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• ▪ 𝑨 1 , 𝑨 2 , ⋯ , 𝑨 𝑂 𝑧 1 , 𝑧 2 , ⋯ , 𝑧 𝑁 (≈ 𝑡 1 , 𝑡 2 , ⋯ , 𝑡 𝑁 ) 𝑡 1 , 𝑡 2 , ⋯ , 𝑡 𝑁 𝑨 1 , 𝑨 2 , ⋯ , 𝑨 𝑂 𝑩 𝑩 𝑡 1 𝑏 11 𝑨 1 𝑥 11 𝑧 1 𝑥 21 𝑏 21 𝑏 12 𝑥 12 𝑡 2 𝑨 2 𝑧 2 𝑥 22 𝑏 22

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• ▪ 𝑡(𝑢) 𝑁 ▪ 𝑛 𝜐 𝑛 𝑎 1 (𝜕) 𝑨 1 (𝑢) 𝑡(𝑢 − 𝜐 1 ) 𝑎 2 (𝜕) 𝑨 2 (𝑢) 𝑡(𝑢 − 𝜐 2 ) 𝒜 𝜕 = 𝒜 𝑢 = = ⋮ ⋮ ⋮ 𝑎 𝑁 𝜕 𝑨 𝑁 𝑢 𝑡(𝑢 − 𝜐 𝑁 ) ∞ ∞ 𝑨 𝑛 𝑢 𝑓 −𝑘𝜕𝑢 𝑒𝑢 = 𝑡 𝑢 − 𝜐 𝑛 𝑓 −𝑘𝜕𝑢 𝑒𝑢 = 𝑓 −𝑘𝜕𝜐 𝑛 𝑇(𝜕) 𝑎 𝑛 𝜕 ≡ −∞ −∞ 𝑏 1 𝑓 −𝑘𝜕𝜐 1 ∞ 𝒜 𝜕 = 𝒃(𝜕)𝑇(𝜕) 𝑡 𝑢 𝑓 −𝑘𝜕𝑢 𝑒𝑢 ⋮ 𝑇 𝜕 ≡ 𝒃(𝜕) = ≡ ⋮ 𝑏 𝑁 −∞ 𝑓 −𝑘𝜕𝜐 𝑁

• 𝑧 𝜄 𝑡 𝑈 𝑛 − 1 − 𝑁 − 1 𝒒 𝒏 = 𝑒 𝑦 , 0, 0 2 𝑇(𝜕) • 𝑒 𝑦 𝑛 − 1 − 𝑁 − 1 𝑒 𝑦 𝜐 𝑛 = − 𝑑 sin 𝜄 𝑡 𝑦 2 𝒜 𝜕 = [𝑎 1 𝜕 , 𝑎 2 𝜕 , ⋯ , 𝑎 6 (𝜕)] • 𝑛 − 1 − 𝑁 − 1 2𝜌𝑒 𝑦 𝒜 𝜕 = 𝒃(𝜕)𝑇(𝜕) 𝑏 𝑛 (𝜕) = exp 𝑘 sin 𝜄 𝑡 2 𝜇 𝜔 = 2𝜌𝑒 𝑦 𝒃(𝜕) = 𝑓 −𝑘 𝑁−1 𝜔 𝑈 1, 𝑓 𝑘𝜔 , 𝑓 𝑘2𝜔 , ⋯ , 𝑓 𝑘 𝑁−1 𝜔 sin 𝜄 𝑡 2 𝜇

