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CM 2021/11/26 PyData.Tokyo Meetup #24 @dropout009
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TVISION INSIGHTS Twitter: @dropout009 Speaker Deck: dropout009 Blog: https://dropout009.hatenablog.com/
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A B C D E F CM CM
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• 𝑖 𝑖 = 1, … , 𝑁 • 𝑗 𝑗 = 1, … , 𝐽 • 𝑘! 𝑗 CM 𝒌 = (𝑘" , … , 𝑘! , … 𝑘# ) • 𝑅 • 𝑅$ 𝑖 CM 1 0 • 𝑃𝑟𝑖𝑐𝑒! 𝑗 CM 1 • 𝐵𝑢𝑑𝑔𝑒𝑡 max 𝒌&' 𝔼 𝑅 ∣ 𝒌 = 𝔼 1 𝑁 < $(" ) 𝑅$ ∣ 𝒌 𝑠. 𝑡. < !(" # 𝑃𝑟𝑖𝑐𝑒!×𝑘! ≤ 𝐵𝑢𝑑𝑔𝑒𝑡
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𝔼 𝑅 ∣ 𝒌 = 𝔼 1 𝑁 ( !"# $ 𝑅! ∣ 𝒌 = 1 𝑁 ( !"# $ 𝔼 𝑅! ∣ 𝒌 = 1 𝑁 ( !"# $ Pr(𝑅! = 1 ∣ 𝒌)
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Pr(𝑅! = 1 ∣ 𝒌) = 1 − . %"# & 1 − 𝑝!% '! = 1 − . %"# & 𝑞 !% '! 𝑞!" 𝑞!" = 1 − 𝑝!" 𝑝!" 1
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max 𝒌&' 1 − 1 𝑁 < $(" ) B !(" # 𝑞 $! *! s. t. < !(" # 𝑝𝑟𝑖𝑐𝑒!×𝑘! ≤ 𝐵𝑢𝑑𝑔𝑒𝑡 • 𝑖 𝑖 = 1, … , 𝑁 • 𝑗 𝑗 = 1, … , 𝐽 • 𝑘! 𝑗 CM 𝒌 = (𝑘", … , 𝑘!, … 𝑘#) • 𝑞$! 𝑗 CM 𝑖 • 𝑃𝑟𝑖𝑐𝑒! 𝑗 CM 1 • 𝐵𝑢𝑑𝑔𝑒𝑡
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2 2 CM 2 CM 0 2 1 2 1 A 1 − 2 3 ! = 5 9 1 − 1 3 ! = 7 9 1 − 2 3 1 3 = 7 9 B 1 − 1 3 ! = 7 9 1 − 2 3 ! = 5 9 1 − 1 3 2 3 = 7 9 6 9 6 9 𝟕 𝟗 0 1 A 1 3 2 3 B 2 3 1 3 (𝑝!")
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Python
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Python scipy K
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0 1 A 1 3 2 3 B 2 3 1 3 1 7/9 (𝑝!")
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𝑝$' ∼ Beta 4, 200 𝑝$" ∼ Beta 3, 200 max (7",7#) 𝔼 𝑅 ∣ (𝑘' , 𝑘" ) = 1 − 1 𝑁 < $(" ) 𝑞$' *"𝑞$" *# s. t. 𝑘' + 𝑘" ≤ 100 0 0, 1
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𝔼 𝑅 ∣ 𝒌 = 1 − 1 𝑁 < $(" ) B !(" # 𝑞 $! *! O 𝔼 𝑅 ∣ 𝒌 = 1 − 1 𝑁 < $(" ) B !(" # P 𝑞 $! *! P 𝑞$! = 1 − 1 𝐿! < :(" ;! 𝑅$!: 𝑗 𝐿! CM 𝑖 𝑗 𝑙 CM 1 0
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𝑝!" ∼ Beta 1, 49 𝐹!" ∼ Binomial 𝐿, 𝑝!" > 𝑞!" = 1 − 𝐹!" 𝐿 ? 𝔼 𝑅 ∣ 𝒌 = = 1 − 1 𝑁 E !#$ % F "#$ & > 𝑞 !" '! 𝑖 𝑗 CM
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CM 𝐿 𝐹"# ∼ Binomial 𝐿, 𝑝"# 0 𝑞"# = 1 − 𝐹"# 𝐿
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𝔼 P 𝑞$! = 𝑞$! 2 𝔼 P 𝑞$! = = 𝔼 P 𝑞$! = + Var P 𝑞$! ≥ 𝑞$! = U 𝑞$! = = P 𝑞$! = − V Var P 𝑞$! 𝑞$! = P 𝑞$! = P 𝑞$! G Var > 𝑞!" = > 𝑞!"(1 − > 𝑞!") 𝐿
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1 𝑞!% ' 2 𝔼 P 𝑞$! * ≈ 𝔼 𝑞$! * + 𝑘𝑞$! *>" P 𝑞$! − 𝑞$! + 1 2 𝑘 𝑘 − 1 𝑞$! *>= P 𝑞$! − 𝑞$! = = 𝑞$! * + 𝑘𝑞$! *>" 𝔼 P 𝑞$! − 𝑞$! + 1 2 𝑘 𝑘 − 1 𝑞$! *>=𝔼 P 𝑞$! − 𝑞$! = = 𝑞$! * + 1 2 𝑘 𝑘 − 1 𝑞$! *>=Var P 𝑞$! P 𝑞$! * 𝑞$! * 2 U 𝑞$! * = P 𝑞$! * − 1 2 𝑘 𝑘 − 1 X 𝑞$! *>= V Var P 𝑞$! 𝑞$! * ⾒
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CM 𝐿
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• CM • • ⾒
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CM 𝔼 𝐴 ∣ 𝒌 = 𝔼 1 𝑁 < $(" ) 𝐴$ ∣ 𝒌 = 1 𝑁 < $(" ) 𝔼 𝐴$ ∣ 𝒌 = 1 𝑁 < $(" ) Pr(𝐴$ = 1 ∣ 𝒌) = 1 𝑁 < $(" ) < ?(' ∑*! Pr(𝐴$ = 1 ∣ 𝑓) Pr(𝐹$ = 𝑓 ∣ 𝒌) CM 𝑘 𝑖 𝑓 𝑖 𝑓 CM CM
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Python