{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0042",
  "pilot_number": "P250-0167",
  "number": 167,
  "family_id": "qf-mod47-mle-argmax",
  "title": {
    "zh-Hant": "第 167 題：有限模型最大似然估計量",
    "en": "Question 167: Retain Every Likelihood Maximizer before Discussing Uniqueness"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0042",
    "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-argmax；資料={\"theta_values\":[\"a16\",\"b16\"],\"observations\":[\"x16\",\"y16\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x16\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(x16, [[a16, 3/4], [b16, 1/4]], 3/4, [a16], 是, MLE_argmax_need_not_be_unique)",
      "explanation_zh": "Wave6第十層核對=(x16, [[a16, 3/4], [b16, 1/4]], 3/4, [a16], 是, MLE_argmax_need_not_be_unique)；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-argmax",
      "seed": 202611164042,
      "parameters": {
        "theta_values": [
          "a16",
          "b16"
        ],
        "observations": [
          "x16",
          "y16"
        ],
        "probability_rows": [
          [
            [
              3,
              4
            ],
            [
              1,
              4
            ]
          ],
          [
            [
              1,
              4
            ],
            [
              3,
              4
            ]
          ]
        ],
        "observed": "x16",
        "domain": "Exact_finite_maximum_likelihood",
        "operation": "mle-argmax",
        "semantic_template": "qf-mod47-mle-argmax/{\"observations\":[\"x16\",\"y16\"],\"observed\":\"x16\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a16\",\"b16\"]}"
      }
    },
    "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"
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    "question_id": "q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0042",
    "batch_id": "batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
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      "family_id": "qf-mod47-mle-argmax",
      "seed": 202611164042,
      "semantic_template": "qf-mod47-mle-argmax/{\"observations\":[\"x16\",\"y16\"],\"observed\":\"x16\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a16\",\"b16\"]}"
    },
    "fields": {
      "title": "Retain Every Likelihood Maximizer before Discussing Uniqueness",
      "stem": "Use complete theta labels=[a16, b16], observations=[x16, y16] 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=x16. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(x16, [[a16, 3/4], [b16, 1/4]], 3/4, [a16], Yes, MLE_argmax_need_not_be_unique)",
        "explanation": "Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x16, [[a16, 3/4], [b16, 1/4]], 3/4, [a16], Yes, MLE_argmax_need_not_be_unique)."
      },
      "skills": [
        "Retain Every Likelihood Maximizer before Discussing Uniqueness"
      ],
      "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-argmax；資料={\"theta_values\":[\"a16\",\"b16\"],\"observations\":[\"x16\",\"y16\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"observed\":\"x16\"}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete theta labels=[a16, b16], observations=[x16, y16] 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=x16. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "Θ=[a16, b16]; Ω=[x16, y16]",
      "after": "observed=x16, column=0",
      "relation": "implies",
      "rule_id": "fixed_observation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定已觀察的結果，只沿同一欄比較不同參數的機率。",
        "en": "Fix the observed outcome and compare its one probability column across parameter values."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "不要把同一列的其他可能觀察當成另一個參數。",
          "en": "Do not confuse other observations in a row with other parameter values."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "P_a16(Ω) := sum(3/4, 1/4)",
      "after": "P_a16(Ω) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核對這個參數的整列仍是合法機率 law。",
        "en": "First check that this parameter's row remains a normalized probability law."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P_a16(Ω)，不以小數近似替代。",
          "en": "Recompute P_a16(Ω) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3/4",
          "1/4"
        ],
        "output": "1",
        "label": "P_a16(Ω)"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "row for a16=[3/4, 1/4]",
      "after": "L(a16;x16)=P_a16(x16)=3/4",
      "relation": "implies",
      "rule_id": "likelihood_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "取觀測標籤在該列對應的機率，不乘任意 prior。",
