{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-estimator-variance",
  "semantic_family_id": "sf-estimator-variance",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "估計量變異數"
  },
  "summary": {
    "zh-Hant": "估計量變異數逐theta衡量sampling law相對自身期望的二次離差，不等同偏誤或MSE。"
  },
  "statement": {
    "zh-Hant": "sampling law、support與mean全部綁同一theta，精確核Var_theta(T)=E_theta[(T-E_thetaT)^2]=E_theta[T^2]-(E_thetaT)^2。零變異只代表該theta下幾乎必然常數，不代表等於theta或無偏。"
  },
  "formula": {
    "tex": "V_theta(T)=E_theta(T^2)-E_theta(T)^2",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><msub><mi>V</mi><mi>theta</mi></msub><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>=</mo><mrow><mrow><msub><mi>E</mi><mi>theta</mi></msub><mo>(</mo><msup><mi>T</mi><mn>2</mn></msup><mo>)</mo></mrow><mo>−</mo><msup><mrow><msub><mi>E</mi><mi>theta</mi></msub><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "subscript",
          "base": {
            "type": "symbol",
            "name": "V",
            "glyph": "V"
          },
          "subscript": {
            "type": "symbol",
            "name": "theta",
            "glyph": "theta"
          }
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "T",
            "glyph": "T"
          }
        ]
      },
      "rhs": {
        "type": "subtraction",
        "left": {
          "type": "function_call",
          "callee": {
            "type": "subscript",
            "base": {
              "type": "symbol",
              "name": "E",
              "glyph": "E"
            },
            "subscript": {
              "type": "symbol",
              "name": "theta",
              "glyph": "theta"
            }
          },
          "arguments": [
            {
              "type": "power",
              "base": {
                "type": "symbol",
                "name": "T",
                "glyph": "T"
              },
              "exponent": {
                "type": "number",
                "value": 2
              }
            }
          ]
        },
        "right": {
          "type": "power",
          "base": {
            "type": "function_call",
            "callee": {
              "type": "subscript",
              "base": {
                "type": "symbol",
                "name": "E",
                "glyph": "E"
              },
              "subscript": {
                "type": "symbol",
                "name": "theta",
                "glyph": "theta"
              }
            },
            "arguments": [
              {
                "type": "symbol",
                "name": "T",
                "glyph": "T"
              }
            ]
          },
          "exponent": {
            "type": "number",
            "value": 2
          }
        }
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e54f42fdb9be1bf53b9348fd001c18bde1963f1c5ef1375cf42e770f40be35ff"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "共同有限樣本空間、完整逐theta exact laws與同一total statistic。"
    },
    {
      "id": "a2",
      "zh-Hant": "所有二次矩有限且以exact rational arithmetic計算。"
    }
  ],
  "symbols": [
    {
      "token": "V_theta",
      "role": "estimator_variance",
      "role_zh": "估計量變異數",
      "scope": "representative_formula"
    },
    {
      "token": "T",
      "role": "statistical_estimator",
      "role_zh": "估計量",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-sampling-distribution",
      "title_zh": "抽樣分布",
      "reason_zh": "逐theta estimator law。"
    },
    {
      "id": "mko-variance",
      "title_zh": "有限機率模型的變異數",
      "reason_zh": "中心二次矩。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "variance是theta-indexed family。",
      "二次矩identity可作精確交叉核對。",
      "variance單獨不推unbiased、consistency或MSE。"
    ],
    "common_misconception_zh": "把零variance當成無偏，或把variance直接稱為MSE。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限sampling laws；跨樣本大小一致性另需序列模型。",
    "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MSE與有效性比較。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-estimator-variance.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-11-06 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"
  }
}