PAC-Bayes confidence margin

Put a non-vacuous, distribution-free upper bound on your governance head's risk using the McAllester-1999 PAC-Bayes inequality — the runnable implementation is pinned below.

Headline number: n=100,000, KL=0.5, δ=0.05, R̂=0.05 → risk bound ≈ 0.057 (non-vacuous).

How confident can you be that a passing Λ-gate generalizes beyond the sample you measured it on? PAC-Bayes answers with a tail bound. The implementation is real and linked at an immutable szl-cookbook revision: recipes/knot-calculus-v1/code/src/pac-bayes-bound.ts.

Honest scope. The closed-form arithmetic is proved in Lean (TH13governanceHead_PACBayes_bound); the probabilistic Pr ≥ 1−δ quantifier is the documented residual sorry. The bound is a bound, not a guarantee of optimality, and Λ remains Conjecture 1.

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The inequality

With probability ≥ 1−δ over an i.i.d. n-sample, for posterior Q and prior P over governance policies,

[ R(Q) \le \hat{R}(Q) + \sqrt{\frac{\mathrm{KL}(Q,|,P) + \ln!\frac{2\sqrt{n}}{\delta}}{2n}}. ]

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Prerequisites

git clone https://github.com/szl-holdings/szl-cookbook.git
cd szl-cookbook
git checkout 1be98cc0fed44fba98bfb89a1056c6f3364ae736
cd recipes/knot-calculus-v1/code && npm install

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Quickstart (runnable, in this repo)

cd recipes/knot-calculus-v1/code
npx tsx tests/demo.ts          # step 3 prints the PAC-Bayes bound for the worked example

Or call the function directly:

import { pacBayesBound } from "./src/pac-bayes-bound";
const r = pacBayesBound({ empiricalRisk: 0.05, klDivergence: 0.5, sampleSize: 100_000, delta: 0.05 });
console.log(r);
// => { slack: ~0.0070, upperBound: ~0.0571, nonVacuous: true }

Reproduce in pure Python (no deps):

import math
def pac_bayes(R_hat, KL, n, delta):
    slack = math.sqrt((KL + math.log(2 * math.sqrt(n) / delta)) / (2 * n))
    ub = R_hat + slack
    return {"slack": slack, "upperBound": ub, "nonVacuous": ub < 1.0}
print(pac_bayes(0.05, 0.5, 100_000, 0.05))
# => {'slack': 0.00705…, 'upperBound': 0.05705…, 'nonVacuous': True}

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Full walkthrough

Step 1 — Estimate the empirical risk R̂(Q)

Replay your receipts (e.g., a compliance regime from recipe 03) and count the fraction where the gate's decision disagreed with the ground-truth label. That fraction is R̂(Q) on bounded 0–1 loss.

Step 2 — Estimate KL(Q‖P)

P is the doctrine prior over policies (the locked axis priors); Q is your tuned posterior. For a diagonal-Gaussian parameterization, KL is the usual closed form. Keep KL small by staying close to the doctrine prior — the bound rewards that directly.

Step 3 — Choose n and δ

n is your evaluation sample size; δ is the failure probability (0.05 is standard). The slack shrinks like (1/\sqrt{n}): going from n=10⁴ to n=10⁵ roughly cuts the slack by ~3×.

Step 4 — Read the margin

The confidence margin is lambda_floor − upperBound_on_failure. If your Λ floor is 0.9 and the bound says failure risk ≤ 0.057, you have a real, distribution-free safety margin — not a vibe.

Step 5 — Non-vacuity check

nonVacuous must be true (upperBound < 1). A vacuous bound means you need more samples or a posterior closer to the prior. Lotfi et al. 2023 show non-vacuous bounds are achievable even for LLM-scale heads.

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Lean obligation

Theorem	File	Status
TH13 governanceHead_PACBayes_bound	lutar-lean/Lutar/PACBayes.lean	closed-form proved; Pr-quantifier open

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See also

• 03 — Fine-tune a compliance regime — produces R̂ and KL.
• 08 — Receipt knot algebra — same demo program.
• Code: pac-bayes-bound.ts

Cite this recipe

@misc{szl_cookbook_pac_bayes_2026,
  title        = {PAC-Bayes confidence margin (SZL Cookbook recipe 09)},
  author       = {{SZL Holdings}},
  year         = {2026},
  howpublished = {\url{https://github.com/szl-holdings/szl-cookbook/blob/main/recipes/09-pac-bayes-confidence-margin.md}},
  note         = {McAllester-1999; TH13 closed-form proved, Pr-quantifier open. Λ = Conjecture 1.}
}

References: McAllester 1999, COLT; McAllester 2003, Machine Learning 51:5–21; Lotfi et al. 2023, arXiv:2312.17173 (NeurIPS); Amari 1985 (Springer LNS 28), 2016 (Springer).

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Doctrine v11 LOCKED — 749/14/163 — kernel c7c0ba17 · Λ = Conjecture 1 · SLSA L1 (honest)