{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-estimator-bias",
  "semantic_family_id": "sf-estimator-bias",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "估計量偏誤"
  },
  "summary": {
    "zh-Hant": "估計量在theta的偏誤是其theta抽樣分布期望減去目標參數；無偏要求對所有theta成立。"
  },
  "statement": {
    "zh-Hant": "在共同有限樣本空間與同一theta-independent total estimator上，逐theta由sampling distribution計算E_theta[T]並減theta。單一參數、sample或observed estimate不足以證unbiased；無偏也不推出零變異、一致、充分或最小MSE。"
  },
  "formula": {
    "tex": "B_theta(T)=E_theta(T)-theta",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><msub><mi>B</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><mi>T</mi><mo>)</mo></mrow><mo>−</mo><mi>theta</mi></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "subscript",
          "base": {
            "type": "symbol",
            "name": "B",
            "glyph": "B"
          },
          "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": "symbol",
              "name": "T",
              "glyph": "T"
            }
          ]
        },
        "right": {
          "type": "symbol",
          "name": "theta",
          "glyph": "theta"
        }
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ff23696a173bed580048b2fa83aa854cf5475bf1c6b85ea8553bed8d98518391"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "theta值、逐theta完整exact laws與同一total statistic全部明示。"
    },
    {
      "id": "a2",
      "zh-Hant": "本輪目標函數固定為g(theta)=theta；其他目標須另行明示。"
    }
  ],
  "symbols": [
    {
      "token": "B_theta",
      "role": "estimator_bias",
      "role_zh": "估計量偏誤",
      "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-expectation",
      "title_zh": "有限模型的期望值",
      "reason_zh": "E_theta[T]。"
    },
    {
      "id": "mko-parametric-statistical-model",
      "title_zh": "參數統計模型",
      "reason_zh": "參數空間與逐theta laws。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "bias是theta-indexed family。",
      "unbiased要求整個參數空間的bias全為零。",
      "observed estimate不是bias，其他統計性質各需獨立證書。"
    ],
    "common_misconception_zh": "看到一次estimate等於theta便稱估計量無偏。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限參數模型與exact rational expectations；不宣稱漸近性質。",
    "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bias-variance decomposition。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-estimator-bias.md 整理；2026-10-14 脈衝 R03（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"
  }
}