{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-bayes-theorem",
  "semantic_family_id": "sf-bayes-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "貝氏定理"
  },
  "summary": {
    "zh-Hant": "正分母條件下，Bayes定理以共同intersection與priors/likelihoods精確反轉條件機率。"
  },
  "statement": {
    "zh-Hant": "完整有限原子law上用精確有理質量。二事件與partition形式都核正conditioning denominator；partition必須互斥且exhaustive，total probability得到的evidence mass也須正。Bayes不表示因果。"
  },
  "formula": {
    "tex": "B(E)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>B</mi><mo>(</mo><mi>E</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "B",
        "glyph": "B"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "E",
          "glyph": "E"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "f4d0764e76c03392279e25d661f9acc1b4d7e93d5d2a1bc11c5a30dd252dc465"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限樣本空間及各原子精確非負有理質量總和1。"
    },
    {
      "id": "a2",
      "zh-Hant": "所有實際使用的conditional denominator嚴格正。"
    }
  ],
  "symbols": [
    {
      "token": "B",
      "role": "bayes_update",
      "role_zh": "Bayes更新",
      "scope": "representative_formula"
    },
    {
      "token": "E",
      "role": "evidence_event",
      "role_zh": "證據事件",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-conditional-probability",
      "title_zh": "條件機率",
      "reason_zh": "正分母定義。"
    },
    {
      "id": "mko-event",
      "title_zh": "事件",
      "reason_zh": "hypothesis與evidence為事件。"
    },
    {
      "id": "mko-probability",
      "title_zh": "機率",
      "reason_zh": "joint、prior與evidence mass。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "雙事件公式共享同一intersection。",
      "partition形式由total probability正規化。",
      "沒有prior/likelihood不得反轉，且不推出causation。"
    ],
    "common_misconception_zh": "忽略零分母、漏prior，或把P(E|H)直接當P(H|E)。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限law；不作因果推論。",
    "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": "加入參數模型與posterior distributions。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-bayes-theorem.md 整理；2026-10-14 脈衝 R06（使用者手動接續 2026-11-02 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"
  }
}