{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-confidence-interval-coverage",
  "semantic_family_id": "sf-finite-confidence-interval-coverage",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限模型信賴區間覆蓋率"
  },
  "summary": {
    "zh-Hant": "有限模型中預先固定的interval-valued procedure逐參數覆蓋真值的機率，定義其frequentist coverage。"
  },
  "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。"
  },
  "formula": {
    "tex": "I(T,x,t)=c",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>I</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>c</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "I",
          "glyph": "I"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "T",
            "glyph": "T"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          },
          {
            "type": "symbol",
            "name": "t",
            "glyph": "t"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "c",
        "glyph": "c"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "b7401c85f2ac372d5cc061e89bf51b4ea7a57fce5ba76ce7ff9bec7e2f11833d"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "theta與sample carriers finite unique，逐theta law exact normalized。"
    },
    {
      "id": "a2",
      "zh-Hant": "interval procedure total且closed endpoints有序。"
    },
    {
      "id": "a3",
      "zh-Hant": "任何selection rule均須成為procedure的一部分後重新計算coverage。"
    }
  ],
  "symbols": [
    {
      "token": "I",
      "role": "confidence_interval_procedure",
      "role_zh": "信賴區間procedure",
      "scope": "representative_formula"
    },
    {
      "token": "T",
      "role": "finite_parameter_set",
      "role_zh": "有限參數集合",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "sample_outcome",
      "role_zh": "樣本結果",
      "scope": "representative_formula"
    },
    {
      "token": "t",
      "role": "true_parameter",
      "role_zh": "真參數",
      "scope": "representative_formula"
    },
    {
      "token": "c",
      "role": "coverage_probability",
      "role_zh": "覆蓋機率",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-sampling-distribution",
      "title_zh": "抽樣分布",
      "reason_zh": "sample law。"
    },
    {
      "id": "mko-statistical-estimator",
      "title_zh": "統計估計量",
      "reason_zh": "data-dependent procedure背景。"
    },
    {
      "id": "mko-parametric-statistical-model",
      "title_zh": "參數統計模型",
      "reason_zh": "theta-indexed laws。"
    },
    {
      "id": "mko-probability",
      "title_zh": "機率",
      "reason_zh": "coverage sum。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "偏序與全序",
      "reason_zh": "closed interval membership。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "total precommitted procedure與laws先核。",
      "pointwise coverage table與worst-case minimum完整計算。",
      "average、posterior與selection-adjusted procedure清楚分界。"
    ],
    "common_misconception_zh": "把95% coverage解讀成觀察資料後theta有95% posterior probability，或忽略selection。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "finite coverage table不推出asymptotic validity或任何特定資料集的posterior statement。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入test inversion、randomized intervals與selective inference。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-confidence-interval-coverage.md 整理；2026-10-15 脈衝 R03（使用者手動接續 2026-11-18 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}