{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-estimator-mean-squared-error",
  "semantic_family_id": "sf-estimator-mean-squared-error",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "估計量均方誤差"
  },
  "summary": {
    "zh-Hant": "scalar parameter theta下，估計量T的MSE_theta(T)=E_theta[(T-theta)^2]由完整sampling distribution逐theta定義。"
  },
  "statement": {
    "zh-Hant": "每個theta都必須有完整、正規化的finite sampling law，直接計算sum_t P_theta(T=t)(t-theta)^2；不可只由bias與variance間接取得，因bias–variance decomposition是下一節點。多theta table逐law分開。MSE=0恰當T=theta在P_theta下almost surely。單一observed estimate或sample squared error不是MSE；unbiased不推出zero MSE，MSE也不等同variance或推出consistency。本輪限scalar parameter與exact rational laws。"
  },
  "formula": {
    "tex": "M(T,t)=e",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>M</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>e</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "M",
          "glyph": "M"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "T",
            "glyph": "T"
          },
          {
            "type": "symbol",
            "name": "t",
            "glyph": "t"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "e",
        "glyph": "e"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "b819b4ca2455159568420aca0c56e3a81df368cd10ed8f74ca5112dd609ee8d8"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "support values、theta與probabilities皆為exact rationals，probabilities非負且總和一。"
    },
    {
      "id": "a2",
      "zh-Hant": "多theta family完整列出每個parametric sampling distribution。"
    }
  ],
  "symbols": [
    {
      "token": "M",
      "role": "mean_squared_error_operator",
      "role_zh": "均方誤差算子",
      "scope": "representative_formula"
    },
    {
      "token": "T",
      "role": "scalar_estimator",
      "role_zh": "純量估計量",
      "scope": "representative_formula"
    },
    {
      "token": "t",
      "role": "target_parameter",
      "role_zh": "目標參數",
      "scope": "representative_formula"
    },
    {
      "token": "e",
      "role": "exact_expected_squared_error",
      "role_zh": "精確平方誤差期望",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-estimator-bias",
      "title_zh": "估計量偏誤",
      "reason_zh": "boundary comparison only。"
    },
    {
      "id": "mko-estimator-variance",
      "title_zh": "估計量變異數",
      "reason_zh": "boundary comparison only。"
    },
    {
      "id": "mko-expectation",
      "title_zh": "有限分布的期望值",
      "reason_zh": "expected squared error。"
    },
    {
      "id": "mko-sampling-distribution",
      "title_zh": "抽樣分布",
      "reason_zh": "完整sampling law。"
    },
    {
      "id": "mko-parametric-statistical-model",
      "title_zh": "參數統計模型",
      "reason_zh": "逐theta laws。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "MSE按theta逐點定義。",
      "完整finite law直接加權squared errors。",
      "zero-MSE、observed error與unbiased boundaries分離。"
    ],
    "common_misconception_zh": "拿單一sample誤差當MSE，或用unbiased直接推MSE=0。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "不提前使用bias–variance decomposition，也不推出consistency或asymptotic optimality。",
    "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bias–variance decomposition。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-estimator-mean-squared-error.md 整理；2026-10-15 脈衝 R05（使用者手動接續 2026-11-14 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"
  }
}