{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:5WWUY2LL2Y2VQLFVRR6XEI4GQP","short_pith_number":"pith:5WWUY2LL","canonical_record":{"source":{"id":"1611.08580","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-11-25T20:34:36Z","cross_cats_sorted":[],"title_canon_sha256":"40bb3ef63d906e053e75b1666ad6911bc132cd8073d4ff01d883a9898ca168cd","abstract_canon_sha256":"9f859036b8b7dc758d81097825a06918ca355696b886d79316e4d82fe5194c76"},"schema_version":"1.0"},"canonical_sha256":"edad4c696bd635582cb58c7d72238683e7e8798cc174210e07e3aec9eb5b0ea5","source":{"kind":"arxiv","id":"1611.08580","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.08580","created_at":"2026-05-18T00:43:01Z"},{"alias_kind":"arxiv_version","alias_value":"1611.08580v2","created_at":"2026-05-18T00:43:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.08580","created_at":"2026-05-18T00:43:01Z"},{"alias_kind":"pith_short_12","alias_value":"5WWUY2LL2Y2V","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5WWUY2LL2Y2VQLFV","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5WWUY2LL","created_at":"2026-05-18T12:30:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:5WWUY2LL2Y2VQLFVRR6XEI4GQP","target":"record","payload":{"canonical_record":{"source":{"id":"1611.08580","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-11-25T20:34:36Z","cross_cats_sorted":[],"title_canon_sha256":"40bb3ef63d906e053e75b1666ad6911bc132cd8073d4ff01d883a9898ca168cd","abstract_canon_sha256":"9f859036b8b7dc758d81097825a06918ca355696b886d79316e4d82fe5194c76"},"schema_version":"1.0"},"canonical_sha256":"edad4c696bd635582cb58c7d72238683e7e8798cc174210e07e3aec9eb5b0ea5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:01.644850Z","signature_b64":"SHbkd6dJgG6OLek8x4ChCQc1T3VaQiC9ryaK3tVVYINk6B6XTLJlqnMrep8+n3Vlp51VrzwZlun+UaLApPuRDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"edad4c696bd635582cb58c7d72238683e7e8798cc174210e07e3aec9eb5b0ea5","last_reissued_at":"2026-05-18T00:43:01.644218Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:01.644218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.08580","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":"2026-05-18T00:43:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8O1Sj08xxkehHrBKfZmbGYdeuGz6x9fLndBz5KCrid1SDPDEnJpa3kr2YMUM3IHSAFuHiz11qxiERhWx8N5ZCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T16:14:44.240366Z"},"content_sha256":"ca0822e551889d3c2cc9929aa0561e08a9a696b0f40de87a6551ab9bff674aa2","schema_version":"1.0","event_id":"sha256:ca0822e551889d3c2cc9929aa0561e08a9a696b0f40de87a6551ab9bff674aa2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:5WWUY2LL2Y2VQLFVRR6XEI4GQP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On computing distributions of products of random variables via Gaussian multiresolution analysis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Gregory Beylkin, Ignas Satkauskas, Lucas Monzon","submitted_at":"2016-11-25T20:34:36Z","abstract_excerpt":"We introduce a new approximate multiresolution analysis (MRA) using a single Gaussian as the scaling function, which we call Gaussian MRA (GMRA). As an initial application, we employ this new tool to accurately and efficiently compute the probability density function (PDF) of the product of independent random variables. In contrast with Monte-Carlo (MC) type methods (the only other universal approach known to address this problem), our method not only achieves accuracies beyond the reach of MC but also produces a PDF expressed as a Gaussian mixture, thus allowing for further efficient computat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08580","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":"2026-05-18T00:43:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U/6Oorgc/5GPuoamAKVojcErScwKW679FnpzqOlwnsYxSofrojXmsP/38vJiOOmTobWhpj9R9bUurdIjVBiuAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T16:14:44.240871Z"},"content_sha256":"0a8f7291a780c99c21361899a481de767e2936ab81324c08afa76f146cff5fb4","schema_version":"1.0","event_id":"sha256:0a8f7291a780c99c21361899a481de767e2936ab81324c08afa76f146cff5fb4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5WWUY2LL2Y2VQLFVRR6XEI4GQP/bundle.json","state_url":"https://pith.science/pith/5WWUY2LL2Y2