{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-parametric-statistical-model-01",
  "mko_id": "mko-parametric-statistical-model",
  "title": {
    "zh-Hant": "兩個參數值、四個樣本點的完整模型",
    "en": "A complete model with two parameter values and four sample points"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-parametric-statistical-model-01",
    "source": {
      "mko_id": "mko-parametric-statistical-model",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "ef431eac244935b2a2629f424d522b6d9b36a17dede2b668b4402aed44ac715d",
      "english_sha256": "d659a31945072a4e32f774a17f8534156a1e73ada04c4f726fdb98c5f15f979b"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "逐參數驗law並區分likelihood與posterior。",
      "en": "Validate every parameter law and distinguish likelihood from posterior."
    },
    "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."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "\\Theta=\\{1/4,3/4\\},\\quad\\Omega=\\{00,01,10,11\\}",
        "after_tex": "P_\\theta:\\Omega\\to[0,1]\\quad(\\theta\\in\\Theta)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先固定共同carrier，不隨參數刪樣本。",
          "en": "Fix the common carrier without deleting samples as theta changes."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩列都必須有四個entry。",
            "en": "Both rows require four entries."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "P_{1/4}=(9,3,3,1)/16",
        "after_tex": "(9+3+3+1)/16=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第一列四項非負且歸一。",
          "en": "All four entries of the first row are nonnegative and normalized."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "沒有把weight9當機率9。",
            "en": "The weight9 is not itself probability 9."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "P_{3/4}=(1,3,3,9)/16",
        "after_tex": "(1+3+3+9)/16=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二列也獨立驗完，不由對稱外觀跳過。",
          "en": "Validate the second row explicitly rather than skipping it by appearance."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "完整參數集合只有這兩列，已全部核完。",
            "en": "These two rows exhaust the parameter set."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "mko_definition"
        ],
        "before_tex": "x=01\\text{ fixed}",
        "after_tex": "L(1/4;01)=3/16,\\quad L(3/4;01)=3/16",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "固定觀察後，likelihood的自變數變成theta。",
          "en": "After fixing the observation, theta is the likelihood's argument."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "mko_definition"
          ],
          "explanation": {
            "zh-Hant": "不得把x與theta角色顛倒。",
            "en": "Do not interchange the roles of x and theta."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "\\sum_{\\theta\\in\\Theta}L(\\theta;01)",
        "after_tex": "3/16+3/16=3/8\\ne1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "likelihood橫跨參數不必歸一為機率law。",
          "en": "Likelihood values across parameters need not normalize to a probability law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這個3/8明確反駁likelihood本身就是posterior。",
            "en": "The sum3/8 explicitly refutes treating likelihood itself as a posterior."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "x=00",
        "after_tex": "L(1/4;00)=9/16>1/16=L(3/4;00)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "另一觀察對兩參數提供不同likelihood。",
          "en": "Another observation gives different likelihoods for the two parameters."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "較大likelihood不等於已計出參數機率。",
            "en": "A larger likelihood is not a computed parameter probability."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "P_{1/4}(00)=9/16\\ne1/16=P_{3/4}(00)",
        "after_tex": "P_{1/4}\\ne P_{3/4}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整模型列確實不同；差異由明示樣本事件見證。",
          "en": "The model laws are distinct, witnessed by an explicit sample event."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "此比較來自已知完整laws，不由一次observed01推出。",
            "en": "This comparison uses the complete laws, not one observed 01."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "mko_definition"
        ],
        "before_tex": "\\theta\\mapsto P_\\theta\\text{ with both rows normalized}",
        "after_tex": "\\{P_\\theta:\\theta\\in\\Theta\\}\\text{ is a finite parametric model}",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "共同carrier和所有參數law均合法，模型資格完成。",
          "en": "The common carrier and every parameter law qualify the finite model."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "mko_definition"
          ],
          "explanation": {
            "zh-Hant": "未供prior，故不自行做Bayes更新。",
            "en": "No prior was supplied, so no Bayes update is performed."
          }
        },
        "leaf_basis": "mko_definition"
      }
    ],
    "result": {
      "zh-Hant": "兩列構成合法有限參數模型；L(theta;01)兩值皆3/16但不是posterior。",
      "en": "The rows define a valid finite parametric model; both L(theta;01) values are 3/16, not a posterior."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "01觀察下posterior就是(3/16,3/16)。",
          "en": "The posterior at 01 is(3/16,3/16)."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "總和3/8而非1，且無prior。",
          "en": "Its sum is 3/8, not 1, and no prior is supplied."
        }
      },
      {
        "claim": {
          "zh-Hant": "每theta可默默用不同carrier。",
          "en": "Each theta may silently use a different carrier."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本模型定義要求同一四樣本carrier。",
          "en": "This model requires the same four-sample carrier."
        }
      },
      {
        "claim": {
          "zh-Hant": "一個01就證兩個law相同。",
          "en": "One01 proves the laws identical."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "00的機率9/16與1/16不同。",
          "en": "Their00 probabilities differ:9/16 versus1/16."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "完整核有限parameter set與每列非負、總和1的law table。likelihood L(theta;x)=P_theta(x)視theta為自變數；它不是posterior，未供prior時不作Bayes。identifiability不由單一觀察推得。",
          "en": "Validate the finite parameter set and every nonnegative normalized row of the complete law table. In L(theta;x)=P_theta(x), likelihood varies theta while the observation x is fixed. It is not a posterior; without a prior no Bayes update is performed. One observation does not establish identifiability."
        },
        "source": {
          "mko_id": "mko-parametric-statistical-model",
          "sha256": "ef431eac244935b2a2629f424d522b6d9b36a17dede2b668b4402aed44ac715d",
          "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。",
          "en": "Finite laws have nonnegative probabilities summing to one; a statistic depends only on data, and expectation sums all weighted outcomes."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "(9+3+3+1)/16",
        "rhs": "1"
      },
      {
        "id": "N02",
        "lhs": "(1+3+3+9)/16",
        "rhs": "1"
      },
      {
        "id": "N03",
        "lhs": "3/16+3/16",
        "rhs": "3/8"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限完整table；不作posterior或最優性推論。",
      "en": "A complete finite table; no posterior or optimality inference."
    },
    "md_symbols": [
      [
        "θ",
        "固定但未知參數",
        "fixed unknown parameter"
      ]
    ],
    "md_error_controls": [
      {
        "title": "逐列 / Rowwise",
        "zh": "分別檢查兩個分母16的完整列。",
        "en": "Check both complete denominator 16 rows separately."
      },
      {
        "title": "固定觀察 / Fixed observation",
        "zh": "likelihood table先固定x再遍歷全部theta。",
        "en": "Fix x before traversing every theta for the likelihood table."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "d00df7b6f9cdd89591f628108246c88f499ea10b7d1142e3b840f769915b49dc",
    "original_source_sha256": "fcaef8fae4f87c9c47807672f655dbe899166728be3799d64ec5e07d2e5c74f2",
    "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-S062",
      "BR142-P019",
      "BR142-S060",
      "BR142-S061",
      "BR142-S059",
      "BR142-S057",
      "BR142-S058",
      "BR142-E001-REMOVE-DISPLAY"
    ],
    "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."
  }
}