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𝑓 2.
𝑓
𝒇
𝑃
𝑄
𝐷𝑓 (𝑃 ∥ 𝑄)
𝑄
𝑃
𝑓
𝑓
𝑄
𝑃
𝛺
𝑓(1) = 0
𝑓
𝑃
𝑄
𝑃
𝑄
𝑓
𝑑𝑃 𝐷𝑓 (𝑃 ∥ 𝑄) = ∫ 𝑓 ( ) 𝑑𝑄 𝑑𝑄 Ω 𝛺 𝑝
𝑞
𝜇
𝑃
𝑑𝑃 = 𝑝𝑑𝜇, 𝑑𝑄 = 𝑞𝑑𝜇
𝑄 𝑓
𝑝(𝑥) 𝐷𝑓 (𝑃 ∥ 𝑄) = ∫ 𝑓 ( ) 𝑞(𝑥)𝑑𝜇(𝑥) 𝑞(𝑥) Ω 𝑓
𝜒
𝑓 𝑓 𝑓 𝑓 𝑓(𝑡) 𝑡 ln 𝑡 , − ln 𝑡 2
(√𝑡 − 1) , 2(1 − √𝑡)
(𝜒 2 )
|𝑡 − 1| 𝜒
(𝑡 − 1)2 , 𝑡 2 − 1
2
1+𝛼 4 (1 − 𝑡 2 ) , if 𝛼 ≠ ±1 2 {1 − 𝛼 𝑡 ln 𝑡 , if 𝛼 = 1 − ln 𝑡 , if 𝛼 = −1
𝛼
𝑓
𝑃
𝑄
𝑑𝑃 𝑑𝑃 𝐷𝑓 (𝑃 ∥ 𝑄) = ∫ 𝑓 ( ) 𝑑𝑄 ≥ 𝑓 (∫ 𝑑𝑄) = 𝑓(1) = 0 𝑑𝑄 𝑑𝑄
𝑃 𝑄 𝐷𝑓 (𝑃 ∥ 𝑄) ≥ 𝐷𝑓 (𝑃𝜅 ∥ 𝑄𝜅 )
𝑃𝜅
𝑄𝜅
𝜅
*𝑃, 𝑄+
0≤𝜆≤1 𝐷𝑓 (𝜆𝑃1 + (1 − 𝜆)𝑃2 ∥ 𝜆𝑄1 + (1 − 𝜆)𝑄2 ) ≤ 𝜆𝐷𝑓 (𝑃1 ∥ 𝑄1 ) + (1 − 𝜆)𝐷𝑓 (𝑃2 ∥ 𝑄2 ) ℝ2+
𝑝
(𝑝, 𝑞) ↦ 𝑞𝑓 ( ) 𝑞
F-divergence.pdf
dP. dQ dQ) = f(1) = 0. P Q Pκ Qκ κ. Df. (P ⥠Q) ⥠Df. (Pκ ⥠Qκ. ) *P,Q+. 0 ⤠λ ⤠1. Df. (λP1 + (1 â λ)
P2
⥠λQ1 + (1 â λ)Q2. ) ⤠λDf. (P1 ⥠Q1. ) + (1 â λ)Df. (
P2
⥠Q2. ) R+. 2. (p, q) ⦠qf (. p. q. ) Page 2 of 2. F-divergence.
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