Pointwise coverage, averaging and selection

Goal and scope

Compute pointwise and worst-case coverage of a fixed interval procedure.

Frequentist coverage is not an observed-data posterior; selection must be incorporated anew.

Givens and symbols

• work

• G1 — Theta={0,1}, Omega={a,b,c}, P0=(1/2,1/4,1/4), P1=(1/4,1/4,1/2). Precommit I(a)=[0,0], I(b)=[0,1], I(c)=[0,0].

• I(x) — precommitted interval procedure
• C(θ) — pointwise coverage
• A — selection event

Rules and sources

• mko_definition · cited — Supply the complete normalized finite sampling law for every theta. The precommitted procedure I(x)=[L(x),U(x)] is defined at every sample, independent of the unknown theta, and has L(x)≤U(x). Its pointwise coverage is sum_x P_theta(x)*1{L(x)≤theta≤U(x)}; its worst-case coverage is the minimum over the finite theta carrier. An average-only boundary must have distinct pointwise coverages and a target with average≥target>minimum. Posterior probability uses a prior and observed data and is not this coverage. Selection included in the full procedure can be analyzed again; the error is ignoring selection while reusing old coverage. Zero selected mass makes the corresponding conditional coverage undefined. · mko-finite-confidence-interval-coverage
• work · prerequisite — C(theta)=sum_x Ptheta(x)1{L(x)≤theta≤U(x)}; worst case is the minimum; positive-mass selection uses the conditional intersection quotient.

Main derivation

P_0,P_1

1/2+1/4+1/4=1,\quad1/4+1/4+1/2=1

I(a),I(b),I(c)

L(x)\le U(x)\quad\text{for all }x\in\Omega

\theta=0:\quad\mathbf1\{0\in I(x)\}

(1,1,1),\quad C(0)=1

\theta=1:\quad\mathbf1\{1\in I(x)\}

(0,1,0),\quad C(1)=P_1(b)=1/4

\min_{\theta\in\Theta}C(\theta)

\min(1,1/4)=1/4

\overline C=(C(0)+C(1))/2,\quad q=1/2

\overline C=5/8\ge1/2>1/4=\min C

A=\{a,b\}\text{ selected},\quad\theta=1

P_1(1\in I(X)\mid A)=\frac{P_1(b)}{P_1(a)+P_1(b)}=1/2

x_{\rm obs}=b,\quad I(b)=[0,1]

\mathbf1\{\theta\in I(b)\}=1\quad(\theta=0,1)

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — A posterior needs a prior and data update; no prior was supplied.
• S07 · premise_audit · G1, S06, work — The selection denominator 1/2 is positive; zero selection mass would leave it undefined.
• S06 · premise_audit · G1, S05, work — The average is a specified equal-weight summary, not an automatic frequentist guarantee or prior.
• S05 · premise_audit · G1, S04, work — Both theta values have been fully computed.
• S04 · premise_audit · G1, S03, work — Even coincident numerical masses do not remove the need to use the correct law.
• S03 · premise_audit · G1, S02, work — Endpoint equality must be included.
• S02 · premise_audit · G1, S01, work — The two singleton intervals[0,0] are valid.
• S01 · premise_audit · G1, work — The procedure cannot change with true theta.

Error controls and negative controls

Complete procedure

Fix intervals at a,b,c before weighting either law.

Recompute selection

Compute selection numerators and denominators under the same theta law.

• Average5/8 means every theta has coverage≥1/2. → reject — At theta=1 coverage is only 1/4.
• After selecting a,b coverage remains1/4. → reject — The recomputed conditional coverage at theta=1 is 1/2.
• After b the posterior probability of theta in I(b) is 1/4. → reject — I(b) contains all of Theta, so every valid posterior gives 1; the1/4 is procedural pointwise coverage.

Result and limitations

C(0)=1 and C(1)=1/4, with worst case1/4; average5/8 does not guarantee1/2 for every parameter.

Verification and source binding

3 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 4107037a3f83077f1cc2a7fa2b5b455a7cdbabb0e3e44d2397157a1118baea32
• original source SHA-256: e02e1138c369c28009b17a65c10da3c0ca5a8b024ccd40f5a3df260cadf99b8b
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)