{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0095",
  "pilot_number": "P1000-0620",
  "number": 620,
  "family_id": "qf-mod47-mle-boundary",
  "title": {
    "zh-Hant": "第 620 題：有限模型最大似然估計量",
    "en": "Question 620: Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0095",
    "batch_id": "batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-maximum-likelihood-estimator"
    ],
    "title_zh": "有限模型最大似然估計量",
    "stem_zh": "完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a19\",\"b19\"],\"observations\":[\"x19\",\"y19\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x19\",\"tie_order\":[\"a19\",\"b19\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(tie, x19, [[a19, 1/2], [b19, 1/2]], 1/2, [a19, b19], [a19, b19], [x19, [a19, b19], a19], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)",
      "explanation_zh": "Wave6第十層核對=(tie, x19, [[a19, 1/2], [b19, 1/2]], 1/2, [a19, b19], [a19, b19], [x19, [a19, b19], a19], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)；依完整明示資料獨立重算；theta-independent deterministic tie rule形成total estimator；MLE不是posterior且不假設unique或推出unbiased/consistent/asymptotic claims。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限模型最大似然估計量"
      ]
    },
    "variant": {
      "family_id": "qf-mod47-mle-boundary",
      "seed": 202611164095,
      "parameters": {
        "mode": "tie",
        "theta_values": [
          "a19",
          "b19"
        ],
        "observations": [
          "x19",
          "y19"
        ],
        "probability_rows": [
          [
            [
              1,
              2
            ],
            [
              1,
              2
            ]
          ],
          [
            [
              1,
              2
            ],
            [
              1,
              2
            ]
          ]
        ],
        "observed": "x19",
        "tie_order": [
          "a19",
          "b19"
        ],
        "domain": "Exact_finite_maximum_likelihood",
        "operation": "mle-boundary",
        "semantic_template": "qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x19\",\"y19\"],\"observed\":\"x19\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a19\",\"b19\"],\"tie_order\":[\"a19\",\"b19\"]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、一般Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0095",
    "batch_id": "batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4026a6e5c5406e5d3ff7604c0a73dc63f13a30435d386e5df02b410918c5c51d",
      "family_id": "qf-mod47-mle-boundary",
      "seed": 202611164095,
      "semantic_template": "qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x19\",\"y19\"],\"observed\":\"x19\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a19\",\"b19\"],\"tie_order\":[\"a19\",\"b19\"]}"
    },
    "fields": {
      "title": "Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule",
      "stem": "Use complete theta labels=[a19, b19], observations=[x19, y19] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x19, mode=tie, fixed tie order=[a19, b19]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(tie, x19, [[a19, 1/2], [b19, 1/2]], 1/2, [a19, b19], [a19, b19], [x19, [a19, b19], a19], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)",
        "explanation": "Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x19, [[a19, 1/2], [b19, 1/2]], 1/2, [a19, b19], [a19, b19], [x19, [a19, b19], a19], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."
      },
      "skills": [
        "Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "完整finite parametric laws、observed likelihood與全部argmax責任。操作=mle-boundary；資料={\"mode\":\"tie\",\"theta_values\":[\"a19\",\"b19\"],\"observations\":[\"x19\",\"y19\"],\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"observed\":\"x19\",\"tie_order\":[\"a19\",\"b19\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete theta labels=[a19, b19], observations=[x19, y19] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x19, mode=tie, fixed tie order=[a19, b19]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "Θ=[a19, b19]; Ω=[x19, y19]",
      "after": "[[1/2, 1/2], [1/2, 1/2]]",
      "relation": "implies",
      "rule_id": "full_parametric_model",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列是一個固定參數的law，每欄是一個樣本結果；重排標籤時同時重排機率。",
        "en": "Each row is one parameter law and each column an outcome; reorder probabilities together with labels."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "P_a19(Ω) := sum(1/2, 1/2)",
      "after": "P_a19(Ω) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核對此law的完整總質量。",
        "en": "Check this law's complete total mass first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P_a19(Ω)，不以小數近似替代。",
          "en": "Recompute P_a19(Ω) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1",
        "label": "P_a19(Ω)"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "P_b19(Ω) := sum(1/2, 1/2)",
      "after": "P_b19(Ω) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核對此law的完整總質量。",
        "en": "Check this law's complete total mass first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P_b19(Ω)，不以小數近似替代。",
          "en": "Recompute P_b19(Ω) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/2",
          "1/2"
        ],
        "output": "1",
        "label": "P_b19(Ω)"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "tie priority=[a19, b19]",
      "after": "one fixed ordering, independent of the observed outcome",
