{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261103-r04-parametric-statistical-model-0091", "pilot_number": "R0034-17866", "number": 17866, "family_id": "qf-mod34-statmodel-parameter-observation-boundary", "title": { "zh-Hant": "第 17866 題：參數統計模型", "en": "Question 17866: Equal Numeric Indices Do Not Give Parameters and Observations the Same Role" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261103-r04-parametric-statistical-model-0091", "batch_id": "batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-parametric-statistical-model" ], "title_zh": "參數統計模型", "stem_zh": "共同有限sample space與每個parameter的完整law row。操作=statmodel-parameter-observation-boundary；資料={\"parameter_count\":2,\"outcome_count\":3,\"laws\":[[[1,19],[1,19],[17,19]],[[1,2],[1,3],[1,6]]],\"parameter\":1,\"outcome\":0}。請依 [parameter,outcome,parameter_role,observation_role,law_value,same_role,boundary] 欄序回答。", "answer": { "kind": "expression", "value": "(1, 0, parameter_index, observed_outcome, 1/2, 否, parameter_not_observation_and_likelihood_not_posterior)", "explanation_zh": "Wave5第三層核對=(1, 0, parameter_index, observed_outcome, 1/2, 否, parameter_not_observation_and_likelihood_not_posterior)；依完整明示資料重算；每列非負且精確總和1，parameter與observation分離；likelihood是parameter函數而非posterior；不暗示prior或identifiability。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "參數統計模型" ] }, "variant": { "family_id": "qf-mod34-statmodel-parameter-observation-boundary", "seed": 202611034091, "parameters": { "parameter_count": 2, "outcome_count": 3, "laws": [ [ [ 1, 19 ], [ 1, 19 ], [ 17, 19 ] ], [ [ 1, 2 ], [ 1, 3 ], [ 1, 6 ] ] ], "parameter": 1, "outcome": 0, "domain": "Complete_finite_parametric_law_table", "operation": "statmodel-parameter-observation-boundary", "semantic_template": "qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,19],[1,19],[17,19]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須由主端獨立重算；不構成人工審定、sampled rank upper bound、infinite-domain uniformity、finite-prefix series proof、posterior/identifiability、graph connectivity、nonexhaustive maximum、Lean或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261103-r04-parametric-statistical-model-0091", "batch_id": "batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "bd983da8d546de485ac633f722a90283f8ddc5ade7c9e8fbe9f10e869a9c322f", "family_id": "qf-mod34-statmodel-parameter-observation-boundary", "seed": 202611034091, "semantic_template": "qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,19],[1,19],[17,19]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}" }, "fields": { "title": "Equal Numeric Indices Do Not Give Parameters and Observations the Same Role", "stem": "Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/19, 1/19, 17/19], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=0. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).", "answer": { "kind": "expression", "value": "(1, 0, parameter_index, observed_outcome, 1/2, No, parameter_not_observation_and_likelihood_not_posterior)", "explanation": "Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 0, parameter_index, observed_outcome, 1/2, No, parameter_not_observation_and_likelihood_not_posterior)." }, "skills": [ "Equal Numeric Indices Do Not Give Parameters and Observations the Same Role" ], "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": "共同有限sample space與每個parameter的完整law row。操作=statmodel-parameter-observation-boundary；資料={\"parameter_count\":2,\"outcome_count\":3,\"laws\":[[[1,19],[1,19],[17,19]],[[1,2],[1,3],[1,6]]],\"parameter\":1,\"outcome\":0}。請依 [parameter,outcome,parameter_role,observation_role,law_value,same_role,boundary] 欄序回答。", "en": "Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/19, 1/19, 17/19], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=0. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "parameter0 complete outcome mass sum := sum(1/19, 1/19, 17/19)", "after": "parameter0 complete outcome mass sum = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 parameter0 complete outcome mass sum，不以小數近似替代。", "en": "Recompute parameter0 complete outcome mass sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/19", "1/19", "17/19" ], "output": "1", "label": "parameter0 complete outcome mass sum" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "original parameter row=0;SAME outcome carrier Fin(3);original aligned rational entries=[[1, 19], [1, 19], [17, 19]]", "after": "reduced rational candidate entries=[1/19, 1/19, 17/19];sum=1;nonnegative=是;row qualified=是", "relation": "implies", "rule_id": "statmodel_every_parameter_law_same_observation_space", "conditions": [], "explanation": { "zh-Hant": "每個parameter都要同一sample space上的完整非負總和1之law；未合格候選列保留數值與False，不因叫law table就自動正常化。", "en": "Every parameter needs a complete nonnegative unit-mass law on the same sample space.Invalid candidate rows retain their values and false flags;the name law table does not normalize them." