{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0011",
  "pilot_number": "P1000-0561",
  "number": 561,
  "family_id": "qf-mod47-mle-law-likelihood",
  "title": {
    "zh-Hant": "第 561 題：有限模型最大似然估計量",
    "en": "Question 561: The Observed Likelihood Uses Every Qualified Parameter Law"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0011",
    "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-law-likelihood；資料={\"theta_values\":[\"a10\",\"b10\"],\"observations\":[\"x10\",\"y10\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x10\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([a10, b10], [x10, y10], [[3/4, 1/4], [1/4, 3/4]], x10, [[a10, 3/4], [b10, 1/4]], complete_normalized_finite_model_likelihood_table)",
      "explanation_zh": "Wave6第十層核對=([a10, b10], [x10, y10], [[3/4, 1/4], [1/4, 3/4]], x10, [[a10, 3/4], [b10, 1/4]], complete_normalized_finite_model_likelihood_table)；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": 2,
      "skills": [
        "有限模型最大似然估計量"
      ]
    },
    "variant": {
      "family_id": "qf-mod47-mle-law-likelihood",
      "seed": 202611164011,
      "parameters": {
        "theta_values": [
          "a10",
          "b10"
        ],
        "observations": [
          "x10",
          "y10"
        ],
        "probability_rows": [
          [
            [
              3,
              4
            ],
            [
              1,
              4
            ]
          ],
          [
            [
              1,
              4
            ],
            [
              3,
              4
            ]
          ]
        ],
        "observed": "x10",
        "domain": "Exact_finite_maximum_likelihood",
        "operation": "mle-law-likelihood",
        "semantic_template": "qf-mod47-mle-law-likelihood/{\"observations\":[\"x10\",\"y10\"],\"observed\":\"x10\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a10\",\"b10\"]}"
      }
    },
    "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-0011",
    "batch_id": "batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "c3a7e26bd805333fca2b745d7f5a8f761db1260c7da3b73325d289ebb023e0bf",
      "family_id": "qf-mod47-mle-law-likelihood",
      "seed": 202611164011,
      "semantic_template": "qf-mod47-mle-law-likelihood/{\"observations\":[\"x10\",\"y10\"],\"observed\":\"x10\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a10\",\"b10\"]}"
    },
    "fields": {
      "title": "The Observed Likelihood Uses Every Qualified Parameter Law",
      "stem": "Use complete theta labels=[a10, b10], observations=[x10, y10] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x10. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([a10, b10], [x10, y10], [[3/4, 1/4], [1/4, 3/4]], x10, [[a10, 3/4], [b10, 1/4]], complete_normalized_finite_model_likelihood_table)",
        "explanation": "Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a10, b10], [x10, y10], [[3/4, 1/4], [1/4, 3/4]], x10, [[a10, 3/4], [b10, 1/4]], complete_normalized_finite_model_likelihood_table)."
      },
      "skills": [
        "The Observed Likelihood Uses Every Qualified Parameter Law"
      ],
      "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-law-likelihood；資料={\"theta_values\":[\"a10\",\"b10\"],\"observations\":[\"x10\",\"y10\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x10\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete theta labels=[a10, b10], observations=[x10, y10] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x10. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "Θ=[a10, b10]; Ω=[x10, y10]",
      "after": "[[3/4, 1/4], [1/4, 3/4]]",
      "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_a10(Ω) := sum(3/4, 1/4)",
      "after": "P_a10(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_a10(Ω)，不以小數近似替代。",
          "en": "Recompute P_a10(Ω) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3/4",
          "1/4"
        ],
        "output": "1",
        "label": "P_a10(Ω)"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "P_b10(Ω) := sum(1/4, 3/4)",
      "after": "P_b10(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_b10(Ω)，不以小數近似替代。",
          "en": "Recompute P_b10(Ω) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "3/4"
        ],
        "output": "1",
        "label": "P_b10(Ω)"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "parameter=a10; observed=x10",
      "after": "L(a10;x10)=3/4",
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "parameter=b10; observed=x10",
      "after": "L(b10;x10)=1/4",
      "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": "[[a10, 3/4], [b10, 1/4]]",
      "after": "maximum=3/4; all argmax=[a10]",
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "likelihood indexed by fixed observed outcome",
      "after": "not a posterior probability over parameters",
      "relation": "implies",
      "rule_id": "likelihood_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "不同參數的likelihood不需要加總為1；本題列出的原law才須正規化。",
        "en": "Likelihoods across parameters need not sum to1; the original laws are the normalized rows."
      },
      "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": [
      [
        "a10",
        "b10"
      ],
      [
        "x10",
        "y10"
      ],
      [
        [
          "3/4",
          "1/4"
        ],
        [
          "1/4",
          "3/4"
        ]
      ],
      "x10",
      [
        [
          "a10",
          "3/4"
        ],
        [
          "b10",
          "1/4"
        ]
      ],
      "complete_normalized_finite_model_likelihood_table"
    ]
  },
  "conclusion": {
    "canonical_value": "([a10, b10], [x10, y10], [[3/4, 1/4], [1/4, 3/4]], x10, [[a10, 3/4], [b10, 1/4]], complete_normalized_finite_model_likelihood_table)",
    "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": "每個參數的有限律各自正規化；固定觀測後比較同一欄，likelihood 不是後驗。",
    "en": "Each parameter law is normalized. Fix the observation and compare its column; likelihood is not a posterior."
  },
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      "batch_sha256": "9faaeb838f1f456732b156364ce54a44a9a4a7595503e262949b88edcff4f18f",
      "original_line": 11,
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      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations.en.json",
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        "english_path": "public/data/i18n/editions/mko-finite-maximum-likelihood-estimator.en.json"
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    "original_candidate_line_sha256": "c40f742aa204fe398c53d7071d4540d4ecfa1c7740a8cde082aebdfbcd0239de"
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。",
      "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI review or general formal proof. Original questions, answers, MKOs, formulas, ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-003-02",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
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    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
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    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}