      "relation": "implies",
      "rule_id": "fixed_tie_rule",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先宣告固定tie優先序，之後只從完整argmax中選第一個。",
        "en": "Declare one fixed tie order, then choose its first member within each complete argmax."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "parameter=a19; observed=x19",
      "after": "L(a19;x19)=1/2",
      "relation": "implies",
      "rule_id": "fixed_likelihood_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定觀測後沿參數比較同一欄；這裡不引入prior。",
        "en": "Fix the observation and compare its column across parameters, without introducing a prior."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "parameter=b19; observed=x19",
      "after": "L(b19;x19)=1/2",
      "relation": "implies",
      "rule_id": "fixed_likelihood_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定觀測後沿參數比較同一欄；這裡不引入prior。",
        "en": "Fix the observation and compare its column across parameters, without introducing a prior."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "[[a19, 1/2], [b19, 1/2]]",
      "after": "maximum=1/2; all argmax=[a19, b19]",
      "relation": "implies",
      "rule_id": "complete_argmax",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留每個達最大值的參數；全零欄的完整argmax仍是全部參數。",
        "en": "Retain every maximizer; for an all-zero column the complete argmax is still all parameters."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "argmax(x19)=[a19, b19]; priority=[a19, b19]",
      "after": "estimator(x19)=a19",
      "relation": "implies",
      "rule_id": "deterministic_tie_selection",
      "conditions": [],
      "explanation": {
        "zh-Hant": "選擇器是單值，不表示原本argmax一定是單元素集。",
        "en": "The selector is single-valued; the original argmax need not be a singleton."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "parameter=a19; observed=y19",
      "after": "L(a19;y19)=1/2",
      "relation": "implies",
      "rule_id": "fixed_likelihood_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定觀測後沿參數比較同一欄；這裡不引入prior。",
        "en": "Fix the observation and compare its column across parameters, without introducing a prior."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "parameter=b19; observed=y19",
      "after": "L(b19;y19)=1/2",
      "relation": "implies",
      "rule_id": "fixed_likelihood_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定觀測後沿參數比較同一欄；這裡不引入prior。",
        "en": "Fix the observation and compare its column across parameters, without introducing a prior."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "[[a19, 1/2], [b19, 1/2]]",
      "after": "maximum=1/2; all argmax=[a19, b19]",
      "relation": "implies",
      "rule_id": "complete_argmax",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留每個達最大值的參數；全零欄的完整argmax仍是全部參數。",
        "en": "Retain every maximizer; for an all-zero column the complete argmax is still all parameters."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "argmax(y19)=[a19, b19]; priority=[a19, b19]",
      "after": "estimator(y19)=a19",
      "relation": "implies",
      "rule_id": "deterministic_tie_selection",
      "conditions": [],
      "explanation": {
        "zh-Hant": "選擇器是單值，不表示原本argmax一定是單元素集。",
        "en": "The selector is single-valued; the original argmax need not be a singleton."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "defined on 2 outcomes",
      "after": "total on all 2 listed outcomes",
      "relation": "implies",
      "rule_id": "total_estimator",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整carrier中的每個觀測都有輸出，包括共同零質量觀測。",
        "en": "Every listed outcome has an output, including an outcome null under all laws."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "boundary mode=tie; argmax=[a19, b19]",
      "after": "no automatic posterior, unbiasedness, consistency or asymptotic optimum",
      "relation": "implies",
      "rule_id": "estimation_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "結尾false否定的是額外品質宣稱，不是否定argmax的存在。",
        "en": "The final false flag declines additional quality claims; it does not deny that argmax exists."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "保留全部參數與樣本標籤；null、空表、零分布及並列最優集合不能混同。",
          "en": "Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "tie",
      "x19",
      [
        [
          "a19",
          "1/2"
        ],
        [
          "b19",
          "1/2"
        ]
      ],
      "1/2",
      [
        "a19",
        "b19"
      ],
      [
        "a19",
        "b19"
      ],
      [
        "x19",
        [
          "a19",
          "b19"
        ],
        "a19"
      ],
      false,
      "MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal"
    ]
  },
  "conclusion": {
    "canonical_value": "(tie, x19, [[a19, 1/2], [b19, 1/2]], 1/2, [a19, b19], [a19, b19], [x19, [a19, b19], a19], 否, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)",
    "explanation": {
      "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。",
      "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings."
    }
  },
  "definition_note": {
    "zh-Hant": "並列或全零 likelihood 仍有完整 argmax；結尾否定的是後驗或估計品質宣稱。",
    "en": "Ties and all-zero likelihoods still have a complete argmax. The final No declines posterior or estimator-quality claims."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations/questions.jsonl",
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}