• 𝑏 𝑛 (𝜕) = exp(−𝑘𝒍 𝑈 𝒒) = exp 𝑘 2𝜌 𝜇 𝒗 𝑈 𝒒 𝑛 ▪ • ▪ ▪ 𝜌 0 𝜕 −𝜌 𝜕

• y = 𝑔(𝑦) 𝑦 ▪ 𝑧 ▪ 𝑎 2 𝜕 𝒜 𝜕 = 𝒃(𝜕)𝑇(𝜕) 𝑇 𝜕 𝑎 2 (𝜕) = 𝑏 1 (𝜕) 𝑎 1 (𝜕) 𝑏 2 (𝜕) 𝑇(𝜕) 𝑎 1 𝜕 𝑏 1 𝑓 −𝑘𝜕𝜐 1 ⋮ 𝒃(𝜕) = ≡ ⋮ 𝑏 𝑁 𝑓 −𝑘𝜕𝜐 𝑁 𝒜 𝜕 ∼ 𝑂 𝑑 (0, 𝒃 𝜕 𝒃 𝐼 (𝜕)) 𝑇 𝜕 ∼ 𝑂 𝑑 (0,1)

• 𝑂 𝑁 𝑇 1 (𝜕) 𝑎 1 (𝜕) 𝑎 2 (𝜕) 𝑇 2 (𝜕) 𝑇 1 (𝜕) 𝑎 1 (𝜕) 𝑇 2 (𝜕) 𝑎 2 (𝜕) 𝑎 3 (𝜕) 𝒕 𝜕 = 𝒜 𝜕 = ⋮ ⋮ 𝑇 𝑁 (𝜕) 𝑎 𝑂 (𝜕)

• 𝑂 ▪ ▪ 𝑂 𝒜 𝑡 (𝜕) = 𝒃 𝑗 𝜕 𝑇 𝑗 𝜕 = 𝑩(𝜕)𝒕(𝜕) 𝒜 𝑡 𝜕 = 𝒃(𝜕)𝑇(𝜕) 𝑗=1 𝑎 𝑡1 (𝜕) 𝑇 1 (𝜕) 𝑎 𝑡2 (𝜕) 𝑁 × 𝑂 𝑇 2 (𝜕) 𝒜 𝑡 𝜕 = 𝒕(𝜕) = ⋮ 𝑩 𝜕 = [𝒃 1 (𝜕), ⋯ , 𝒃 𝑂 (𝜕)] ⋮ 𝑎 𝑡𝑁 (𝜕) 𝑇 𝑂 (𝜕) 𝒃 𝑜 (𝜕) 𝑜

• 𝒜 = 𝑩𝒕 + 𝒘 ▪ 𝑞 𝒘 = 𝑂(𝒘|𝟏, 𝑳) 𝑞 𝒜; 𝚰 = 𝑂 𝒜 𝑩𝒕, 𝑳 𝚰 = {𝑩, 𝒕, 𝑳} 𝑞(𝒜; 𝚰) 𝒜 = 𝑩𝒕 + 𝒘 𝑩 𝑂 {𝜄 1 , ⋯ , 𝜄 𝑂 } ▪ 𝚫 𝑂 {𝛿 1 , ⋯ , 𝛿 𝑂 } 𝑞 𝒜; 𝚰 = 𝑂 𝒜 𝟏, 𝑩𝚫𝑩 𝐼 + 𝑳 𝑞 𝒘 = 𝑂(𝒘|𝟏, 𝑳) 𝚰 = {𝑩, 𝚫, 𝑳} 𝑞(𝒜; 𝚰) 𝑞 𝒕 = 𝑂(𝒕|𝟏, 𝚫) 𝒜 = 𝑩𝒕 + 𝒘 𝚫 = 𝐹 𝒕𝒕 𝐼 (= diag 𝛿 1 , ⋯ , 𝛿 𝑂 ) 𝑞(𝚰) 𝑞 𝚰|𝒜 = 𝑞 𝒜 𝚰 𝑞 𝚰 𝑞 𝑨

• ▪ ▪ 𝒜 ▪ 𝜀(𝑢) ℎ(𝑢) ℎ(𝑢) ℎ 𝑢 = ℎ 𝑢 ∗ 𝜀 𝑢

• ▪ 𝑤(𝑢) ℎ(𝑢) 𝑡(𝑢) 𝑨(𝑢) 𝑨 𝑢 = ℎ 𝑢 ∗ 𝑡 𝑢 + 𝑤(𝑢)

