https://www.youtube.com/channel/UCEr2VZ6DI4gATRtVJw49ZFw

1. ήʔϜཧ࿦#4*$ԋश
ऑ
‫׬‬શϕΠδΞϯ‫ߧۉ‬

2. ‫׬‬શϕΠδΞϯ‫ߧۉ‬
໰୊
ղ౴

3. ‫׬‬શϕΠδΞϯ‫ߧۉ‬
‫׬‬શϕΠδΞϯ‫ߧۉ‬ઓུͷ૊ ͱ৴೦ͷମ‫ܥ‬ ͷ૊ Ͱ
͕ ͷ΋ͱͰஞ࣍߹ཧతͰ͋Γ
͕ ʹ੔߹తͰ͋Δ΋ͷΛ ऑ
‫׬‬શϕΠδΞϯ‫͏͍ͱߧۉ‬
ߦಈઓུͷஞ࣍߹ཧੑల։‫ܗ‬ήʔϜ ʹ͓͍ͯ
৴೦ͷମ‫ܥ‬ ͕༩͑ΒΕ͍ͯΔͱ͢Δ
͜ͷͱ͖
ઓུͷ૊ ͕
ϓϨΠϠʔ ͷ৘ใू߹ ʹ͓͍ͯҎԼΛຬͨ͢ͱ͖
͸৴೦ͷମ‫ܥ‬ ͷ΋ͱͰ৘ใू߹ ʹ͓͍ͯஞ࣍߹ཧతͰ͋Δͱ͍͏
৴೦ͷମ‫ͱܥ‬੔߹ੑల։‫ܗ‬ήʔϜ ʹ͓͍ͯ
ઓུͷ૊ ͕༩͑ΒΕ͍ͯΔͱ͢Δ
͜ͷͱ͖
৴೦ͷମ‫ܥ‬ ͕
೚ҙͷϓϨΠϠʔ ͷ೚ҙͷ৘ใू߹ ʹ͓͍ͯ
Ͱ͋Ε͹
͢΂ͯͷ ͷ఺ ʹ͍ͭͯ
ͱͳΔͱ͖
͸ઓུͷ૊ ʹ͓͍ͯ੔߹తͰ͋Δͱ͍͏
b μ (b, μ)
b μ μ b
Γ μ
b = (b1
, ⋯, bn
) i ui
l
b μ ui
l
Hi
(ui
l, μ, (bi,ui
l, b−i,ui
l)) ≥ Hi
(ui
l, μ, (b′

i,ui
l, b−i,ui
l)), ∀b′

i,ui
l ∈ Bi,ui
l
Γ b = (b1
, ⋯, bn
)
μ i ui
l
Prob(ui
l |b) 0 ui
l x*
μ(x*) =
Prob(x*|b)
∑x∈ui
l
Prob(x|b)
μ b
ߦಈઓུʹΑͬͯ౸ୡՄೳͳ
‫ܦ‬࿏ʹ͓͚Δ߹ཧੑͷΈߟ͑Δ

4. ҎԼͷήʔϜʹ͓͚Δ
ߦಈઓུͰͷ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
໰୊
1
2
2
a
b
d
e
ϓϨΠϠʔͷརಘ ϓϨΠϠʔͷརಘ
৘ใू߹
(2, 1)
(1, − 1)
(3, − 2)
(−1, − 1)
(0, 2)
c
d
e

5. ҎԼͷήʔϜʹ͓͚Δ
ߦಈઓུͰͷ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ղ౴
1
2
2
a
b
d
e
ϓϨΠϠʔͷརಘ ϓϨΠϠʔͷརಘ
৘ใू߹
(2, 1)
(1, − 1)
(3, − 2)
(−1, − 1)
(0, 2)
c
d
e
s
1 − s
p
q
1 − p − q
r
1 − r
ϓϨΠϠʔͷߦಈઓུΛ Ͱද͠
ϓϨΠϠʔͷߦಈઓུΛ Ͱද͢
·ͨ
৴೦ͷମ‫ܥ‬
ͳ͓
Λຬͨ͢
(p, q, 1 − p − q)
(r, 1 − r)
μ = (1, (s, 1 − s))
p, q ≥ 0, 0 ≤ p + q ≤ 1 0 ≤ r ≤ 1 0 ≤ s ≤ 1
1

