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SLIDE 1

❆ ◆❡✉r❛❧ Pr♦❜❛❜✐❧✐st✐❝ ▲❛♥❣✉❛❣❡ ▼♦❞❡❧

P❛♣❡r Pr❡s❡♥t❛t✐♦♥ ✭❨ ❇❡♥❣✐♦✱ ❡t✳ ❛❧✳ ✷✵✵✸✮ ❩❡♠✐♥❣ ▲✐♥

❉❡♣❛rt♠❡♥t ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❛t ❯♥✐✈❡rs✐t② ♦❢ ❱✐r❣✐♥✐❛

▼❛r❝❤ ✶✾ ✷✵✶✺

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SLIDE 2

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

❇❛❝❦❣r♦✉♥❞ ▲❛♥❣✉❛❣❡ ♠♦❞❡❧s ◆❡✉r❛❧ ◆❡t✇♦r❦s ◆❡✉r❛❧ ▲❛♥❣✉❛❣❡ ▼♦❞❡❧ ▼♦❞❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❘❡s✉❧ts

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SLIDE 3

❘❡✈✐❡✇ ♦❢ ▲❛♥❣✉❛❣❡ ▼♦❞❡❧s

◮ Pr❡❞✐❝t P(✇❚ ✶ ) = P(✇✶, ✇✷, ✇✸, . . . , ✇❚)

◮ ❆s ❛ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✿ P(✇ ❚

✶ ) = ❚ ✐=✶ P(✇t|✇ t−✶ ✶

) ❚♦♦ ♠❛♥② ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s✦ ❙✐♠♣❧✐❢② ✉s✐♥❣ ♥✲❣r❛♠ ♠♦❞❡❧✿ P ✇t ✇ ❚

P ✇t ✇ ❚

❚ ♥ ✶

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SLIDE 4

❘❡✈✐❡✇ ♦❢ ▲❛♥❣✉❛❣❡ ▼♦❞❡❧s

◮ Pr❡❞✐❝t P(✇❚ ✶ ) = P(✇✶, ✇✷, ✇✸, . . . , ✇❚)

◮ ❆s ❛ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✿ P(✇ ❚

✶ ) = ❚ ✐=✶ P(✇t|✇ t−✶ ✶

)

◮ ❚♦♦ ♠❛♥② ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s✦ ❙✐♠♣❧✐❢② ✉s✐♥❣ ♥✲❣r❛♠

♠♦❞❡❧✿

◮ P(✇t|✇ ❚

✶ ) ≈ P(✇t|✇ ❚ ❚−♥+✶)

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SLIDE 5

Pr♦❜❧❡♠s

◮ ❲❡ ✇❛♥t t♦ ♦♣t✐♠✐③❡ ♦♥ ❧❛r❣❡ ♥✱ ❜✉t ✈♦❝❛❜✉❧❛r② s✐③❡

❱ > ✶✵✵✵✵✵✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ✭❛s ♦❢ ✷✵✵✸✮ ✇❛s t♦ ✉s❡ ✷ ♦r ✸✲❣r❛♠ ♠♦❞❡❧s✳

