{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-finite-confidence-interval-coverage-01",
  "mko_id": "mko-finite-confidence-interval-coverage",
  "title": {
    "zh-Hant": "逐參數coverage、平均與selection",
    "en": "Pointwise coverage, averaging and selection"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-finite-confidence-interval-coverage-01",
    "source": {
      "mko_id": "mko-finite-confidence-interval-coverage",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "5ac0ccb03c88f011ec1edd7c258fe04dc72e982b042a8dcea53ac59deb386363",
      "english_sha256": "e2602185873b2f9b57fed1ac6efd6d390016ea0a0ff7efc9dc970097b8fca2cf"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "完整計算固定區間程序的pointwise及worst-case coverage。",
      "en": "Compute pointwise and worst-case coverage of a fixed interval procedure."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "Θ={0,1}，Ω={a,b,c}，P0=(1/2,1/4,1/4)，P1=(1/4,1/4,1/2)。預先固定I(a)=[0,0]、I(b)=[0,1]、I(c)=[0,0]。",
          "en": "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]."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "P_0,P_1",
        "after_tex": "1/2+1/4+1/4=1,\\quad1/4+1/4+1/2=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整兩law均歸一；每樣本的區間皆已預定。",
          "en": "Both complete laws normalize; the interval at each sample is precommitted."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不能依真theta另換區間程序。",
            "en": "The procedure cannot change with true theta."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "I(a),I(b),I(c)",
        "after_tex": "L(x)\\le U(x)\\quad\\text{for all }x\\in\\Omega",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個closed interval都有有序端點，procedure total。",
          "en": "All three closed intervals have ordered endpoints and the procedure is total."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩個單點區間[0,0]合法。",
            "en": "The two singleton intervals[0,0] are valid."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "\\theta=0:\\quad\\mathbf1\\{0\\in I(x)\\}",
        "after_tex": "(1,1,1),\\quad C(0)=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "零在所有closed intervals內，故完整coverage一。",
          "en": "Zero belongs to every closed interval, yielding coverage one."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "端點等號必須包含。",
            "en": "Endpoint equality must be included."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "\\theta=1:\\quad\\mathbf1\\{1\\in I(x)\\}",
        "after_tex": "(0,1,0),\\quad C(1)=P_1(b)=1/4",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "一只在b的區間內，取第二law的b質量。",
          "en": "One belongs only to the interval at b; use its mass under the second law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "即使 P_0(b) 與 P_1(b) 數值相同，也不能因此混淆 law 的角色；此處必須使用 P_1(b)。",
            "en": "Even coincident numerical masses do not remove the need to use the correct law."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "\\min_{\\theta\\in\\Theta}C(\\theta)",
        "after_tex": "\\min(1,1/4)=1/4",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "有限參數集合的worst-case coverage取最小，不取平均。",
          "en": "Worst-case coverage on the finite parameter set is the minimum, not the average."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩個theta都已完整計算。",
            "en": "Both theta values have been fully computed."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "\\overline C=(C(0)+C(1))/2,\\quad q=1/2",
        "after_tex": "\\overline C=5/8\\ge1/2>1/4=\\min C",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "明列average-only邊界：平均達標但pointwise保證失敗。",
          "en": "This explicitly witnesses average-only validity: the average meets the target while worst-case coverage fails."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "此平均只是指定均權摘要，不是自動頻率保證或prior。",
            "en": "The average is a specified equal-weight summary, not an automatic frequentist guarantee or prior."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "A=\\{a,b\\}\\text{ selected},\\quad\\theta=1",
        "after_tex": "P_1(1\\in I(X)\\mid A)=\\frac{P_1(b)}{P_1(a)+P_1(b)}=1/2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "若只在a、b報告，需重新算conditional coverage，與原1/4不同。",
          "en": "Reporting only at a,b requires recomputed conditional coverage, differing from original1/4."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "selection分母1/2正；零選中質量則條件coverage未定義。",
            "en": "The selection denominator 1/2 is positive; zero selection mass would leave it undefined."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "x_{\\rm obs}=b,\\quad I(b)=[0,1]",
        "after_tex": "\\mathbf1\\{\\theta\\in I(b)\\}=1\\quad(\\theta=0,1)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "觀察後這個固定區間對兩個允許theta都包含，不能把程序coverage1/4當此區間posterior。",
          "en": "After observing b this fixed interval contains either allowed theta; procedure coverage1/4 is not its posterior."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "posterior需要prior與資料更新；本例未供prior。",
            "en": "A posterior needs a prior and data update; no prior was supplied."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "C(0)=1、C(1)=1/4，worst case1/4；平均5/8不給1/2的全參數保證。",
      "en": "C(0)=1 and C(1)=1/4, with worst case1/4; average5/8 does not guarantee1/2 for every parameter."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "平均5/8故每theta coverage≥1/2。",
          "en": "Average5/8 means every theta has coverage≥1/2."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "theta=1的coverage只有1/4。",
          "en": "At theta=1 coverage is only 1/4."
