Pith Number

pith:UPHINIGV

pith:2026:UPHINIGVGYFBV6YV7WTKNB43VV

not attested not anchored not stored refs pending

Direct Bethe Free Energy Minimization for Bayesian Neural Networks

Pavel Prochazka

Direct Bethe free energy minimization for BNNs produces losses strictly between MAP and ELBO, enables joint empirical Bayes in one gradient pass, and matches reference methods on UCI benchmarks at single-pass cost.

arxiv:2605.08446 v3 · 2026-05-08 · cs.LG

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{UPHINIGVGYFBV6YV7WTKNB43VV}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

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

Both objectives sit strictly between MAP and the ELBO (L_MAP ≤ L_Bethe ≤ L_ELBO), removing the structural Jensen gap that no choice of variational family can close. The Z-consistent prior formulation makes the prior precision a differentiable parameter, enabling empirical Bayes in a single gradient pass.

C2weakest assumption

Restricting the weight posterior to a last-layer Gaussian yields analytically tractable losses and that the factor graph is tree-structured so the Bethe free energy is exact; if these do not hold the closed-form claims and inequality may fail.

C3one line summary

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-07-07T01:16:48.338728Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

a3ce86a0d5360a1afb15fda6a6879bad4a9198458c13427d8d0e7fce9becd3b2

Aliases

arxiv: 2605.08446 · arxiv_version: 2605.08446v3 · doi: 10.48550/arxiv.2605.08446 · pith_short_12: UPHINIGVGYFB · pith_short_16: UPHINIGVGYFBV6YV · pith_short_8: UPHINIGV

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/UPHINIGVGYFBV6YV7WTKNB43VV \
  | 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: a3ce86a0d5360a1afb15fda6a6879bad4a9198458c13427d8d0e7fce9becd3b2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "63fad6c1f6fe122d50a2fd4278de1d1fa40ed93e635af233d81b62f6742be210",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-08T20:12:48Z",
    "title_canon_sha256": "1385fe557fea6e4e2f1a7e135cd60fd09890f611f999ce2a91dbcbcd6292a37d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.08446",
    "kind": "arxiv",
    "version": 3
  }
}