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12-summary.html

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/* 12 — Summary: five take-aways, each with its core formula */
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 {zh:'原則：選最不武斷的分佈',en:'Principle: the least prejudiced distribution',col:P.cyan,
  items:[['最大熵原理','\\hat p=\\arg\\max_{p}\\,H[p]\\quad\\text{s.t. constraints}']],
  text:'在所有滿足已知限制的候選分佈中，挑出熵最高、最不偏袒任何位置的那一個。只使用證據支持的資訊。'},
 {zh:'工具：微分熵 = 無偏好分數',en:'Tool: differential entropy as an honesty score',col:P.violet,
  items:[['微分熵','H=-\\!\\int p(x)\\ln p(x)\\,dx'],['直覺','\\text{flatter}\\ \\Rightarrow\\ \\text{larger }H']],
  text:'分佈越平坦、越公平，H 越高；機率越集中，代表私自加了更多偏愛，H 越低。'},
 {zh:'只知道範圍 → 均勻分佈',en:'Range only: the uniform distribution',col:P.lime,
  items:[['最大熵解','\\hat p(x)=\\dfrac{1}{x_2-x_1}'],['最大熵值','H_{\\max}=\\ln(x_2-x_1)']],
  text:'由拉格朗日乘子法求導，解不含 x，只能是水平線；再用面積 = 1 決定高度。[0, 4] 上 H = ln 4 ≈ 1.386。'},
 {zh:'限制改變，答案跟著改變',en:'New constraints, new answer',col:P.amber,
  items:[['只知範圍','\\text{Uniform}:\\ \\dfrac{1}{x_2-x_1}'],['＋均值 μ、變異數 σ²','\\text{Gaussian}:\\ \\mathcal N(\\mu,\\sigma^2)'],['x ≥ 0 ＋均值 μ','\\text{Exponential}:\\ \\dfrac{1}{\\mu}e^{-x/\\mu}']],
  text:'嚴格尊重所有已知限制，對限制之外的一切保持徹底的謙遜。'},
 {zh:'避開四個常見誤解',en:'Avoid the four classic pitfalls',col:P.coral,items:[],
  text:['① 熵最大 ≠ 資訊最多，而是額外武斷最少。','② 高峰 ≠ 更精確，可能只是缺乏證據的豪賭。','③ 最大熵 ≠ 永遠均勻，答案取決於限制清單。','④ 微分熵依賴尺度，比較必須在同一定義域上進行。']}];
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