• 𝑡(𝑢) ℎ(𝑢) 𝑨(𝑢) 𝑇(𝜕) 𝐼(𝜕) 𝑎(𝜕)

• ▪ ▪ = ∗ 𝜀 𝑢 TSP 𝑢 iTSP 𝑢

• ▪ ℎ 𝑢 ∗ TSP(𝑢) ∗ ℎ 𝑢 ∗ TSP 𝑢 ∗ iTSP 𝑢 = ℎ(𝑢) ℎ 𝑢 ∗ TSP 𝑢 iTSP 𝑢

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• ▪ 𝑂 𝑁 𝑁 = 𝑂 𝑂 𝑿 = 𝑩 −1 𝒛 ≈ 𝒕 𝒜 = 𝑩𝒕 = 𝒃 𝑗 𝑡 𝑗 𝒛 = 𝑿𝒜 = 𝑿𝑩𝒕 𝑗=1 𝒜 = 𝑩𝒕 𝒛 = 𝑿𝒜 𝑏 11 𝑥 11 𝑡 1 𝑨 1 𝑧 1 𝑏 21 𝑥 21 𝑏 12 𝑥 12 𝑡 2 𝑨 2 𝑧 2 𝑏 22 𝑥 22

• ▪ • ▪ 𝑧 1 𝑧 2 = 𝑥 11 𝑥 12 𝑨 1 𝑨 2 = 𝒙 1 𝑨 1 + 𝒙 2 𝑨 2 𝑥 21 𝑥 22 𝒜 𝒛 = 𝑿 ICA 𝒜 𝒛 = 𝑿 PCA 𝒜

• ▪ • ▪ 𝒁 𝜕 = 𝑿(𝜕)𝒂(𝜕) 𝒂(𝜕) 𝒁(𝜕) 𝒁(𝜕′) 𝒂(𝜕′) 𝒁 𝜕′ = 𝑿(𝜕′)𝒂(𝜕′)

• ▪ 𝒚 𝑢𝑔 = [𝑦 𝑢𝑔1 , 𝑦 𝑢𝑔2 , ⋯ , 𝑦 𝑢𝑔𝑁 ] 𝑛 = 2 𝑔 𝑛 = 1 𝑢 𝑛 = 1 𝑔 𝑛 = 2 𝑜 = 1 𝑢

• ▪ 𝒚 𝑢𝑔 = [𝑦 𝑢𝑔1 , 𝑦 𝑢𝑔2 , ⋯ , 𝑦 𝑢𝑔𝑁 ] 𝑛 = 2 𝑜 = 2 𝑔 𝑛 = 1 𝑢 𝑛 = 1 𝑔 𝑛 = 2 𝑢

• ▪ 𝒚 𝑢𝑔 = [𝑦 𝑢𝑔1 , 𝑦 𝑢𝑔2 , ⋯ , 𝑦 𝑢𝑔𝑁 ] 𝑛 = 2 𝑜 = 2 𝑔 𝑛 = 1 𝑢 𝑛 = 1 𝑔 𝑛 = 2 𝑜 = 1 𝑢

• ▪ 𝑢 𝑔 𝑙 𝑨 𝑢𝑔 = 𝑙 ▪ 𝑰 𝑔𝑒 𝑔 𝑒 𝑒 1 𝑒 2 −1 𝑒 3 𝒚 𝑢𝑔 ∼ 𝑂 𝑑 𝒚 𝑢𝑔 𝟏, 𝜇 𝑢𝑔 𝑰 𝑔𝑒 𝑨𝑢𝑔 𝑢 𝑔 −1 , 𝜉 0 𝐼 + 𝜗𝑱 𝑰 𝑔𝑒 ∼ 𝑋 𝒃 𝑔𝑒 𝒃 𝑔𝑒 𝑑 𝑔 𝑒 𝑒 𝑗