VQLFVRR6XEI4GQP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5WWUY2LL2Y2VQLFVRR6XEI4GQP/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":"2026-05-27T16:14:44Z","links":{"resolver":"https://pith.science/pith/5WWUY2LL2Y2VQLFVRR6XEI4GQP","bundle":"https://pith.science/pith/5WWUY2LL2Y2VQLFVRR6XEI4GQP/bundle.json","state":"https://pith.science/pith/5WWUY2LL2Y2VQLFVRR6XEI4GQP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5WWUY2LL2Y2VQLFVRR6XEI4GQP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:5WWUY2LL2Y2VQLFVRR6XEI4GQP","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":"9f859036b8b7dc758d81097825a06918ca355696b886d79316e4d82fe5194c76","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-11-25T20:34:36Z","title_canon_sha256":"40bb3ef63d906e053e75b1666ad6911bc132cd8073d4ff01d883a9898ca168cd"},"schema_version":"1.0","source":{"id":"1611.08580","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.08580","created_at":"2026-05-18T00:43:01Z"},{"alias_kind":"arxiv_version","alias_value":"1611.08580v2","created_at":"2026-05-18T00:43:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.08580","created_at":"2026-05-18T00:43:01Z"},{"alias_kind":"pith_short_12","alias_value":"5WWUY2LL2Y2V","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_16","alias_value":"5WWUY2LL2Y2VQLFV","created_at":"2026-05-18T12:30:01Z"},{"alias_kind":"pith_short_8","alias_value":"5WWUY2LL","created_at":"2026-05-18T12:30:01Z"}],"graph_snapshots":[{"event_id":"sha256:0a8f7291a780c99c21361899a481de767e2936ab81324c08afa76f146cff5fb4","target":"graph","created_at":"2026-05-18T00:43:01Z","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":"We introduce a new approximate multiresolution analysis (MRA) using a single Gaussian as the scaling function, which we call Gaussian MRA (GMRA). As an initial application, we employ this new tool to accurately and efficiently compute the probability density function (PDF) of the product of independent random variables. In contrast with Monte-Carlo (MC) type methods (the only other universal approach known to address this problem), our method not only achieves accuracies beyond the reach of MC but also produces a PDF expressed as a Gaussian mixture, thus allowing for further efficient computat","authors_text":"Gregory Beylkin, Ignas Satkauskas, Lucas Monzon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-11-25T20:34:36Z","title":"On computing distributions of products of random variables via Gaussian multiresolution analysis"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08580","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:ca0822e551889d3c2cc9929aa0561e08a9a696b0f40de87a6551ab9bff674aa2","target":"record","created_at":"2026-05-18T00:43:01Z","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":"9f859036b8b7dc758d81097825a06918ca355696b886d79316e4d82fe5194c76","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-11-25T20:34:36Z","title_canon_sha256":"40bb3ef63d906e053e75b1666ad6911bc132cd8073d4ff01d883a9898ca168cd"},"schema_version":"1.0","source":{"id":"1611.08580","kind":"arxiv","version":2}},"canonical_sha256":"edad4c696bd635582cb58c7d72238683e7e8798cc174210e07e3aec9eb5b0ea5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"edad4c696bd635582cb58c7d72238683e7e8798cc174210e07e3aec9eb5b0ea5","first_computed_at":"2026-05-18T00:43:01.644218Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:01.644218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SHbkd6dJgG6OLek8x4ChCQc1T3VaQiC9ryaK3tVVYINk6B6XTLJlqnMrep8+n3Vlp51VrzwZlun+UaLApPuRDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:01.644850Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.08580","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ca0822e551889d3c2cc9929aa0561e08a9a696b0f40de87a6551ab9bff674aa2","sha256:0a8f7291a780c99c21361899a481de767e2936ab81324c08afa76f146cff5fb4"],"state_sha256":"d83674aa794f5f6c7a18bfabeee5aeb1858847e6c5b7703e40319ea2acd02562"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lV7CnznLk1wR63edxzuGDhgIkjA5zhmrR8SBs03gw+7E/CbDrnJRi9CfTe+skDa/soakJDiOtt828eCAemC8Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T16:14:44.244148Z","bundle_sha256":"74e15a88499701a91d3a65ac972c4fffd69c9241f00f478b09597e90a54cd925"}}