{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-finite-maximum-likelihood-estimator-01",
  "mko_id": "mko-finite-maximum-likelihood-estimator",
  "title": {
    "zh-Hant": "含正質量 tie 與全零似然欄的 MLE",
    "en": "MLE with a positive-mass tie and an all-zero likelihood column"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-finite-maximum-likelihood-estimator-01",
    "source": {
      "mko_id": "mko-finite-maximum-likelihood-estimator",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "44f2514c3f39a056585e86bd48bcfd2e135dc3ee54fdd4074dcdae1dc0a763f3",
      "english_sha256": "5680fae17b0a3ab4910838f362adf9700b79e52b642fa927e622c9ddccb3cd5a"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "完整算argmax及固定tie rule的total estimator。",
      "en": "Compute every argmax and a total estimator using a fixed tie rule."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "Θ={0,1,2}，Ω={a,b,c,d,z}。三列law依序為(1/2,1/8,1/8,1/4,0)、(1/8,1/2,1/8,1/4,0)、(1/8,1/8,1/2,1/4,0)。固定tie優先次序2、0、1，不依未知theta。",
          "en": "Theta={0,1,2}, Omega={a,b,c,d,z}. Rows are(1/2,1/8,1/8,1/4,0),(1/8,1/2,1/8,1/4,0),(1/8,1/8,1/2,1/4,0). Fix tie priority2,0,1 independently of unknown theta."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "P_0,P_1,P_2",
        "after_tex": "\\sum_xP_j(x)=1/2+1/8+1/8+1/4=1\\quad(j=0,1,2)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三列皆為非負機率，carrier共有五點包含z。",
          "en": "All three rows are nonnegative laws on the five-point carrier including z."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "零質量樣本仍屬total estimator定義域。",
            "en": "A zero-mass sample still belongs to the estimator's total domain."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "x=a,\\quad L(\\theta;a)",
        "after_tex": "(L(0;a),L(1;a),L(2;a))=(1/2,1/8,1/8),\\quad A(a)=\\{0\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐參數比完整三值，最大唯一在0。",
          "en": "Compare all three parameter likelihoods; the unique maximum is at 0."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "argmax是參數集合而非最大值1/2。",
            "en": "Argmax is a parameter set, not the maximum value 1/2."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "x=b",
        "after_tex": "L(\\theta;b)=(1/8,1/2,1/8),\\quad A(b)=\\{1\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二觀察有唯一maximizer1。",
          "en": "The second observation uniquely maximizes at 1."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不從a的結果沿用參數0。",
            "en": "Do not reuse parameter0 from the a case."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "x=c",
        "after_tex": "L(\\theta;c)=(1/8,1/8,1/2),\\quad A(c)=\\{2\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第三觀察完整比三值，maximizer2。",
          "en": "Compare all three values for the third observation; the maximizer is 2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "c已完成其獨立case。",
            "en": "The c case has been independently covered."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "x=d",
        "after_tex": "L(\\theta;d)=(1/4,1/4,1/4),\\quad A(d)=\\{0,1,2\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "d在每law有正質量，提供真實tie witness。",
          "en": "d has positive mass in every law and supplies a genuine tie witness."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不得默認MLE唯一。",
            "en": "Do not assume a unique MLE."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "x=z",
        "after_tex": "L(\\theta;z)=(0,0,0),\\quad A(z)=\\{0,1,2\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "在觀察 z 對應的全零似然欄中，所有參數都達到最大值 0，仍有完整 argmax set。",
          "en": "In the all-zero column every parameter attains maximum0, giving the full argmax set."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不把0/0正規化為posterior。",
            "en": "Do not normalize0/0 into a posterior."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "\\widehat\\theta(x)=\\text{first element of }A(x)\\text{ in }(2,0,1)",
        "after_tex": "(\\widehat\\theta(a),\\widehat\\theta(b),\\widehat\\theta(c),\\widehat\\theta(d),\\widehat\\theta(z))=(0,1,2,2,2)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "固定tie order對兩種tie都選2，完成全部樣本。",
          "en": "The fixed order selects2 for both ties and completes all samples."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "tie order預先固定，不可偷看真theta。",
            "en": "The tie order is precommitted and cannot inspect true theta."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "\\forall x\\in\\Omega:\\ \\widehat\\theta(x)\\in A(x)",
        "after_tex": "\\widehat\\theta\\text{ is a total finite MLE}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個outcome的選值皆在完整argmax中。",
          "en": "At every outcome the selected value belongs to its full argmax."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "total性含不可能樣本z，但不對其作觀察後機率宣稱。",
            "en": "Totality includes impossible sample z without asserting an observed-data probability there."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "argmax在d與z皆為Θ；固定order得到total table(0,1,2,2,2)。",
      "en": "At d and z the argmax is Theta; the fixed order gives total table(0,1,2,2,2)."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "全零似然欄沒有任何 maximizer。",
          "en": "The all-zero column has no maximizer."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "三參數皆達最大0。",
          "en": "All three parameters attain maximum0."
