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09-quiz.html
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/* ===== Page script ===== */
/* ===== Shared helpers for KP18: DAG drawing, parent sets, CPT parameter counting ===== */
const C1='#63b3ed',C2='#ff7b72';
const NM=['x₁','x₂','x₃','x₄','x₅','x₆'];
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const SUPM={'0':'⁰','1':'¹','2':'²','3':'³','4':'⁴','5':'⁵','6':'⁶','7':'⁷','8':'⁸','9':'⁹','-':'⁻'};
const sup=n=>String(n).split('').map(c=>SUPM[c]||c).join('');
/* ===== KP19: the four-variable chain x -> y -> z -> w with the lecture's CPTs ===== */
const CPC={px:.6,py:[.3,.4],pz:[.6,.25],pw:[.3,.45]}; /* P(child=1 | parent=0/1) */
const NC=['x','y','z','w'],POS4=[[.07,.5],[.36,.5],[.64,.5],[.93,.5]];
const bern=(p,t)=>t?p:1-p;
function jointC(v,th){th=th||CPC;return bern(th.px,v[0])*bern(th.py[v[0]],v[1])*bern(th.pz[v[1]],v[2])*bern(th.pw[v[2]],v[3])}
/* exact posterior P(q=1 | evidence) by full enumeration; ev maps node index -> 0/1 */
function postC(q,ev,th){let n1=0,n0=0;for(let m=0;m<16;m++){const v=[(m>>3)&1,(m>>2)&1,(m>>1)&1,m&1];
if(Object.keys(ev).some(k=>v[k]!==ev[k]))continue;const j=jointC(v,th);if(v[q])n1+=j;else n0+=j}return n1/(n1+n0)}
/* 09 — Three-question self-check */
const Q=[
{q:'在情境 (a) 的下傳預測中，節點 z 算出 P(z=1 | x=1) = 0.46，請問它在計算時使用了哪些資訊？',
o:['A) 原始 x = 1 的觀察值與整條鏈的聯合機率表','B) 僅使用了 y 傳來的機率分佈（0.4 與 0.6）加上 z 自身的 CPT','C) 僅使用了 x 與 w 的邊際先驗機率','D) 必須先知道下游 w 是否發生才能計算'],a:1,
ok:'答對了！(0.25)(0.4) + (0.6)(0.6) = 0.46：只用 y 傳來的 0.4 / 0.6 與 z 自己的 CPT。',
no:'再想一想：z 只需要上一站 y 傳來的分佈，加上自己的條件機率表，不必回頭看 x，更不需要下游的 w。'},
{q:'在取值數 L = 10 的樹狀網路中，利用訊息傳遞分解求和相比於暴力窮舉法（Big-O 量級），運算次數縮減了多少倍？',
o:['A) 10 倍（100,000 降至 10,000）','B) 100 倍（100,000 降至 1,000）','C) 1,000 倍（100,000 降至 100）','D) 10,000 倍（100,000 降至 10）'],a:2,
ok:'正確！L⁵ = 100,000 對 L² = 100，相差 L³ = 1,000 倍（第 6 頁可拖曳 L 驗證）。',
no:'暴力法是 L⁵ = 10⁵，樹狀分解是 L² = 10²，相差 L³ = 1,000 倍。注意這是忽略常數的量級比較。'},
{q:'為什麼在多重連通（包含無向環路）的貝氏網路中，精確機率推論會是 NP-hard？',
o:['A) 因為節點數超過 10 個後記憶體不足','B) 因為圖形無法畫出箭頭','C) 因為環路會造成訊息在多條路徑間相互依賴與迴圈反饋，無法簡單單向接力而不重複計算','D) 因為每個節點的條件機率表無法用小數表示'],a:2,
ok:'太棒了！第 7 頁的菱形圖就是例子：A 的資訊沿兩條路徑到達 D，單向接力會被重複計算。',
no:'關鍵在結構而不是記憶體或表示法：有迴圈時，同一份資訊會沿多條路徑傳遞並互相依賴，簡單的單向訊息傳遞就會重複計算。'}];
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