✶✵ ❣r❛♠ ♠♦❞❡❧s ❤❛s ✶✵✵✵✵✵✶✵ ✶ ✶✵✺✵ ✶ ♣❛r❛♠❡t❡rs

❉♦❡s ♥♦t t❛❦❡ ✐♥ ❛❝❝♦✉♥t ♦❢ s✐♠✐❧❛r✐t② ❜❡t✇❡❡♥ ✇♦r❞s

❆ ❝❛t ✐s ✇❛❧❦✐♥❣ ✐♥ t❤❡ ❜❡❞r♦♦♠ ❚❤❡ ❞♦❣ ✇❛s r✉♥♥✐♥❣ ✐♥ ❛ r♦♦♠

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SLIDE 6

Pr♦❜❧❡♠s

◮ ❲❡ ✇❛♥t t♦ ♦♣t✐♠✐③❡ ♦♥ ❧❛r❣❡ ♥✱ ❜✉t ✈♦❝❛❜✉❧❛r② s✐③❡

❱ > ✶✵✵✵✵✵✳

◮ ❙t❛t❡ ♦❢ t❤❡ ❛rt ✭❛s ♦❢ ✷✵✵✸✮ ✇❛s t♦ ✉s❡ ✷ ♦r ✸✲❣r❛♠ ♠♦❞❡❧s✳

◮ ✶✵ ❣r❛♠ ♠♦❞❡❧s ❤❛s ✶✵✵✵✵✵✶✵ − ✶ = ✶✵✺✵ − ✶ ♣❛r❛♠❡t❡rs

❉♦❡s ♥♦t t❛❦❡ ✐♥ ❛❝❝♦✉♥t ♦❢ s✐♠✐❧❛r✐t② ❜❡t✇❡❡♥ ✇♦r❞s

❆ ❝❛t ✐s ✇❛❧❦✐♥❣ ✐♥ t❤❡ ❜❡❞r♦♦♠ ❚❤❡ ❞♦❣ ✇❛s r✉♥♥✐♥❣ ✐♥ ❛ r♦♦♠

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SLIDE 7

Pr♦❜❧❡♠s

◮ ❲❡ ✇❛♥t t♦ ♦♣t✐♠✐③❡ ♦♥ ❧❛r❣❡ ♥✱ ❜✉t ✈♦❝❛❜✉❧❛r② s✐③❡

❱ > ✶✵✵✵✵✵✳

◮ ❙t❛t❡ ♦❢ t❤❡ ❛rt ✭❛s ♦❢ ✷✵✵✸✮ ✇❛s t♦ ✉s❡ ✷ ♦r ✸✲❣r❛♠ ♠♦❞❡❧s✳

◮ ✶✵ ❣r❛♠ ♠♦❞❡❧s ❤❛s ✶✵✵✵✵✵✶✵ − ✶ = ✶✵✺✵ − ✶ ♣❛r❛♠❡t❡rs

◮ ❉♦❡s ♥♦t t❛❦❡ ✐♥ ❛❝❝♦✉♥t ♦❢ s✐♠✐❧❛r✐t② ❜❡t✇❡❡♥ ✇♦r❞s

◮ ❆ ❝❛t ✐s ✇❛❧❦✐♥❣ ✐♥ t❤❡ ❜❡❞r♦♦♠ ◮ ❚❤❡ ❞♦❣ ✇❛s r✉♥♥✐♥❣ ✐♥ ❛ r♦♦♠

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SLIDE 8

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

❇❛❝❦❣r♦✉♥❞ ▲❛♥❣✉❛❣❡ ♠♦❞❡❧s ◆❡✉r❛❧ ◆❡t✇♦r❦s ◆❡✉r❛❧ ▲❛♥❣✉❛❣❡ ▼♦❞❡❧ ▼♦❞❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❘❡s✉❧ts

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SLIDE 9

◆❡✉r❛❧ ◆❡t✇♦r❦s

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SLIDE 10

◆❡✉r❛❧ ◆❡t✇♦r❦s

❢ : R✹ → R✺❀ ❊①❛♠♣❧❡✿ ❋✶(①) = ❢ (①) = t❛♥❤(❆①)✱ ❆ ∈ R✺×✹ ❣ : R✺ → R❀ ❊①❛♠♣❧❡✿ ❋✷(①) = ❣(①) = t❛♥❤(❇①)✱ ❇ ∈ R✶×✺ ❆ ❛♥❞ ❇ ❛r❡ ♣❛r❛♠❡t❡rs✦ ❖✉r ♥❡t✇♦r❦ ✐s ❥✉st ❋ ① ❋✷ ❋✶ ①

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SLIDE 11

◆❡✉r❛❧ ◆❡t✇♦r❦s

❢ : R✹ → R✺❀ ❊①❛♠♣❧❡✿ ❋✶(①) = ❢ (①) = t❛♥❤(❆①)✱ ❆ ∈ R✺×✹ ❣ : R✺ → R❀ ❊①❛♠♣❧❡✿ ❋✷(①) = ❣(①) = t❛♥❤(❇①)✱ ❇ ∈ R✶×✺ ❆ ❛♥❞ ❇ ❛r❡ ♣❛r❛♠❡t❡rs✦ ❖✉r ♥❡t✇♦r❦ ✐s ❥✉st ❋(①) = ❋✷(❋✶(①))

