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12-bernoulli-compact-form.html

<!doctype html><html lang="zh-Hant"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width,initial-scale=1"><title>KP16 12 Bernoulli Compact Form</title>
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</head><body><div class="wrap"><div class="stage" id="stage"><canvas id="c" width="1600" height="900"></canvas><div id="math-layer"></div><div class="hint">互動：點擊 / 拖曳 • Click / Drag</div></div></div>
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<script>
/* ===== Page script ===== */

const P1=[0.8,0.3,0.7], Q1=[0.4,0.6,0.5];
const NAMES=['x₁ 帶書','x₂ 戴眼鏡','x₃ 上課時間'];
const NAMES_EN=['carries a book','wears glasses','class hour'];
let fi=0, xi=1;

addFormula({id:'def1',tex:String.raw`P(x_i=1\mid\omega_1)=p_i\,,\quad P(x_i=0\mid\omega_1)=1-p_i`,x:1165,y:222,w:680,size:18,align:'center',color:C1});
addFormula({id:'def2',tex:String.raw`P(x_i=1\mid\omega_2)=q_i\,,\quad P(x_i=0\mid\omega_2)=1-q_i`,x:1165,y:258,w:680,size:18,align:'center',color:C2});

addFormula({id:'compact1',tex:String.raw`P(x_i\mid\omega_1)=p_i^{\,x_i}(1-p_i)^{\,1-x_i}`,x:420,y:414,w:640,size:28,align:'center',color:C1});
addFormula({id:'compact2',tex:String.raw`P(x_i\mid\omega_2)=q_i^{\,x_i}(1-q_i)^{\,1-x_i}`,x:1180,y:414,w:640,size:28,align:'center',color:C2});

addFormula({id:'live1',tex:'',x:420,y:694,w:600,size:21,align:'center',color:C1});
addFormula({id:'live2',tex:'',x:1180,y:694,w:600,size:21,align:'center',color:C2});

addFormula({id:'prod1',tex:String.raw`p(\boldsymbol{x}\mid\omega_1)=\prod_{i=1}^{l}p_i^{\,x_i}(1-p_i)^{\,1-x_i}`,x:420,y:818,w:640,size:19,align:'center',color:C1});
addFormula({id:'prod2',tex:String.raw`p(\boldsymbol{x}\mid\omega_2)=\prod_{i=1}^{l}q_i^{\,x_i}(1-q_i)^{\,1-x_i}`,x:1180,y:818,w:640,size:19,align:'center',color:C2});

function buildLive(cls){
  const p = cls===1 ? P1[fi] : Q1[fi];
  const sym = cls===1 ? 'p' : 'q';
  const omega = cls===1 ? '1' : '2';
  const result = xi===1 ? p : (1-p);
  if(xi===1){
    return String.raw`P(x_i\mid\omega_${omega})=${sym}_i^{1}(1-${sym}_i)^{0}=${sym}_i=${result.toFixed(2)}`;
  }
  return String.raw`P(x_i\mid\omega_${omega})=${sym}_i^{0}(1-${sym}_i)^{1}=1-${sym}_i=${result.toFixed(2)}`;
}

function render(){
  reset();
  title('伯努利緊緻式：二元特徵機率寫成一行','Bernoulli Compact Form: One Line for a Binary-Feature Likelihood',12);

  /* ---------- Panel A: setup ---------- */
  rr(60,170,1480,140,22,P.panel,P.cyan,2);
  txt('情境設定 Setup',90,198,18,P.ink,'left',800);
  wrap('二元特徵 xᵢ ∈ {0,1} 只有兩種結果：對 ω₁ 而言，「出現」(xᵢ=1) 的機率是 pᵢ，「不出現」(xᵢ=0) 的機率是 1−pᵢ；對 ω₂ 則用 qᵢ、1−qᵢ 描述。',90,226,660,24,15.5,P.ink,600);

  /* ---------- Panel B: compact form ---------- */
  rr(60,324,1480,170,22,P.panel2,'#ffffff14',2);
  txt('壓縮成一行：Bernoulli 緊緻式　The Bernoulli compact form',90,352,19,P.amber,'left',800);
  wrapCenter('xᵢ=1 時，(1−pᵢ)⁰=1，只剩下 pᵢ；xᵢ=0 時，pᵢ⁰=1，只剩下 1−pᵢ —— 一個式子同時涵蓋兩種情況！',800,462,1380,24,15.5,P.lime,650);

  /* ---------- Panel C: interactive collapse ---------- */
  rr(60,510,1480,240,24,P.panel,'#ffffff14',2);
  txt('互動：選一個特徵、切換 0 / 1，看壓縮公式即時代入　Pick a feature and toggle its bit',90,538,18,P.ink,'left',800);

  const fx=[90,400,710];
  NAMES.forEach((n,i)=>btn(n,fx[i],564,300,44,()=>{fi=i;render()},fi===i,P.cyan,18));
  btn(`xᵢ = ${xi}`,1120,564,320,44,()=>{xi=1-xi;render()},true,xi===1?P.lime:P.coral,20);

  card(90,622,660,112,C1);
  txt(`ω₁　${NAMES[fi]}（${NAMES_EN[fi]}）　pᵢ = ${P1[fi].toFixed(2)}`,120,650,16,C1,'left',750);
  card(850,622,660,112,C2);
  txt(`ω₂　${NAMES[fi]}（${NAMES_EN[fi]}）　qᵢ = ${Q1[fi].toFixed(2)}`,880,650,16,C2,'left',750);

  setTex('live1', buildLive(1));
  setTex('live2', buildLive(2));

  /* ---------- Panel D: product over all features ---------- */
  txt('套用條件獨立假設（第 2 頁），把每個特徵的壓縮式連乘起來：Combine across l features via conditional independence',90,754,14.5,P.muted,'left',650);
  rr(60,772,1480,84,22,P.panel2,'#ffffff14',2);

  showF(['def1','def2','compact1','compact2','live1','live2','prod1','prod2']);
}
render();
</script></body></html>