Pith Number

pith:4GGQQUM6

pith:2026:4GGQQUM6BGEXQ45ADURYCLJTAB

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Bayesian Hierarchical Models and the Maximum Entropy Principle

Brendon J. Brewer

When conditional priors in hierarchical models are maximum entropy distributions, the marginal prior is also maximum entropy but constrained on a function of the parameters.

arxiv:2603.10252 v2 · 2026-03-10 · stat.ML · cs.LG · physics.data-an · stat.ME

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\pithnumber{4GGQQUM6BGEXQ45ADURYCLJTAB}

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4 Citations open

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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

when the prior given the hyperparameters is a canonical distribution (a maximum entropy distribution with moment constraints), the dependent marginal prior also has a maximum entropy property, with a different constraint. This constraint is on the marginal distribution of some function of the unknown quantities.

C2weakest assumption

That the conditional prior given hyperparameters is exactly a canonical maximum entropy distribution with moment constraints, and that the hierarchical structure produces the claimed dependence in the marginal.

C3one line summary

Bayesian hierarchical models with canonical maxent conditional priors induce marginal priors that are themselves maxent distributions subject to a constraint on a function of the parameters.

Receipt and verification

First computed	2026-07-10T00:18:44.138614Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e18d08519e09897873a01d23812d33005cfcf45146a0829f73c2f6db28edabeb

Aliases

arxiv: 2603.10252 · arxiv_version: 2603.10252v2 · doi: 10.48550/arxiv.2603.10252 · pith_short_12: 4GGQQUM6BGEX · pith_short_16: 4GGQQUM6BGEXQ45A · pith_short_8: 4GGQQUM6

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/4GGQQUM6BGEXQ45ADURYCLJTAB \
  | 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: e18d08519e09897873a01d23812d33005cfcf45146a0829f73c2f6db28edabeb

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9a01f32a494d78e1eba5d26c6be1e49f4706b17dd5352d82283829a0ad4685c2",
    "cross_cats_sorted": [
      "cs.LG",
      "physics.data-an",
      "stat.ME"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "stat.ML",
    "submitted_at": "2026-03-10T22:14:23Z",
    "title_canon_sha256": "a1c50d7b279c5c634dd9152914274b9d932575b81f7e0ed5be89bf25ae70bd63"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2603.10252",
    "kind": "arxiv",
    "version": 2
  }
}