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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