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SLIDE 12

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

❇❛❝❦❣r♦✉♥❞ ▲❛♥❣✉❛❣❡ ♠♦❞❡❧s ◆❡✉r❛❧ ◆❡t✇♦r❦s ◆❡✉r❛❧ ▲❛♥❣✉❛❣❡ ▼♦❞❡❧ ▼♦❞❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❘❡s✉❧ts

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SLIDE 13

◆❡✉r❛❧ ◆❡t✇♦r❦

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SLIDE 14

▼♦❞❡❧

ˆ P(✇t = ✐|✇t−✶

) = ❢ (✐, ❣(✇t−✶

t−♥+✶))

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SLIDE 15

▼♦❞❡❧

❣(✇t

✶) = (❈(✇✶), ❈(✇✷), . . . ❈(✇❚))

❈(✇✐) ❣✐✈❡s t❤❡ ✇✐✲t❤ r♦✇ ♦❢ |❱ | × ♠ ♠❛tr✐① ✏▲♦♦❦✉♣ t❛❜❧❡✑

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SLIDE 16

▼♦❞❡❧

②(①) = ❜ + ❲① + ❯ t❛♥❤(❞ + ❍①) s(①) =

❡①

  • ✐(❡①)✐

❢ (✐, ①) = s(②(①))✐ ❍ ✐s ❛ ❤ × (♥ − ✶)♠ ♠❛tr✐① ❞ ✐s ❛ ❤ ❧❡♥❣t❤ ✈❡❝t♦r ❯ ✐s ❛ |❱ | × ❤ ♠❛tr✐① ❲ ✐s ❛ |❱ | × (♥ − ✶)♠ ♠❛tr✐① ❜ ✐s ❛ |❱ | ❧❡♥❣t❤ ✈❡❝t♦r

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SLIDE 17

▼♦❞❡❧

ˆ P(✇t = ✐|✇t−✶

) = s(②(❈(✇t−✶

t−♥+✶)))✐

P❛r❛♠❡t❡rs✿ θ = (❜, ❞, ❲ , ❯, ❍, ❈) ❖(|❱ |(♥♠ + ❤)) ♣❛r❛♠❡t❡rs ▲✐♥❡❛r ✐♥ ♥✲❣r❛♠ s✐③❡ ❛♥❞ ✈♦❝❛❜ s✐③❡✦

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SLIDE 18

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

❇❛❝❦❣r♦✉♥❞ ▲❛♥❣✉❛❣❡ ♠♦❞❡❧s ◆❡✉r❛❧ ◆❡t✇♦r❦s ◆❡✉r❛❧ ▲❛♥❣✉❛❣❡ ▼♦❞❡❧ ▼♦❞❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❘❡s✉❧ts

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SLIDE 19

❚r❛✐♥✐♥❣

θ ← θ + ǫ∂ ˆ P ∂θ

P ❝❛♥ ❜❡ ❢♦✉♥❞ ❛♥❛❧②t✐❝❛❧❧② ✉s✐♥❣ t❤❡ ❝❤❛✐♥ r✉❧❡

✐s t❤❡ ❧❡❛r♥✐♥❣ r❛t❡

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SLIDE 20

❚r❛✐♥✐♥❣

θ ← θ + ǫ∂ ˆ P ∂θ

◮ ∂ ˆ P ∂θ ❝❛♥ ❜❡ ❢♦✉♥❞ ❛♥❛❧②t✐❝❛❧❧② ✉s✐♥❣ t❤❡ ❝❤❛✐♥ r✉❧❡ ◮ ǫ ✐s t❤❡ ❧❡❛r♥✐♥❣ r❛t❡

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SLIDE 21

❉❛t❛ P❛r❛❧❧❡❧✐s♠

◮ ❆ss✉♠❡ s❤❛r❡❞ ♠❡♠♦r② ♣r♦❝❡ss♦r✱ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦sts ❛r❡

❧♦✇

◮ ❙♣❧✐t ❞❛t❛ ✉♣ ❛♥❞ ❤❛✈❡ ❡❛❝❤ ❈P❯ ✉♣❞❛t❡ t❤❡ ♣❛r❛♠❡t❡rs

tr✐❝❦✿ ❛s②♥❝❤r♦♥♦✉s❧② ✉♣❞❛t❡ ♣❛r❛♠❡t❡rs ▼❛② ❧♦s❡ ♣❛r❛♠❡t❡rs✱ ❜✉t ♦❝❝❛s✐♦♥❛❧ ♥♦✐s❡ ❞✐❞ ♥♦t ✐♠♣❛❝t ♣❡r❢♦r♠❛♥❝❡