        }
      },
      {
        "claim": {
          "zh-Hant": "在 d 的 likelihood 上，唯一達最大值的參數為 2。",
          "en": "At d, the likelihood has the unique maximizing parameter 2."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "argmax有三值；2只由tie order選出。",
          "en": "Argmax has three values;2 is selected only by the tie order."
        }
      },
      {
        "claim": {
          "zh-Hant": "用真theta作tie優先順序。",
          "en": "Use true theta to prioritize ties."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "estimator規則不能依未知theta。",
          "en": "The estimator rule cannot depend on unknown theta."
        }
      }
    ],
    "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須有對完整observation carrier的normalized finite law。對observed x建立全參數likelihood table與完整argmax set；固定、theta-independent deterministic tie order後，對全部observations形成total estimator。tie與all-zero likelihood須各有明示witness。MLE最大化likelihood而非posterior，不假設unique，也不推出unbiased、consistent或asymptotically optimal。",
          "en": "Every theta has a normalized finite law on the complete observation carrier. For an observed x construct every parameter likelihood and the full argmax set. A fixed deterministic parameter-independent tie order gives a total estimator on all observations. Ties and all-zero likelihoods need explicit witnesses. MLE maximizes likelihood, not posterior probability; uniqueness, unbiasedness, consistency and asymptotic optimality do not follow automatically."
        },
        "source": {
          "mko_id": "mko-finite-maximum-likelihood-estimator",
          "sha256": "44f2514c3f39a056585e86bd48bcfd2e135dc3ee54fdd4074dcdae1dc0a763f3",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "likelihood L(theta;x)=Ptheta(x)，argmax包含所有達最大參數；固定完整參數排列可決定tie。",
          "en": "Likelihood is L(theta;x)=Ptheta(x); argmax contains all maximizing parameters; a fixed full parameter permutation breaks ties."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "1/2+1/8+1/8+1/4",
        "rhs": "1"
      },
      {
        "id": "N02",
        "lhs": "max(1/2,1/8,1/8)",
        "rhs": "1/2"
      },
      {
        "id": "N03",
        "lhs": "max(0,0,0)",
        "rhs": "0"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限exact模型，不推無偏、一致或posterior。",
      "en": "A finite exact model; no unbiasedness, consistency or posterior inference."
    },
    "md_symbols": [
      [
        "A(x)",
        "完整argmax set",
        "full argmax set"
      ],
      [
        "hat θ",
        "tie-broken MLE",
        "tie-broken MLE"
      ]
    ],
    "md_error_controls": [
      {
        "title": "全列argmax / Full argmax",
        "zh": "每個樣本都比較0、1、2三參數。",
        "en": "Compare all three parameters0,1,2 for every sample."
      },
      {
        "title": "兩種tie / Two ties",
        "zh": "分開記d的正likelihood tie與z的全零tie。",
        "en": "Distinguish the positive-likelihood tie at d from the all-zero tie at z."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "b70f7d0e33dd447f7ea1ecc0d95b7e12332b17dc0445dcd95ea962c56a1233e5",
    "original_source_sha256": "fc9738f97fb73ca2a370a55d0540a368591e138d436c52e5271233f12285e7d6",
    "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-P075",
      "BR142-P077",
      "BR142-P079",
      "BR142-P078",
      "BR142-S218",
      "BR142-S219",
      "BR142-S220",
      "BR142-S221",
      "BR142-P076"
    ],
    "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."
  }
}