{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-estimator-mean-squared-error-01",
  "mko_id": "mko-estimator-mean-squared-error",
  "title": {
    "zh-Hant": "不用分解公式的直接MSE計算",
    "en": "Direct MSE without the decomposition formula"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-estimator-mean-squared-error-01",
    "source": {
      "mko_id": "mko-estimator-mean-squared-error",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "b86734a34e0811ecbcceb6973118f1feff5e437465869c8706bc6a00c4cce3cb",
      "english_sha256": "89d9b8b6f81c362c4f68e6b4cbaf80938cc9168434c135366f56330c231e6f5f"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "逐theta直接加權loss，保留bias-variance為後續。",
      "en": "Weight losses directly at every theta, leaving decomposition for later."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "Θ={1/4,3/4}，Ω={00,01,10,11}；P_(1/4)=(9,3,3,1)/16，P_(3/4)=(1,3,3,9)/16。",
          "en": "Theta={1/4,3/4}, Omega={00,01,10,11}; P_(1/4)=(9,3,3,1)/16 and P_(3/4)=(1,3,3,9)/16."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "U(x)=(x1+x2)/4，依Ω順序的total table為(0,1/4,1/4,1/2)，目標參數為g(theta)=theta。",
          "en": "U(x)=(x1+x2)/4 has total table(0,1/4,1/4,1/2) in Omega order, with target g(theta)=theta."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "G2",
          "work"
        ],
        "before_tex": "U\\in\\{0,1/4,1/2\\}",
        "after_tex": "P_{1/4}^U=(9,6,1)/16,\\quad P_{3/4}^U=(1,6,9)/16",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先完整列兩個抽樣laws，各列歸一。",
          "en": "First list both full normalized sampling laws."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "work"
          ],
          "explanation": {
            "zh-Hant": "後續loss與權重必須來自同theta。",
            "en": "Losses and weights must use the same theta."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "G2",
          "S01",
          "work"
        ],
        "before_tex": "\\theta=1/4,\\quad(U-\\theta)^2",
        "after_tex": "(1/16,0,1/16)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對三個support值分別減1/4再平方。",
          "en": "Subtract1/4 and square at each of the three support values."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "目標theta不是mean1/8。",
            "en": "The target theta is not the mean 1/8."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "G2",
          "S02",
          "work"
        ],
        "before_tex": "\\operatorname{MSE}_{1/4}(U)=\\sum_uP_{1/4}(U=u)(u-1/4)^2",
        "after_tex": "(9/16)(1/16)+(6/16)0+(1/16)(1/16)=5/128",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "直接loss加權，不借bias-variance identity。",
          "en": "Weight losses directly, without the bias-variance identity."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "中間loss雖零，其質量仍完整保留。",
            "en": "The middle mass remains represented despite zero loss."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "G2",
          "S03",
          "work"
        ],
        "before_tex": "\\theta=3/4,\\quad(U-\\theta)^2",
        "after_tex": "(9/16,1/4,1/16)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "換theta後重新算全部loss，不能沿用前一列。",
          "en": "Recompute every loss after changing theta rather than reusing the preceding row."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "1/4與3/4的差平方為1/4。",
            "en": "The squared difference between1/4 and 3/4 is 1/4."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "G2",
          "S04",
          "work"
        ],
        "before_tex": "\\operatorname{MSE}_{3/4}(U)",
        "after_tex": "(1/16)(9/16)+(6/16)(1/4)+(9/16)(1/16)=21/128",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二個完整law直接給MSE21/128。",
          "en": "The second full law gives MSE21/128 directly."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "沒有把兩theta的MSE平均當pointwise MSE。",
            "en": "No averaging across theta replaces pointwise MSE."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "G2",
          "S05",
          "work"
        ],
        "before_tex": "\\theta=1/4,\\quad x=01",
        "after_tex": "(U(01)-\\theta)^2=0\\ne5/128",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "一次loss為零，期望loss仍正，構成明確比較。",
          "en": "One loss is zero while expected loss is positive, giving a concrete distinction."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不是只用物件名稱差異否證數值相等。",
            "en": "The distinction is numerical, not merely terminological."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "G2",
          "S06",
          "work"
        ],
        "before_tex": "\\operatorname{MSE}_\\theta(U)=0",
        "after_tex": "(u-\\theta)^2=0\\text{ at every positive-mass }u",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "非負有限和為零當且僅當每個正權項零。",
          "en": "A finite nonnegative sum vanishes exactly when every positive-weight term vanishes."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "已列出但機率權重為 0 的樣本點，即使有有限且非零的 loss，對 MSE 加權和的貢獻仍為 0。",
            "en": "At a listed sample point of probability mass zero, even a finite nonzero loss contributes 0 to the MSE weighted sum."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "G2",
          "S07",
          "work"
        ],
        "before_tex": "(u-\\theta)^2=0\\iff u=\\theta",
        "after_tex": "\\operatorname{MSE}_\\theta(U)=0\\iff U=\\theta\\quad P_\\theta\\text{-a.s.}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "得到零MSE的精確a.s.語義，不要求每個零質量原子。",
          "en": "Obtain the exact a.s. meaning of zero MSE, without constraining null atoms."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "本例兩MSE皆正，亦不推一致性。",
            "en": "Both MSEs here are positive, and no consistency claim follows."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "直接計得MSE_(1/4)=5/128、MSE_(3/4)=21/128，沒有使用分解公式。",
      "en": "Direct calculation gives MSE_(1/4)=5/128 and MSE_(3/4)=21/128 without decomposition."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "01的loss零，故MSE零。",
          "en": "The loss at 01 is zero, so MSE is zero."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "第一law完整MSE為5/128。",
          "en": "The first full-law MSE is 5/128."
