT6XOTW2KPF767C53","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Data Analysis Recipes: Products of multivariate Gaussians in Bayesian inferences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["astro-ph.IM","physics.data-an"],"primary_cat":"stat.CO","authors_text":"Adrian M. Price-Whelan (Flatiron), Boris Leistedt (Imperial) (NYU), David W. Hogg (NYU) (MPIA) (Flatiron)","submitted_at":"2020-05-28T18:00:00Z","abstract_excerpt":"A product of two Gaussians (or normal distributions) is another Gaussian. That's a valuable and useful fact! Here we use it to derive a refactoring of a common product of multivariate Gaussians: The product of a Gaussian likelihood times a Gaussian prior, where some or all of those parameters enter the likelihood only in the mean and only linearly. That is, a linear, Gaussian, Bayesian model. This product of a likelihood times a prior pdf can be refactored into a product of a marginalized likelihood (or a Bayesian evidence) times a posterior pdf, where (in this case) both of these are also Gau"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.14199","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/2005.14199/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:06:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wXTPnzYANUNgErr5omxX85VrtG7nll+lWzsDqbpVI5fvtvjCWoP7KlILf83/lmRn7KVr+y6MNA/DIwtld9h8Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T07:32:17.261280Z"},"content_sha256":"5a4f60928a1a6ca98e61eb72494765e1261205e5c3b64da475a93c4aee923ab1","schema_version":"1.0","event_id":"sha256:5a4f60928a1a6ca98e61eb72494765e1261205e5c3b64da475a93c4aee923ab1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OQ4DUP2RTRT6XOTW2KPF767C53/bundle.json","state_url":"https://pith.science/pith/OQ4DUP2RTRT6XOTW2KPF767C53/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OQ4DUP2RTRT6XOTW2KPF767C53/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":"2026-08-12T07:32:17Z","links":{"resolver":"https://pith.science/pith/OQ4DUP2RTRT6XOTW2KPF767C53","bundle":"https://pith.science/pith/OQ4DUP2RTRT6XOTW2KPF767C53/bundle.json","state":"https://pith.science/pith/OQ4DUP2RTRT6XOTW2KPF767C53/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OQ4DUP2RTRT6XOTW2KPF767C53/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:OQ4DUP2RTRT6XOTW2KPF767C53","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":"24bf78abaa56526774e3907e6df56164b564e811fc9900926ced3fad02098ca9","cross_cats_sorted":["astro-ph.IM","physics.data-an"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2020-05-28T18:00:00Z","title_canon_sha256":"d7550cb1d9a0a4b4d2e0afae1593cc1805033eced7d333f3d3cc951d9e4c78d7"},"schema_version":"1.0","source":{"id":"2005.14199","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.14199","created_at":"2026-07-05T01:06:35Z"},{"alias_kind":"arxiv_version","alias_value":"2005.14199v1","created_at":"2026-07-05T01:06:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.14199","created_at":"2026-07-05T01:06:35Z"},{"alias_kind":"pith_short_12","alias_value":"OQ4DUP2RTRT6","created_at":"2026-07-05T01:06:35Z"},{"alias_kind":"pith_short_16","alias_value":"OQ4DUP2RTRT6XOTW","created_at":"2026-07-05T01:06:35Z"},{"alias_kind":"pith_short_8","alias_value":"OQ4DUP2R","created_at":"2026-07-05T01:06:35Z"}],"graph_snapshots":[{"event_id":"sha256:5a4f60928a1a6ca98e61eb72494765e1261205e5c3b64da475a93c4aee923ab1","target":"graph","created_at":"2026-07-05T01:06:35Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2005.14199/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A product of two Gaussians (or normal distributions) is another Gaussian. That's a valuable and useful fact! Here we use it to derive a refactoring of a common product of multivariate Gaussians: The product of a Gaussian likelihood times a Gaussian prior, where some or all of those parameters enter the likelihood only in the mean and only linearly. That is, a linear, Gaussian, Bayesian model. This product of a likelihood times a prior pdf can be refactored into a product of a marginalized likelihood (or a Bayesian evidence) times a posterior pdf, where (in this case) both of these are also Gau","authors_text":"Adrian M. Price-Whelan (Flatiron), Boris Leistedt (Imperial) (NYU), David W. Hogg (NYU) (MPIA) (Flatiron)","cross_cats":["astro-ph.IM","physics.data-an"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2020-05-28T18:00:00Z","title":"Data Analysis Recipes: Products of multivariate Gaussians in Bayesian inferences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.14199","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:c73fab4b7f8a781ca46afd1e4690f0f8274fc3a2a426994f5c277ad3e9d235af","target":"record","created_at":"2026-07-05T01:06:35Z","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":"24bf78abaa56526774e3907e6df56164b564e811fc9900926ced3fad02098ca9","cross_cats_sorted":["astro-ph.IM","physics.data-an"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2020-05-28T18:00:00Z","title_canon_sha256":"d7550cb1d9a0a4b4d2e0afae1593cc1805033eced7d333f3d3cc951d9e4c78d7"},"schema_version":"1.0","source":{"id":"2005.14199","kind":"arxiv","version":1}},"canonical_sha256":"74383a3f519c67ebba76d29e5ffbe2eeebd86110dff79e243282a0455aecadd3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"74383a3f519c67ebba76d29e5ffbe2eeebd86110dff79e243282a0455aecadd3","first_computed_at":"2026-07-05T01:06:35.314453Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:06:35.314453Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hO/b54my5vQcqP98IKeWc1L27UNQaHzsf8HgujNGbmJAUy1MGbdhhJ9JwwdO19mCXzJGu9Qdr9m6Iy7fxLo4Bw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:06:35.315148Z","signed_message":"canonical_sha256_bytes"},"source_id":"2005.14199","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c73fab4b7f8a781ca46afd1e4690f0f8274fc3a2a426994f5c277ad3e9d235af","sha256:5a4f60928a1a6ca98e61eb72494765e1261205e5c3b64da475a93c4aee923ab1"],"state_sha256":"0b4acbdeb77170ba3efaf3f16c5c4bf5117d3f56f910cfcdd44c727f3ac7aed6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ACqt75qpMl9SKtWaSkV9VDbViadwDmkgZTrh3SQM+E1V00lFhpXsBeZ3vCnYrKnOFDVkYcqFYkKLNjUBGoBMBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T07:32:17.267641Z","bundle_sha256":"5005d4318f78516080abe0cb4d32619c4268e013ad2f9ff46169484e27dc91fe"}}