{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-parametric-statistical-model",
  "semantic_family_id": "sf-parametric-statistical-model",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "參數統計模型"
  },
  "summary": {
    "zh-Hant": "參數統計模型為每個theta指定同一有限可測sample space上的完整機率law P_theta。"
  },
  "statement": {
    "zh-Hant": "完整核有限parameter set與每列非負、總和1的law table。likelihood L(theta;x)=P_theta(x)視theta為自變數；它不是posterior，未供prior時不作Bayes。identifiability不由單一觀察推得。"
  },
  "formula": {
    "tex": "M(T)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>M</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "M",
        "glyph": "M"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "T",
          "glyph": "T"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "6803a4d65325163cc8c0bc75748fcad3377d09653785d5de9d81b7f2b8ecca11"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "共同有限sample space與有限parameter set。"
    },
    {
      "id": "a2",
      "zh-Hant": "每個theta的完整精確有理law皆歸一。"
    }
  ],
  "symbols": [
    {
      "token": "M",
      "role": "parametric_statistical_model",
      "role_zh": "參數模型",
      "scope": "representative_formula"
    },
    {
      "token": "T",
      "role": "parameter_set",
      "role_zh": "參數集合",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-joint-distribution",
      "title_zh": "有限聯合分布",
      "reason_zh": "完整有限law。"
    },
    {
      "id": "mko-probability",
      "title_zh": "機率",
      "reason_zh": "非負及歸一。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "theta映到law。"
    },
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "parameter/sample spaces。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "所有theta共享sample space。",
      "逐theta完整核law。",
      "likelihood不是posterior，parameter不是observation。"
    ],
    "common_misconception_zh": "把likelihood當posterior，或讓sample space隨theta暗中改變。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理有限完整table；不宣稱identifiability或inference optimality。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入estimator與sampling distribution。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-parametric-statistical-model.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-11-03 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}