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❆ ◆❡✉r❛❧ Pr♦❜❛❜✐❧✐st✐❝ ▲❛♥❣✉❛❣❡ ▼♦❞❡❧ P❛♣❡r Pr❡s❡♥t❛t✐♦♥ ✭❨ ❇❡♥❣✐♦✱ ❡t✳ ❛❧✳ ✷✵✵✸✮ ❩❡♠✐♥❣ ▲✐♥ ❉❡♣❛rt♠❡♥t ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❛t ❯♥✐✈❡rs✐t② ♦❢ ❱✐r❣✐♥✐❛ ▼❛r❝❤ ✶✾ ✷✵✶✺

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

❚♦♦ ♠❛♥② ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s✦ ❙✐♠♣❧✐❢② ✉s✐♥❣ ♥✲❣r❛♠ ♠♦❞❡❧✿ P ✇ t ✇ ❚ P ✇ t ✇ ❚ ✶ ❚ ♥ ✶ ❘❡✈✐❡✇ ♦❢ ▲❛♥❣✉❛❣❡ ▼♦❞❡❧s ◮ Pr❡❞✐❝t P ( ✇ ❚ ✶ ) = P ( ✇ ✶ , ✇ ✷ , ✇ ✸ , . . . , ✇ ❚ ) ✶ ) = � ❚ ✐ = ✶ P ( ✇ t | ✇ t − ✶ ◮ ❆s ❛ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✿ P ( ✇ ❚ ) ✶

❘❡✈✐❡✇ ♦❢ ▲❛♥❣✉❛❣❡ ▼♦❞❡❧s ◮ Pr❡❞✐❝t P ( ✇ ❚ ✶ ) = P ( ✇ ✶ , ✇ ✷ , ✇ ✸ , . . . , ✇ ❚ ) ✶ ) = � ❚ ✐ = ✶ P ( ✇ t | ✇ t − ✶ ◮ ❆s ❛ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✿ P ( ✇ ❚ ) ✶ ◮ ❚♦♦ ♠❛♥② ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s✦ ❙✐♠♣❧✐❢② ✉s✐♥❣ ♥✲❣r❛♠ ♠♦❞❡❧✿ ◮ P ( ✇ t | ✇ ❚ ✶ ) ≈ P ( ✇ t | ✇ ❚ ❚ − ♥ + ✶ )

❙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♦♦♠ Pr♦❜❧❡♠s ◮ ❲❡ ✇❛♥t t♦ ♦♣t✐♠✐③❡ ♦♥ ❧❛r❣❡ ♥ ✱ ❜✉t ✈♦❝❛❜✉❧❛r② s✐③❡ ❱ > ✶✵✵✵✵✵✳

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

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♦♦♠

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

◆❡✉r❛❧ ◆❡t✇♦r❦s

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

◆❡✉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 ))

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

◆❡✉r❛❧ ◆❡t✇♦r❦

▼♦❞❡❧ P ( ✇ t = ✐ | ✇ t − ✶ ˆ ) = ❢ ( ✐ , ❣ ( ✇ t − ✶ t − ♥ + ✶ )) ✶

▼♦❞❡❧ ❣ ( ✇ t ✶ ) = ( ❈ ( ✇ ✶ ) , ❈ ( ✇ ✷ ) , . . . ❈ ( ✇ ❚ )) ❈ ( ✇ ✐ ) ❣✐✈❡s t❤❡ ✇ ✐ ✲t❤ r♦✇ ♦❢ | ❱ | × ♠ ♠❛tr✐① ✏▲♦♦❦✉♣ t❛❜❧❡✑

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

▼♦❞❡❧ P ( ✇ t = ✐ | ✇ t − ✶ ˆ ) = s ( ② ( ❈ ( ✇ t − ✶ t − ♥ + ✶ ))) ✐ ✶ P❛r❛♠❡t❡rs✿ θ = ( ❜ , ❞ , ❲ , ❯ , ❍ , ❈ ) ❖ ( | ❱ | ( ♥♠ + ❤ )) ♣❛r❛♠❡t❡rs ▲✐♥❡❛r ✐♥ ♥ ✲❣r❛♠ s✐③❡ ❛♥❞ ✈♦❝❛❜ s✐③❡✦

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

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

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

tr✐❝❦✿ ❛s②♥❝❤r♦♥♦✉s❧② ✉♣❞❛t❡ ♣❛r❛♠❡t❡rs ▼❛② ❧♦s❡ ♣❛r❛♠❡t❡rs✱ ❜✉t ♦❝❝❛s✐♦♥❛❧ ♥♦✐s❡ ❞✐❞ ♥♦t ✐♠♣❛❝t ♣❡r❢♦r♠❛♥❝❡ ❉❛t❛ P❛r❛❧❧❡❧✐s♠ ◮ ❆ss✉♠❡ s❤❛r❡❞ ♠❡♠♦r② ♣r♦❝❡ss♦r✱ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦sts ❛r❡ ❧♦✇ ◮ ❙♣❧✐t ❞❛t❛ ✉♣ ❛♥❞ ❤❛✈❡ ❡❛❝❤ ❈P❯ ✉♣❞❛t❡ t❤❡ ♣❛r❛♠❡t❡rs

❉❛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♠❛♥❝❡

❊✈❡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 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 )) ✐