• ▪ 𝑛 = 2 −1 𝒚 𝑢𝑔 ∼ 𝑂 𝑑 𝒚 𝑢𝑔 𝟏, 𝜇 𝑢𝑔 𝑰 𝑔𝑒 𝑨 𝑢𝑔 𝑢 𝑔 (𝑙 → ∞) 𝜌 𝑢𝑔𝑙 𝑛 = 1 𝝆 𝑢𝑔 ∼ HDP(𝛽, 𝛿, 𝜸) 𝑨 𝑢𝑔 ∼ Categorical(𝝆 𝑢𝑔 ) 𝑙

• • × ⋯ ⋯ × ⋯

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• 𝑙 ▪ 𝑙 𝑢 𝑔 𝜇 𝑢𝑔 × ⋯ ⋯ × 𝑙 𝑀 𝜇 𝑢𝑔 𝑙 = 𝑙,𝑚 ℎ 𝑢 𝑙,𝑚 𝜇 𝑢𝑔 𝑥 𝑔 𝑙,𝑚 𝑙,𝑚 𝑥 ℎ 𝑢 𝑚=1 𝑔

• 𝑙 ▪ 𝑙 𝑢 𝑔 𝜇 𝑢𝑔 × ⋯ ⋯ × 𝑙 𝑙 = 𝑥 (𝑙,𝑚 ′ ) ℎ 𝑢 (𝑙,𝑚 ′ ) 𝜇 𝑢𝑔 𝜇 𝑢𝑔 𝑔 𝑚 ′ 𝑙,𝑚 𝑙,𝑚 𝑥 ℎ 𝑢 𝑔

• ▪ 𝑢 𝑔 (1) ~ 𝑂 𝑑 𝟏, 𝜇 𝑢𝑔 1 𝑯 𝑔 1 𝒚 𝑢𝑔 (𝑙) 𝑯 𝑔 (𝑙) 𝒚 𝑢𝑔 ∼ 𝑂 𝑑 𝟏, 𝜇 𝑢𝑔 (2) ~ 𝑂 𝑑 𝟏, 𝜇 𝑢𝑔 2 𝑯 𝑔 2 𝑙 𝒚 𝑢𝑔

• ▪ 𝑢 𝑔 (1) ~ 𝑂 𝑑 𝟏, 𝜇 𝑢𝑔 1 𝑯 𝑔 1 𝒚 𝑢𝑔 𝑙 𝑂 𝑑 𝟏, 𝜇 𝑢𝑔 𝑙 𝑯 𝑔 𝑙 𝒚 𝑢𝑔 ∼ 𝜌 𝑢𝑔 (2) ~ 𝑂 𝑑 𝟏, 𝜇 𝑢𝑔 2 𝑯 𝑔 2 𝑙 𝒚 𝑢𝑔

• ▪ ▪ ∑ 𝑙,𝑚 ℎ 𝑢 𝑙,𝑚 𝑯 𝑔 𝑙 𝑂 𝑑 𝑙,𝑚 ℎ 𝑢 𝑙,𝑚 𝑯 𝑔 𝑙 𝑙 𝑂 𝑑 𝟏, 𝑥 𝜌 𝑢𝑔 𝟏, 𝑥 𝑔 𝑔 𝑙 𝑚 𝑙 𝑚 𝑙 𝜔 𝑢𝑔 𝑙,𝑚 𝑂 𝑑 0, 𝑥 𝑙,𝑚 ℎ 𝑢 𝑙,𝑚 𝑯 𝑔 𝑙,𝑚 𝑥 𝑔 𝑙,𝑚 ℎ 𝑢 𝑙,𝑚 𝑯 𝑔 𝑙 𝑙 𝜌 𝑢𝑔 𝑂 𝑑 𝟏, 𝜔 𝑢𝑔 𝑔 𝑙 𝑚 𝑙 𝑚

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