Pith Number

pith:JSPKBDJR

pith:2026:JSPKBDJRATAY463CUEYY4ZSYFO

not attested not anchored not stored refs pending

Bundles of Probability Schemes

Wai Yan Pong

Bundles record quotients of sample spaces, algebras of random variables, and conditional schemes simultaneously.

arxiv:2605.03902 v2 · 2026-05-05 · math.PR

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\usepackage{pith}
\pithnumber{JSPKBDJRATAY463CUEYY4ZSYFO}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

A bundle simultaneously records a quotient of a sample space, an algebra of random variables, and the family of conditional schemes over the quotient. The two natural linear functors associated with a bundle give a compact construction of conditional expectation and explain its projection properties. From this point of view we recover the laws of total expectation, total variance, total covariance, the weak law of large numbers, and the variance decomposition behind simple linear regression. We also introduce fiber products of bundles and show that they encode conditional independence, sequential random experiments, and discrete-time Markov chains.

C2weakest assumption

That the category of finite probability schemes and probability-preserving maps can be defined so that bundles accurately capture quotients, random variables, and all conditional schemes without information loss or the need for extra structures beyond the stated linear functors.

C3one line summary

Bundles of probability schemes give a categorical construction of conditional expectation that recovers laws of total expectation, variance, and covariance while using fiber products to encode conditional independence and Markov chains.

Receipt and verification

First computed	2026-05-20T00:04:34.204162Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

4c9ea08d3104c18e7b62a1318e66582b987bc09b7ede6334aa6b39e7717b9315

Aliases

arxiv: 2605.03902 · arxiv_version: 2605.03902v2 · doi: 10.48550/arxiv.2605.03902 · pith_short_12: JSPKBDJRATAY · pith_short_16: JSPKBDJRATAY463C · pith_short_8: JSPKBDJR

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/JSPKBDJRATAY463CUEYY4ZSYFO \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 4c9ea08d3104c18e7b62a1318e66582b987bc09b7ede6334aa6b39e7717b9315

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "1552a773eda51bfcaf8d3a182af6ffd23cee0ad8eff6f094c85ab4854b74b447",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-05T15:55:48Z",
    "title_canon_sha256": "a9ca94397db8f93257ce5693eee8425fe2c7aa5cf673d1473c1275a4c3ce067f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.03902",
    "kind": "arxiv",
    "version": 2
  }
}