{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-conditional-expectation",
  "semantic_family_id": "sf-conditional-expectation",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限條件期望"
  },
  "summary": {
    "zh-Hant": "有限機率空間中，相對分割生成sigma代數的條件期望在每個正質量區塊取加權平均並保持區塊積分。"
  },
  "statement": {
    "zh-Hant": "在同一有限機率空間上，條件期望E[X|G]須G可測，且在每個G可測事件上與X積分相同。對分割生成的G，它在每個正質量區塊為加權平均；零質量區塊只幾乎處處決定，不除以零。"
  },
  "formula": {
    "tex": "E(X,G)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>E</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "E",
        "glyph": "E"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        },
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "2f9bb709740a90ea987a2c43d6fccb5b84da52d49040cd2c7de74edb790f6513"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限樣本、精確有理質量、X值與conditioning partition綁同一空間。"
    },
    {
      "id": "a2",
      "zh-Hant": "主要題族使用每區塊正質量；零質量邊界只回報a.s.非唯一，不強迫點值。"
    }
  ],
  "symbols": [
    {
      "token": "E",
      "role": "conditional_expectation",
      "role_zh": "條件期望",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "random_variable",
      "role_zh": "有限隨機變數",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "sigma_algebra",
      "role_zh": "conditioning資訊",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-joint-distribution",
      "title_zh": "有限聯合分布",
      "reason_zh": "有限共同law提供精確質量。"
    },
    {
      "id": "mko-measurable-function",
      "title_zh": "可測函數",
      "reason_zh": "結果須相對G可測。"
    },
    {
      "id": "mko-expectation",
      "title_zh": "期望值",
      "reason_zh": "區塊積分與總期望使用同一運算。"
    },
    {
      "id": "mko-conditional-probability",
      "title_zh": "條件機率",
      "reason_zh": "正質量區塊的加權平均使用條件law。"
    },
    {
      "id": "mko-random-variable",
      "title_zh": "隨機變數",
      "reason_zh": "X定義在同一機率空間。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "partition blocks生成有限sigma代數。",
      "每個正質量block上的條件期望為質量加權平均。",
      "結果在block上常值並保持每個block積分；零質量block只a.s.決定。"
    ],
    "common_misconception_zh": "直接逐點條件化、在零質量block除法，或只核總期望就宣稱完整條件期望。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限partition模型；零質量block不要求任意點值相等，也不外推一般測度論版本。",
    "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": "與filtration共同加入Martingale。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-conditional-expectation.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-10-29 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"
  }
}