{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-sampling-distribution",
  "semantic_family_id": "sf-sampling-distribution",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "抽樣分布"
  },
  "summary": {
    "zh-Hant": "固定theta時，估計量把樣本分布pushforward成統計量的抽樣分布；全部theta形成一族laws。"
  },
  "statement": {
    "zh-Hant": "在共同有限樣本空間，固定同一個不依未知theta的total statistic T，逐theta把各fiber機率加總並重驗非負與總質量一。抽樣分布不是觀察後estimate、likelihood、posterior或參數本身的分布。"
  },
  "formula": {
    "tex": "P_\\theta(T)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>P</mi><mi>θ</mi></msub><mo>(</mo><mi>T</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "subscript",
        "base": {
          "type": "symbol",
          "name": "P",
          "glyph": "P"
        },
        "subscript": {
          "type": "symbol",
          "name": "theta",
          "glyph": "θ"
        }
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "T",
          "glyph": "T"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "b2637b95cdccd1ba09b5d6f30d5159e4feb977bbd529232e55a87541a8b31737"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "全部theta共享固定有限樣本空間與同一total statistic。"
    },
    {
      "id": "a2",
      "zh-Hant": "每個P_theta為完整exact rational law。"
    }
  ],
  "symbols": [
    {
      "token": "T",
      "role": "statistical_estimator",
      "role_zh": "估計量",
      "scope": "representative_formula"
    },
    {
      "token": "P_\\theta^T",
      "role": "sampling_distribution",
      "role_zh": "抽樣分布",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-parametric-statistical-model",
      "title_zh": "參數統計模型",
      "reason_zh": "逐theta樣本laws。"
    },
    {
      "id": "mko-statistical-estimator",
      "title_zh": "統計估計量",
      "reason_zh": "theta-independent total statistic。"
    },
    {
      "id": "mko-finite-discrete-probability-distribution",
      "title_zh": "有限離散機率分布",
      "reason_zh": "pushforward後的law。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "每個theta都有自己的pushforward law。",
      "一個observed estimate只是T在一次樣本上的值。",
      "本輪不提前判無偏、低變異、一致性或posterior。"
    ],
    "common_misconception_zh": "把一次估計值、likelihood或posterior稱為抽樣分布。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限模型與exact pushforward；統計性質留後續節點。",
    "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": "解鎖estimator bias與estimator variance。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-sampling-distribution.md 整理；2026-10-14 脈衝 R06（使用者手動接續 2026-11-05 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}