6. ҎԼͷήʔϜʹ͓͚Δ
ߦಈઓུͰͷ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ղ౴
1
2
2
a
b
d
e
ϓϨΠϠʔͷརಘ ϓϨΠϠʔͷརಘ
৘ใू߹
(2, 1)
(1, − 1)
(3, − 2)
(−1, − 1)
(0, 2)
c
d
e
s
1 − s
p
q
1 − p − q
r
1 − r
ϓϨΠϠʔͷஞ࣍߹ཧతͳߦಈΛߟ͑Δ
৴೦ ͷ΋ͱͰͷ‫ظ‬଴རಘ͸
ͷͱ͖
ͭ·ΓEΛબͿͷ͕࠷ద
ͷͱ͖
͸೚ҙͷ஋Ͱ࠷ద
ͷͱ͖
ͭ·ΓFΛબͿͷ͕࠷ద
(s, 1 − s)
1 ⋅ s ⋅ r + (−1) ⋅ s ⋅ (1 − r) + (−2) ⋅ (1 − s) ⋅ r + (−1) ⋅ (1 − s) ⋅ (1 − r)
= (3s − 1)r − 1
3s − 1 0 ⇔ s
1
3
r = 1
3s − 1 = 0 ⇔ s =
1
3
r
3s − 1 0 ⇔ s
1
3
r = 0
1

7. ҎԼͷήʔϜʹ͓͚Δ
ߦಈઓུͰͷ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ղ౴
1
2
2
a
b
d
e
ϓϨΠϠʔͷརಘ ϓϨΠϠʔͷརಘ
৘ใू߹
(2, 1)
(1, − 1)
(3, − 2)
(−1, − 1)
(0, 2)
c
d
e
s
1 − s
p
q
1 − p − q
r
1 − r
ϓϨΠϠʔͷஞ࣍߹ཧతͳߦಈΛߟ͑Δ
ͷͱ͖
ͭ·ΓEΛબͿͷ͕࠷ద
ϓϨΠϠʔ͕ʮEΛબͿʯ৔߹
ϓϨΠϠʔ͸ʮCΛબͿʯͷ͕ஞ࣍߹ཧత
͔͠͠
͜ͷͱ͖੔߹తͳ৴೦͸ ͱͳΓ
ͱໃ६͢Δ
3s − 1 0 ⇔ s
1
3
r = 1
(s, 1 − s) = (0, 1)
s
1
3
1

8. ҎԼͷήʔϜʹ͓͚Δ
ߦಈઓུͰͷ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ղ౴
1
2
2
a
b
d
e
ϓϨΠϠʔͷརಘ ϓϨΠϠʔͷརಘ
৘ใू߹
(2, 1)
(1, − 1)
(3, − 2)
(−1, − 1)
(0, 2)
c
d
e
s
1 − s
p
q
1 − p − q
r
1 − r
ϓϨΠϠʔͷஞ࣍߹ཧతͳߦಈΛߟ͑Δ
ͷͱ͖
ͭ·ΓFΛબͿͷ͕࠷ద
ϓϨΠϠʔ͕ʮFΛબͿʯ৔߹
ϓϨΠϠʔ͸ʮBΛબͿʯͷ͕ஞ࣍߹ཧత
͔͠͠
͜ͷͱ͖੔߹తͳ৴೦͸ ͱͳΓ
ͱໃ६͢Δ
3s − 1 0 ⇔ s
1
3
r = 0
(s, 1 − s) = (1, 0)
s
1
3
1

9. ҎԼͷήʔϜʹ͓͚Δ
ߦಈઓུͰͷ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ղ౴
1
2
2
a
b
d
e
ϓϨΠϠʔͷརಘ ϓϨΠϠʔͷརಘ
৘ใू߹
(2, 1)
(1, − 1)
(3, − 2)
(−1, − 1)
(0, 2)
c
d
e
s
1 − s
p
q
1 − p − q
r
1 − r
ϓϨΠϠʔͷஞ࣍߹ཧతͳߦಈΛߟ͑Δ
ͷͱ͖
͸೚ҙͷ஋Ͱ࠷ద
͜ͷ৔߹ͷΈ͕
΋͠ϓϨΠϠʔͷ৘ใू߹ʹ౸ୡ͢ΔͷͰ͋Ε͹
‫׬‬શϕΠδΞϯ‫ߧۉ‬ͷ৚݅ͱͳΔ
3s − 1 = 0 ⇔ s =
1
3
r
1

