{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-maximum-likelihood-estimator",
  "semantic_family_id": "sf-finite-maximum-likelihood-estimator",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限模型最大似然估計量"
  },
  "summary": {
    "zh-Hant": "有限參數模型中，MLE選取使觀察likelihood最大的參數，tie需固定且與theta無關。"
  },
  "statement": {
    "zh-Hant": "每個theta須有對完整observation carrier的normalized finite law。對observed x建立全參數likelihood table與完整argmax set；固定、theta-independent deterministic tie order後，對全部observations形成total estimator。tie與all-zero likelihood須各有明示witness。MLE最大化likelihood而非posterior，不假設unique，也不推出unbiased、consistent或asymptotically optimal。"
  },
  "formula": {
    "tex": "L(T,x)=h",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>L</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>h</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "L",
          "glyph": "L"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "T",
            "glyph": "T"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "h",
        "glyph": "h"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "a65d0d8403688b6eb4956401cb4c77f50cd81a7cd1482b707af8394a1fe5f26e"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "參數、observation finite unique，probabilities exact rational且逐theta總和1。"
    },
    {
      "id": "a2",
      "zh-Hant": "tie order是完整parameter permutation且不依unknown theta。"
    }
  ],
  "symbols": [
    {
      "token": "L",
      "role": "likelihood_operator",
      "role_zh": "似然算子",
      "scope": "representative_formula"
    },
    {
      "token": "T",
      "role": "finite_mle",
      "role_zh": "有限MLE",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "observed_sample",
      "role_zh": "觀察樣本",
      "scope": "representative_formula"
    },
    {
      "token": "h",
      "role": "selected_parameter",
      "role_zh": "選定參數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-parametric-statistical-model",
      "title_zh": "參數統計模型",
      "reason_zh": "完整laws。"
    },
    {
      "id": "mko-statistical-estimator",
      "title_zh": "統計估計量",
      "reason_zh": "total estimator。"
    },
    {
      "id": "mko-optimization-problem",
      "title_zh": "最佳化問題",
      "reason_zh": "argmax。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整laws與observed資格先核對。",
      "argmax set與tie estimator分開。",
      "likelihood/posterior與asymptotic claims分界。"
    ],
    "common_misconception_zh": "把MLE當posterior mode或默認argmax唯一。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限exact estimator不推出無偏、一致或漸近效率。",
    "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finite likelihood ratio與risk比較。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-maximum-likelihood-estimator.md 整理；2026-10-15 脈衝 R04（使用者手動接續 2026-11-16 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}