        "en": "Take the probability at the observed label in this row, without introducing a prior."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "這個值仍綁定同一參數與同一觀察。",
          "en": "The value remains bound to the same parameter and observation."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "P_b16(Ω) := sum(1/4, 3/4)",
      "after": "P_b16(Ω) = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核對這個參數的整列仍是合法機率 law。",
        "en": "First check that this parameter's row remains a normalized probability law."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P_b16(Ω)，不以小數近似替代。",
          "en": "Recompute P_b16(Ω) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/4",
          "3/4"
        ],
        "output": "1",
        "label": "P_b16(Ω)"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "row for b16=[1/4, 3/4]",
      "after": "L(b16;x16)=P_b16(x16)=1/4",
      "relation": "implies",
      "rule_id": "likelihood_lookup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "取觀測標籤在該列對應的機率，不乘任意 prior。",
        "en": "Take the probability at the observed label in this row, without introducing a prior."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "這個值仍綁定同一參數與同一觀察。",
          "en": "The value remains bound to the same parameter and observation."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "L(a16;x16)=3/4, maximum=3/4",
      "after": "a16 ∈ argmax: 是",
      "relation": "implies",
      "rule_id": "argmax_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "達到最大值的每一個參數都加入 argmax，不能在第一次找到最大值後漏掉 ties。",
        "en": "Include every parameter that attains the maximum, including ties after the first maximizer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "最大值是一個數，argmax 則是一個參數集合。",
          "en": "The maximum is a number; argmax is a set of parameters."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "L(b16;x16)=1/4, maximum=3/4",
      "after": "b16 ∈ argmax: 否",
      "relation": "implies",
      "rule_id": "argmax_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "達到最大值的每一個參數都加入 argmax，不能在第一次找到最大值後漏掉 ties。",
        "en": "Include every parameter that attains the maximum, including ties after the first maximizer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "最大值是一個數，argmax 則是一個參數集合。",
          "en": "The maximum is a number; argmax is a set of parameters."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "argmax=[a16]",
      "after": "unique maximizing parameter: 是",
      "relation": "implies",
      "rule_id": "cardinality_one",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只在完整 argmax 恰有一個元素時，才可稱本題最大化參數唯一。",
        "en": "Only a complete argmax with exactly one element has a unique maximizing parameter."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "固定 tie-breaker 可產生單一輸出，卻不會把原本多元素的 argmax 改成單元素。",
          "en": "A fixed tie-breaker can select one output without making a multi-element argmax a singleton."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "finite normalized laws",
      "after": "MLE is not a posterior or an automatic quality certificate",
      "relation": "implies",
      "rule_id": "scope_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若某觀察在所有參數下 likelihood 皆為零，所有參數仍同時最大化為 0；本計算不自動推出無偏或一致。",
        "en": "If all likelihoods at an observation are zero, every parameter still maximizes at 0; this computation does not imply unbiasedness or consistency."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "禁止把 null 觀察誤作不合法的全零機率 law。",
          "en": "Do not confuse a null observation column with an invalid all-zero probability-law row."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "observed": "x16",
    "column": 0,
    "likelihoods": [
      [
        "a16",
        "3/4"
      ],
      [
        "b16",
        "1/4"
      ]
    ],
    "maximum": "3/4",
    "argmax": [
      "a16"
    ],
    "unique": true
  },
  "conclusion": {
    "canonical_value": "(x16, [[a16, 3/4], [b16, 1/4]], 3/4, [a16], 是, MLE_argmax_need_not_be_unique)",
    "explanation": {
      "zh-Hant": "最大 likelihood=3/4；完整 argmax=[a16]。",
      "en": "The maximum likelihood is 3/4; the complete argmax is [a16]."
    }
  },
  "definition_note": {
    "zh-Hant": "完整有限參數與觀察集合；固定觀測欄並保留全部 argmax，不加入 prior。",
    "en": "Use complete finite parameter and observation carriers. Fix the observed column, retain the full argmax, and introduce no prior."
  },
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      "zh-Hant": "本稿是來源綁定的教學推導候選；精確重算不等於獨立數學審查或形式化證明。原題、答案、公式、AST 與歷史未被修改。",
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