10. ҎԼͷήʔϜʹ͓͚Δ
ߦಈઓུͰͷ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ղ౴
1
2
2
a
b
d
e
ϓϨΠϠʔͷརಘ ϓϨΠϠʔͷརಘ
৘ใू߹
(2, 1)
(1, − 1)
(3, − 2)
(−1, − 1)
(0, 2)
c
d
e
1
0
p
q
1 − p − q
r
1 − r
ϓϨΠϠʔͷஞ࣍߹ཧతͳߦಈΛߟ͑Δ
ͷͱ͖
͸೚ҙͷ஋Ͱ࠷ద
͜ͷ৔߹ͷΈ͕
΋͠ϓϨΠϠʔͷ৘ใू߹ʹ౸ୡ͢ΔͷͰ͋Ε͹
‫׬‬શϕΠδΞϯ‫ߧۉ‬ͷ৚݅ͱͳΔ
·ͨ
‫
͍͓ͯʹߧۉ‬ ͷ৔߹͔
ͷ৔߹͔͠ͳ͍
ͷ৔߹
੔߹తͳ৴೦͸
͜ͷͱ͖
ϓϨΠϠʔ͸ʮEΛબͿʯ
ϓϨΠϠʔ͕ʮEΛબͿʯͷͰ͋Ε͹
ϓϨΠϠʔ͸ʮCΛબͿʯํ͕རಘΛߴ͘Ͱ͖ΔͷͰ
ϓϨΠϠʔͷஞ࣍߹ཧੑΛຬͨ͞ͳ͍
ͷ৔߹΋ಉ༷
3s − 1 = 0 ⇔ s =
1
3
r
p, q 0
p = q = 0
p 0 q = 0 s = 1
p = 0 q 0
1

11. ҎԼͷήʔϜʹ͓͚Δ
ߦಈઓུͰͷ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ղ౴
1
2
2
a
b
d
e
ϓϨΠϠʔͷརಘ ϓϨΠϠʔͷརಘ
৘ใू߹
(2, 1)
(1, − 1)
(3, − 2)
(−1, − 1)
(0, 2)
c
d
e
s
1 − s
p
q
1 − p − q
r
1 − r
ϓϨΠϠʔͷஞ࣍߹ཧతͳߦಈΛߟ͑Δ
ͷͱ͖
͸೚ҙͷ஋Ͱ࠷ద
͜ͷ৔߹ͷΈ͕
΋͠ϓϨΠϠʔͷ৘ใू߹ʹ౸ୡ͢ΔͷͰ͋Ε͹
‫׬‬શϕΠδΞϯ‫ߧۉ‬ͷ৚݅ͱͳΔ
·ͨ
‫
͍͓ͯʹߧۉ‬ Ͱ͋Ε͹
ϓϨΠϠʔͷ
ʮBΛબͿʯ৔߹ͱʮCΛબͿʯ৔߹ͷ‫ظ‬଴རಘ͸Ұக͢Δ
ʮBΛબͿʯ৔߹͕ߴ͍‫ظ‬଴རಘΛ༩͑ΔͷͰ͋Ε͹
ʮCΛબͿʯʹׂΓ౰͍ͯͯͨ֬཰Λ
ʮBΛબͿʯʹׂΓ౰ͯΕ͹‫ظ‬଴རಘΛߴ͘͢Δ͜ͱ͕Ͱ͖Δ
ͳͷͰ
ҎԼࣜΛຬͨ͢
3s − 1 = 0 ⇔ s =
1
3
r
p, q 0
2r + (1 − r) = 3r + (−1)(1 − r) ⇔ r =
2
3
1