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "parameter1 complete outcome mass sum := sum(1/2, 1/3, 1/6)", "after": "parameter1 complete outcome mass sum = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確計算，保留分母、正負號、索引與嚴格條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 parameter1 complete outcome mass sum，不以小數近似替代。", "en": "Recompute parameter1 complete outcome mass sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/2", "1/3", "1/6" ], "output": "1", "label": "parameter1 complete outcome mass sum" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "original parameter row=1;SAME outcome carrier Fin(3);original aligned rational entries=[[1, 2], [1, 3], [1, 6]]", "after": "reduced rational candidate entries=[1/2, 1/3, 1/6];sum=1;nonnegative=是;row qualified=是", "relation": "implies", "rule_id": "statmodel_every_parameter_law_same_observation_space", "conditions": [], "explanation": { "zh-Hant": "每個parameter都要同一sample space上的完整非負總和1之law；未合格候選列保留數值與False，不因叫law table就自動正常化。", "en": "Every parameter needs a complete nonnegative unit-mass law on the same sample space.Invalid candidate rows retain their values and false flags;the name law table does not normalize them." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "original parameter carrier Fin(2);common outcome carrier Fin(3);ALL original row audits=[[0, [1/19, 1/19, 17/19], 1, 是], [1, [1/2, 1/3, 1/6], 1, 是]]", "after": "whole model qualified=是;common_sample_space;no parameter prior or posterior supplied", "relation": "implies", "rule_id": "statmodel_whole_table_qualification_no_parameter_prior", "conditions": [], "explanation": { "zh-Hant": "把所有列資格合取，parameter索引不因此成為帶機率的隨機變數；沒有給prior或posterior，不跨parameter自動正規化。", "en": "Conjoin all row qualifications.Parameter indices do not thereby become a probability-distributed random variable.No parameter prior or posterior is supplied;do not normalize across parameters by default." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "original parameter index=1;original observed-outcome index=0;candidate full row=[1/2, 1/3, 1/6]", "after": "raw candidate cell=1/2;whole-table qualification remains=是", "relation": "implies", "rule_id": "statmodel_original_row_column_cell_not_model_repair", "conditions": [], "explanation": { "zh-Hant": "按parameter選列、outcome選欄；無效候選表的raw cell不自動成為合格機率，也不能修好整個model。", "en": "Select the row by parameter and the column by outcome.A raw cell in an invalid candidate table is not automatically a qualified probability and does not repair the model." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "numeric labels parameter=1;outcome=0;numeric equality=否;same raw cell=1/2", "after": "parameter role=parameter_index;outcome role=observed_outcome;same semantic role=否;parameter_not_observation_and_likelihood_not_posterior", "relation": "implies", "rule_id": "statmodel_distinct_semantic_roles_even_equal_numbers", "conditions": [], "explanation": { "zh-Hant": "同為整數0也屬不同角色：parameter索引挑law，outcome是該law下的觀察；No是語義不相同，並非數字不相等的測試。", "en": "Even numeric0 can label different roles:a parameter selects a law and an outcome is an observation under it.The No distinguishes semantic roles,not numerical inequality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 1, 0, "parameter_index", "observed_outcome", "1/2", false, "parameter_not_observation_and_likelihood_not_posterior" ] }, "conclusion": { "canonical_value": "(1, 0, parameter_index, observed_outcome, 1/2, 否, parameter_not_observation_and_likelihood_not_posterior)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations/questions.jsonl", "batch_sha256": "b3c8742442ebbc5f202e3e8c123e4c21155e54b508f6546773c7132efeb16683", "original_line": 91, "original_line_sha256": "af299ce6f1f81a3a4eea7fd504eef8d1a634a4ab1bf49eba56b2a21595791410", "question_compact_sha256": "bd983da8d546de485ac633f722a90283f8ddc5ade7c9e8fbe9f10e869a9c322f", "english_row_compact_sha256": "6dda892c2968dc8441ffe332753e198fe4aaed055531e10f101ead268859b55c", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations.en.jsonl" }, "mko_bindings": { "mko-parametric-statistical-model": { "sha256": "ef431eac244935b2a2629f424d522b6d9b36a17dede2b668b4402aed44ac715d", "english_sha256": "d659a31945072a4e32f774a17f8534156a1e73ada04c4f726fdb98c5f15f979b", "original_path": "public/data/mko/parametric-statistical-model.json", "english_path": "public/data/i18n/editions/mko-parametric-statistical-model.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "d5b0037cbaeef26e3828d04bf29e0e0f6e466926098b6b09e3237d1fdbdbd0e2", "original_candidate_sha256": "b0f296502bd3007a68666b782c3aabb7ba855b114ca75b0e22e6c43ecd85c346", "original_candidate_line_sha256": "b0f296502bd3007a68666b782c3aabb7ba855b114ca75b0e22e6c43ecd85c346" }, "historical_source_state": { "review_state": { "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": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human 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-20261026-008-p35", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings 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": { "semantic_review_sha256": "f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336", "question_first_witness_sha256": "22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699" }, "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": { "model_sha256": "3fb71b642117c8d9fdfbbcc9255be289c82cf3283e512377432fba2728a57d2e", "arithmetic_steps": 2 }, "uses_answer_as_premise": false, "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." } }