        }
      },
      {
        "claim": {
          "zh-Hant": "MSE以E[U]而非theta為target。",
          "en": "MSE targets E[U] rather than theta."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "那會是variance；此處目標明列theta。",
          "en": "That would be variance; the declared target is theta."
        }
      },
      {
        "claim": {
          "zh-Hant": "MSE零必須所有零質量原子也命中。",
          "en": "Zero MSE requires hitting even at null atoms."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "非負和只約束正權項，因此是a.s.等式。",
          "en": "The sum constrains positive weights only, giving an a.s. equality."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "Each theta requires a complete normalized finite sampling law. Compute sum over listed t of P_theta(T=t)*(t−theta)^2 directly, without using the later bias-variance decomposition. A multi-theta table keeps each law separate. MSE0 is equivalent to T=theta almost surely under P_theta. One observed estimate/loss does not define the expected loss, though its value may coincide. Unbiasedness does not imply zero MSE, and MSE is neither automatically variance nor a consistency certificate. This slice uses scalar parameters and exact rational laws."
        },
        "source": {
          "mko_id": "mko-estimator-mean-squared-error",
          "sha256": "b86734a34e0811ecbcceb6973118f1feff5e437465869c8706bc6a00c4cce3cb",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限law機率非負且總和一；statistic只依樣本，期望為所有outcomes的加權和。 MSEtheta(U)=sum_u Ptheta(U=u)(u-theta)²；非負有限和為零恰所有正權項零。",
          "en": "Finite laws have nonnegative probabilities summing to one; a statistic depends only on data, and expectation sums all weighted outcomes. MSEtheta(U)=sum_u Ptheta(U=u)(u-theta)²; a nonnegative finite sum is zero exactly when all positive-weight terms vanish."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "(9/16)*(1/16)+(1/16)*(1/16)",
        "rhs": "5/128"
      },
      {
        "id": "N02",
        "lhs": "(1/16)*(9/16)+(6/16)*(1/4)+(9/16)*(1/16)",
        "rhs": "21/128"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整有限scalar模型，不從單次loss估MSE。",
      "en": "A complete finite scalar model, not MSE inferred from one loss."
    },
    "md_symbols": [
      [
        "θ",
        "固定但未知參數",
        "fixed unknown parameter"
      ],
      [
        "U",
        "同一total statistic",
        "one total statistic"
      ]
    ],
    "md_error_controls": [
      {
        "title": "直接路徑 / Direct route",
        "zh": "三個loss逐項列出，不偷用後續decomposition。",
        "en": "List all three losses without invoking later decomposition."
      },
      {
        "title": "參數對齊 / Parameter alignment",
        "zh": "theta=3/4使用(9/16,1/4,1/16)loss配第二law。",
        "en": "At theta=3/4 pair losses(9/16,1/4,1/16) with the second law."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "6b8dc75670c80b7c314c315d9ba0f05efd2bc67d8ee79d74ae4c39ae8edb70e6",
    "original_source_sha256": "5ba358558d2e86f84d7bddf9c595239e23b51aee26b3b923b284b2a70e7901b4",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-S202",
      "BR142-S203",
      "BR142-S200",
      "BR142-S201",
      "BR142-P070",
      "BR142-P069"
    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}