        }
      },
      {
        "claim": {
          "zh-Hant": "選中a、b後coverage仍1/4。",
          "en": "After selecting a,b coverage remains1/4."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "theta=1的conditional coverage重新算為1/2。",
          "en": "The recomputed conditional coverage at theta=1 is 1/2."
        }
      },
      {
        "claim": {
          "zh-Hant": "觀察b後theta在I(b)的posterior機率是1/4。",
          "en": "After b the posterior probability of theta in I(b) is 1/4."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "I(b)包含全部Θ，任何合法posterior都給1；原1/4是程序pointwise coverage。",
          "en": "I(b) contains all of Theta, so every valid posterior gives 1; the1/4 is procedural pointwise coverage."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "完整finite parametric laws逐theta normalized；procedure I(x)=[L(x),U(x)]在全部sample上total、預先固定、不依未知theta，且逐x有L≤U。逐參數coverage為sum_x P_theta(x)1{L(x)≤theta≤U(x)}，worst-case coverage是有限theta集合上的minimum。average-only boundary須明示至少兩個不同pointwise coverages及target，使average≥target而minimum<target；posterior probability不是coverage。selection若已納入完整procedure可重新計算，錯誤在忽略selection仍沿用舊coverage。",
          "en": "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."
        },
        "source": {
          "mko_id": "mko-finite-confidence-interval-coverage",
          "sha256": "5ac0ccb03c88f011ec1edd7c258fe04dc72e982b042a8dcea53ac59deb386363",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "C(theta)=sum_x Ptheta(x)1{L(x)≤theta≤U(x)}；worst case取min；正質量selection條件機率為intersection商。",
          "en": "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."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "(1+1/4)/2",
        "rhs": "5/8"
      },
      {
        "id": "N02",
        "lhs": "min(1,1/4)",
        "rhs": "1/4"
      },
      {
        "id": "N03",
        "lhs": "(1/4)/(1/4+1/4)",
        "rhs": "1/2"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "頻率coverage非觀察後posterior；selection須重新納入。",
      "en": "Frequentist coverage is not an observed-data posterior; selection must be incorporated anew."
    },
    "md_symbols": [
      [
        "I(x)",
        "預先固定區間程序",
        "precommitted interval procedure"
      ],
      [
        "C(θ)",
        "逐參數coverage",
        "pointwise coverage"
      ],
      [
        "A",
        "selection事件",
        "selection event"
      ]
    ],
    "md_error_controls": [
      {
        "title": "完整程序 / Complete procedure",
        "zh": "a、b、c區間先固定再對兩law加權。",
        "en": "Fix intervals at a,b,c before weighting either law."
      },
      {
        "title": "selection重算 / Recompute selection",
        "zh": "選後分子、分母都在同theta law計算。",
        "en": "Compute selection numerators and denominators under the same theta law."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}