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SLIDE 22

❉❛t❛ P❛r❛❧❧❡❧✐s♠

◮ ❆ss✉♠❡ s❤❛r❡❞ ♠❡♠♦r② ♣r♦❝❡ss♦r✱ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦sts ❛r❡

❧♦✇

◮ ❙♣❧✐t ❞❛t❛ ✉♣ ❛♥❞ ❤❛✈❡ ❡❛❝❤ ❈P❯ ✉♣❞❛t❡ t❤❡ ♣❛r❛♠❡t❡rs

◮ tr✐❝❦✿ ❛s②♥❝❤r♦♥♦✉s❧② ✉♣❞❛t❡ ♣❛r❛♠❡t❡rs ◮ ▼❛② ❧♦s❡ ♣❛r❛♠❡t❡rs✱ ❜✉t ♦❝❝❛s✐♦♥❛❧ ♥♦✐s❡ ❞✐❞ ♥♦t ✐♠♣❛❝t

♣❡r❢♦r♠❛♥❝❡

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SLIDE 23

P❛r❛♠❡t❡r P❛r❛❧❧❡❧✐s♠

◮ ❆ss✉♠❡ ❝♦♠♣✉t❡r ❝❧✉st❡r✱ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♦✈❡r❤❡❛❞ ✐s ❧❛r❣❡ ◮ ▲✐♠✐t✐♥❣ ❢❛❝t♦r ❞✉r✐♥❣ ❝♦♠♣✉t❛t✐♦♥ ♦❢

❢ (✐, ①) = s(②(①))✐ = s(❜ + ❲① + ❯ t❛♥❤(❞ + ❍①))✐

❊✈❡r② ♣r♦❝❡ss♦r ❝♦♠♣✉t❡s ❣ ① ✳ ✐✲t❤ ♣r♦❝❡ss♦r ❝♦♠♣✉t❡s ✐✲t❤ ❜❧♦❝❦ ♦❢ ② ① ❯♣❞❛t❡ ❝❡♥tr❛❧ s❡r✈❡r ✇✐t❤ s✉♠✱ r❡❝❡✐✈❡ s✉♠ ❛❝r♦ss ② ① ✳

✾✾✳✼✪ ❝❛❧❝✉❧❛t✐♦♥s ♥♦t r❡♣❡❛t❡❞

✶ ✶✺ t♦t❛❧ t✐♠❡ s♣❡♥t ❞✉r✐♥❣ ♥❡t✇♦r❦ ❝♦♠♠✉♥✐❝❛t✐♦♥s

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SLIDE 24

P❛r❛♠❡t❡r P❛r❛❧❧❡❧✐s♠

◮ ❆ss✉♠❡ ❝♦♠♣✉t❡r ❝❧✉st❡r✱ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♦✈❡r❤❡❛❞ ✐s ❧❛r❣❡ ◮ ▲✐♠✐t✐♥❣ ❢❛❝t♦r ❞✉r✐♥❣ ❝♦♠♣✉t❛t✐♦♥ ♦❢

❢ (✐, ①) = s(②(①))✐ = s(❜ + ❲① + ❯ t❛♥❤(❞ + ❍①))✐

◮ ❊✈❡r② ♣r♦❝❡ss♦r ❝♦♠♣✉t❡s ❣(①)✳ ◮ ✐✲t❤ ♣r♦❝❡ss♦r ❝♦♠♣✉t❡s ✐✲t❤ ❜❧♦❝❦ ♦❢ ②(①) ◮ ❯♣❞❛t❡ ❝❡♥tr❛❧ s❡r✈❡r ✇✐t❤ s✉♠✱ r❡❝❡✐✈❡ s✉♠ ❛❝r♦ss ②(①)✳