12. ҎԼͷήʔϜʹ͓͚Δ
ߦಈઓུͰͷ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ղ౴
1
2
2
a
b
d
e
ϓϨΠϠʔͷརಘ ϓϨΠϠʔͷརಘ
৘ใू߹
(2, 1)
(1, − 1)
(3, − 2)
(−1, − 1)
(0, 2)
c
d
e
s
1 − s
p
q
1 − p − q
r =
2
3
1
3
ϓϨΠϠʔͷஞ࣍߹ཧతͳߦಈΛߟ͑Δ
ͷͱ͖
͸೚ҙͷ஋Ͱ࠷ద
ͷ৔߹
͜ͷͱ͖
ϓϨΠϠʔͷ‫ظ‬଴རಘ͸
͜ΕΑΓ
ϓϨΠϠʔ͸
ʮBΛબͿʯʮCΛબͿʯʹ֬཰Λৼͬͨ΄͏͕
ʮDΛબͿʯʹ֬཰ΛৼΔΑΓྑ͍
3s − 1 = 0 ⇔ s =
1
3
r
p, q 0 r =
2
3
2
2
3
+
(
1 −
2
3 )
= 3
2
3
+ (−1)
(
1 −
2
3 )
=
5
3
0
1

13. ҎԼͷήʔϜʹ͓͚Δ
ߦಈઓུͰͷ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ղ౴
1
2
2
a
b
d
e
ϓϨΠϠʔͷརಘ ϓϨΠϠʔͷརಘ
৘ใू߹
(2, 1)
(1, − 1)
(3, − 2)
(−1, − 1)
(0, 2)
c
d
e
s
1 − s
p
q
0
r =
2
3
1
3
ϓϨΠϠʔͷஞ࣍߹ཧతͳߦಈΛߟ͑Δ
ͷͱ͖
͸೚ҙͷ஋Ͱ࠷ద
ͷ৔߹
͜ͷͱ͖
ϓϨΠϠʔͷ‫ظ‬଴རಘ͸
͜ΕΑΓ
ϓϨΠϠʔ͸
ʮBΛબͿʯʮCΛબͿʯʹ֬཰Λৼͬͨ΄͏͕
ʮDΛબͿʯʹ֬཰ΛৼΔΑΓྑ͍
Ώ͑ʹ
3s − 1 = 0 ⇔ s =
1
3
r
p, q 0 r =
2
3
2
2
3
+
(
1 −
2
3 )
= 3
2
3
+ (−1)
(
1 −
2
3 )
=
5
3
0
p + q = 1
1

14. ҎԼͷήʔϜʹ͓͚Δ
ߦಈઓུͰͷ‫׬‬શϕΠδΞϯ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ղ౴
1
2
2
a
b
d
e
ϓϨΠϠʔͷརಘ ϓϨΠϠʔͷརಘ
৘ใू߹
(2, 1)
(1, − 1)
(3, − 2)
(−1, − 1)
(0, 2)
c
d
e
1
3
p =
1
3
0
r =
2
3
1
3
ϓϨΠϠʔͷஞ࣍߹ཧతͳߦಈΛߟ͑Δ
ͷͱ͖
͸೚ҙͷ஋Ͱ࠷ద ΋࠷ద
ͷ৚݅ͱ৴೦ͷ੔߹ੑΑΓ
Ҏ্ΑΓ
‫׬‬શϕΠδΞϯ‫ߧۉ‬͸
ϓϨΠϠʔͷߦಈઓུ
ϓϨΠϠʔͷߦಈઓུ
৴೦ͷମ‫ܥ‬
3s − 1 = 0 ⇔ s =
1
3
r r =
2
3
p + q = 1
p
p + q
=
1
3
⇔ p =
1
3
, q =
2
3
(p, q, 1 − p − q) =
(
1
3
,
2
3
, 0
)
(r, 1 − r) =
(
2
3
,
1
3 )
μ = (1, (s, 1 − s)) =
(
1,
(
1
3
,
2
3))
1
2
3
q =
2
3

15. ήʔϜཧ࿦#4*$ԋश
ऑ
‫׬‬શϕΠδΞϯ‫ߧۉ‬

16. ϓϨΠϠʔ ͷ৘ใू߹্ʹ͓͚Δ৴೦ͷΈ
i