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08-four-principles.html

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/* 08 — Four master principles */
let sel=0;
const ROW=[
 {zh:'鏈鎖法則永遠成立',en:'The chain rule always holds',col:P.cyan,
  items:[['通用展開式','p(x_1,\\ldots,x_l)=\\prod_{i=1}^{l}p(x_i\\mid x_{i-1},\\ldots,x_1)'],['完全相依的參數量','\\sum_{i=1}^{l}2^{\\,i-1}=2^{l}-1']],
  text:'但光把聯合機率拆開不等於節省參數：大帳單摺成小帳單，總金額依舊沉重。'},
 {zh:'父集合區域性問責',en:'Local accountability via parent sets',col:P.lime,
  items:[['貝氏網路聯合分佈','p(\\boldsymbol x)=p(x_1)\\prod_{i=2}^{l}p(x_i\\mid A_i)'],['本例的二元 CPT 參數量','\\sum_{i}2^{|A_i|}=1+1+2+4+2+4=14']],
  text:'只向關鍵的 Aᵢ 報到，把全域的指數爆炸削平為區域性的小家族（14 對 63，省約 77.8%）。'},
 {zh:'DAG 拓撲保證可算',en:'The DAG guarantees computability',col:P.amber,
  items:[['父集合只含編號更小的節點','A_i\\subseteq\\{x_{i-1},\\ldots,x_1\\}']],
  text:'箭頭有向且絕無迴圈，禁止互相甩鍋，奠定「由因到果」的計算先後順序。'},
 {zh:'條件獨立非真無關',en:'Conditional independence is not "unrelated"',col:C2,
  items:[['知道中介後，x₁ 不再提供新情報','p(x_3\\mid x_2,x_1)=p(x_3\\mid x_2)'],['但未觀察 x₂ 時','\\small P(x_3\\!=\\!1\\mid x_1\\!=\\!1)=0.74\\ne P(x_3\\!=\\!1\\mid x_1\\!=\\!0)=0.26']],
  text:'它是「知曉中介後的情報飽和」，不是真的毫無關係：未觀察中介前，資訊仍透過它傳遞。'}];
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