✾✾✳✼✪ ❝❛❧❝✉❧❛t✐♦♥s ♥♦t r❡♣❡❛t❡❞

✶ ✶✺ t♦t❛❧ t✐♠❡ s♣❡♥t ❞✉r✐♥❣ ♥❡t✇♦r❦ ❝♦♠♠✉♥✐❝❛t✐♦♥s

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SLIDE 25

P❛r❛♠❡t❡r P❛r❛❧❧❡❧✐s♠

◮ ❆ss✉♠❡ ❝♦♠♣✉t❡r ❝❧✉st❡r✱ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♦✈❡r❤❡❛❞ ✐s ❧❛r❣❡ ◮ ▲✐♠✐t✐♥❣ ❢❛❝t♦r ❞✉r✐♥❣ ❝♦♠♣✉t❛t✐♦♥ ♦❢

❢ (✐, ①) = s(②(①))✐ = s(❜ + ❲① + ❯ t❛♥❤(❞ + ❍①))✐

◮ ❊✈❡r② ♣r♦❝❡ss♦r ❝♦♠♣✉t❡s ❣(①)✳ ◮ ✐✲t❤ ♣r♦❝❡ss♦r ❝♦♠♣✉t❡s ✐✲t❤ ❜❧♦❝❦ ♦❢ ②(①) ◮ ❯♣❞❛t❡ ❝❡♥tr❛❧ s❡r✈❡r ✇✐t❤ s✉♠✱ r❡❝❡✐✈❡ s✉♠ ❛❝r♦ss ②(①)✳

◮ ✾✾✳✼✪ ❝❛❧❝✉❧❛t✐♦♥s ♥♦t r❡♣❡❛t❡❞ ◮ ✶ ✶✺ t♦t❛❧ t✐♠❡ s♣❡♥t ❞✉r✐♥❣ ♥❡t✇♦r❦ ❝♦♠♠✉♥✐❝❛t✐♦♥s

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SLIDE 26

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

❇❛❝❦❣r♦✉♥❞ ▲❛♥❣✉❛❣❡ ♠♦❞❡❧s ◆❡✉r❛❧ ◆❡t✇♦r❦s ◆❡✉r❛❧ ▲❛♥❣✉❛❣❡ ▼♦❞❡❧ ▼♦❞❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❘❡s✉❧ts

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SLIDE 27

❇r♦✇♥ ❈♦r♣✉s

◮ ✶✱✶✽✶✱✵✹✶ ❞✐✛❡r❡♥t ✇♦r❞s ❢r♦♠ ❡♥❣❧✐s❤ t❡①ts ❛♥❞ ❜♦♦❦s ◮ ✹✼✺✼✽ ❞✐✛❡r❡♥t ✇♦r❞s✱ ✐♥❝❧✉❞✐♥❣ ❝❛s❡✱ ❛♥❞ ♣✉❝t✉❛t✐♦♥✳ ◮ |❱ | = ✶✻✸✽✸ ❛❢t❡r ✐♥tr♦❞✉❝✐♥❣ ✏r❛r❡ ✇♦r❞s✑ s②♠❜♦❧

◮ ♥♦ ✇♦r❞s ✇✐t❤ ❢r❡q✉❡♥❝② ❧❡ss t❤❛♥ ✹

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SLIDE 28

❇r♦✇♥ ❈♦r♣✉s

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SLIDE 29

❆P ◆❡✇s

◮ ❆ss♦❝✐❛t❡❞ ♣r❡ss ♥❡✇s ❢r♦♠ ✶✾✾✺✲✶✾✾✻✳ ◮ ⑦✶✹ ♠✐❧❧✐♦♥ ✇♦r❞s ◮ ✶✹✽✱✼✷✶ ❞✐✛❡r❡♥t ✇♦r❞s ◮ |❱ | = ✶✼✾✻✹

◮ ◆♦ ✇♦r❞s ✇✐t❤ ❢r❡q✉❡♥❝② < ✹✳ ◮ Pr❡♣r♦❝❡ss❡❞ t♦ ❧♦✇❡r ❝❛s❡ ◮ Pr❡♣r♦❝❡ss❡❞ ♥✉♠❜❡rs✱ r❛r❡ ✇♦r❞s✱ ❛♥❞ ♣r♦♣❡r ♥♦✉♥s t♦ s♣❡❝✐❛❧

s②♠❜♦❧s

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SLIDE 30

❆P ◆❡✇s