{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-estimator-bias-01",
  "mko_id": "mko-estimator-bias",
  "title": {
    "zh-Hant": "完整參數集合上的偏誤表",
    "en": "A bias table over the whole parameter set"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-estimator-bias-01",
    "source": {
      "mko_id": "mko-estimator-bias",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "be64de19751560ba6bd5142b5efa74407be5c35def263eb936120c6b63925da6",
      "english_sha256": "20825b4dd1d176eec9f31fe0ce7332c60f17ef65eae2ce83800f43e122f82a68"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "逐theta求EthetaU-theta，區分觀察誤差。",
      "en": "Compute EthetaU-theta at every theta, distinct from observed error."
    },
    "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^{-1}(0),U^{-1}(1/4),U^{-1}(1/2)",
        "after_tex": "\\{00\\},\\{01,10\\},\\{11\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "同一U的三個fibers供兩law共用。",
          "en": "Both laws use the same three fibres of U."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "work"
          ],
          "explanation": {
            "zh-Hant": "bias計算不能讓U依theta變動。",
            "en": "U cannot change with theta while computing bias."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "G2",
          "S01",
          "work"
        ],
        "before_tex": "P_{1/4}^U=(9/16,6/16,1/16)",
        "after_tex": "E_{1/4}U=(1/4)(6/16)+(1/2)(1/16)=1/8",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整三點抽樣law給第一mean。",
          "en": "The complete three-point sampling law gives the first mean."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "零值項雖不影響mean，仍在law中。",
            "en": "The zero-value mass remains part of the law even though its mean contribution vanishes."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "G2",
          "S02",
          "work"
        ],
        "before_tex": "b(1/4)=E_{1/4}U-1/4",
        "after_tex": "b(1/4)=1/8-1/4=-1/8",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "從同一 law 下的期望 E_theta[U] 減去指定 target theta，而不是減去 observed estimate。",
          "en": "Subtract the specified target theta, not an observed estimate."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "偏誤有號，不能先取絕對值。",
            "en": "Bias is signed; do not take an absolute value first."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "G2",
          "S03",
          "work"
        ],
        "before_tex": "P_{3/4}^U=(1/16,6/16,9/16)",
        "after_tex": "E_{3/4}U=(1/4)(6/16)+(1/2)(9/16)=3/8",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二law重新求mean。",
          "en": "Recompute the mean under the second law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "最後質量9/16不可沿用1/16。",
            "en": "Do not reuse1/16 for the last mass instead of 9/16."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "G2",
          "S04",
          "work"
        ],
        "before_tex": "b(3/4)=E_{3/4}U-3/4",
        "after_tex": "b(3/4)=3/8-3/4=-3/8",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二參數的偏誤也非零。",
          "en": "Bias at the second parameter is also nonzero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩個theta已窮盡參數集合。",
            "en": "These two values exhaust the parameter set."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "G2",
          "S05",
          "work"
        ],
        "before_tex": "\\text{unbiased}\\iff\\forall\\theta\\in\\Theta:\\ b(\\theta)=0",
        "after_tex": "U\\text{ is biased on this model}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任一非零bias已否定無偏，本例兩個都非零。",
          "en": "Any nonzero bias refutes unbiasedness; here both are nonzero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "無偏是全參數條件，不是選一個theta合適即可。",
            "en": "Unbiasedness is a whole-parameter condition, not success at one chosen theta."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "G2",
          "S06",
          "work"
        ],
        "before_tex": "\\theta=1/4,\\quad x=01",
        "after_tex": "U(01)-\\theta=1/4-1/4=0\\ne b(1/4)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "一次零誤差仍能與非零bias共存。",
          "en": "One zero observed error can coexist with nonzero bias."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "觀察誤差沒有對sample law取期望。",
            "en": "The observed error has not been averaged under the sampling law."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "G2",
          "S07",
          "work"
        ],
        "before_tex": "b=( -1/8,-3/8)",
        "after_tex": "b(\\theta)=-\\theta/2\\quad(\\theta\\in\\Theta)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "在已完整兩點參數集合上核對簡式，不外推未給模型。",
          "en": "Verify the compact formula on the complete two-point parameter set without extending an unspecified model."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "一致性與最小MSE仍未被此bias表證明。",
            "en": "Consistency and minimum MSE remain unproved by this bias table."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "bias表為-1/8、-3/8，因此U非無偏；一次estimate命中不改此結論。",
      "en": "Biases are -1/8 and -3/8, so U is biased; one exact observed estimate does not change that."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "觀察01命中theta=1/4，故無偏。",
          "en": "Observed 01 hits theta=1/4, so U is unbiased."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "此law的期望1/8，bias=-1/8。",
          "en": "Under that law the mean is 1/8 and bias-1/8."
        }
      },
      {
        "claim": {
          "zh-Hant": "bias必非負。",
          "en": "Bias must be nonnegative."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本例兩個bias都負。",
          "en": "Both biases here are negative."
        }
      },
      {
        "claim": {
          "zh-Hant": "只核一個theta就證全模型無偏。",
          "en": "One theta check proves modelwide unbiasedness."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "全稱條件涵蓋所有theta；此例另一值也須計。",
          "en": "The universal condition covers every theta, including the other value here."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "Use one common finite sample space and total statistic independent of theta. For each theta compute E_theta[T] from its sampling distribution and subtract theta. One parameter, sample or observed estimate does not establish unbiasedness over the model. Unbiasedness alone does not give zero variance, consistency, sufficiency or minimum MSE."
        },
        "source": {
          "mko_id": "mko-estimator-bias",
          "sha256": "be64de19751560ba6bd5142b5efa74407be5c35def263eb936120c6b63925da6",
          "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。 偏誤定義b(theta)=Etheta[U]-theta；無偏要求所有theta的bias零。",
          "en": "Finite laws have nonnegative probabilities summing to one; a statistic depends only on data, and expectation sums all weighted outcomes. Bias is b(theta)=Etheta[U]-theta; unbiasedness requires zero bias at every theta."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "1/8-1/4",
        "rhs": "-1/8"
      },
      {
        "id": "N02",
        "lhs": "3/8-3/4",
        "rhs": "-3/8"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "目標g(theta)=theta；不談漸近性質。",
      "en": "Target g(theta)=theta; no asymptotic claim."
    },
    "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": "target / Target",
        "zh": "固定減theta，不減觀察值或其他law mean。",
        "en": "Subtract theta, not an observation or another law's mean."
      },
      {
        "title": "全參數 / All parameters",
        "zh": "兩個theta分開保留有號bias。",
        "en": "Retain signed bias separately at both theta values."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "d114601f2cd77f86c632babb6a0f10f4d14465a1b73f2a05799b3a8f9c90cbeb",
    "original_source_sha256": "a5d42a968e40f0a69ed6cc147607b52fd1f7dbdc6be2f7247051d81167c6668f",
    "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-S106",
      "BR142-S107",
      "BR142-S105",
      "BR142-P032",
      "BR142-S104"
    ],
    "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."
  }
}