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❋❧♦♦❞ ❉❛♠❛❣❡ ❛♥❞ ■♥✢✉❡♥❝✐♥❣ ❋❛❝t♦rs✿ ❆ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦ P❡rs♣❡❝t✐✈❡ ❋❧♦♦❞ ❉❛♠❛❣❡ ❛♥❞ ■♥✢✉❡♥❝✐♥❣ ❋❛❝t♦rs✿ ❆ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦ P❡rs♣❡❝t✐✈❡ P●▼ ✷✵✶✷ ✲ ❆♣♣❧✐❝❛t✐♦♥s ❑r✐st✐♥ ❱♦❣❡❧ ✶ ✱ ❈❛rst❡♥ ❘✐❣❣❡❧s❡♥ ✶ ✱ ❇r✉♥♦ ▼❡r③ ✷ ✱ ❍❡✐❞✐ ❑r❡✐❜✐❝❤ ✷ ✱ ❋r❛♥❦ ❙❝❤❡r❜❛✉♠ ✶ ✶ ❯♥✐✈❡rs✐t② ♦❢ P♦ts❞❛♠✱ ✷ ●❋❩ P♦ts❞❛♠ ✷✵✳✵✾✳✷✵✶✷ ❑✳ ❱♦❣❡❧✱ ❈✳ ❘✐❣❣❡❧s❡♥✱ ❇✳ ▼❡r③✱ ❍✳ ❑r❡✐❜✐❝❤✱ ❋✳ ❙❝❤❡r❜❛✉♠ ❯♥✐✈❡rs✐t② ♦❢ P♦ts❞❛♠✱ ●❋❩ P♦ts❞❛♠ P●▼ ✷✵✶✷ ✲ ❆♣♣❧✐❝❛t✐♦♥s

❋❧♦♦❞ ❉❛♠❛❣❡ ❛♥❞ ■♥✢✉❡♥❝✐♥❣ ❋❛❝t♦rs✿ ❆ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦ P❡rs♣❡❝t✐✈❡ Pr♦❜❧❡♠ ◆❛t✉r❛❧ ❍❛③❛r❞s ❊❛rt❤q✉❛❦❡ ❚s✉♥❛♠✐ ❋❧♦♦❞ ✇❛t❡r ▲❛♥❞s❧✐❞❡ ◮ ❝♦♠♣❧❡① ♣r♦❝❡ss❡s✱ ♥♦t ✇❡❧❧ ✉♥❞❡rst♦♦❞ ◮ ♠❛♥② ✐♥✢✉❡♥❝✐♥❣ ❢❛❝t♦rs ◮ ✉♥❝❡rt❛✐♥t② ✐♥ ✭♣♦t❡♥t✐❛❧❧② ♦❜s❡r✈❛❜❧❡✮ ✈❛r✐❛❜❧❡s ❛♥❞ ✐♥ ♠♦❞❡❧❧✐♥❣ ❢r❛♠❡✇♦r❦s ❑✳ ❱♦❣❡❧✱ ❈✳ ❘✐❣❣❡❧s❡♥✱ ❇✳ ▼❡r③✱ ❍✳ ❑r❡✐❜✐❝❤✱ ❋✳ ❙❝❤❡r❜❛✉♠ ❯♥✐✈❡rs✐t② ♦❢ P♦ts❞❛♠✱ ●❋❩ P♦ts❞❛♠ P●▼ ✷✵✶✷ ✲ ❆♣♣❧✐❝❛t✐♦♥s

❋❧♦♦❞ ❉❛♠❛❣❡ ❛♥❞ ■♥✢✉❡♥❝✐♥❣ ❋❛❝t♦rs✿ ❆ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦ P❡rs♣❡❝t✐✈❡ Pr♦❜❧❡♠ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦s ❞❡s❝r✐❜❡ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ s②st❡♠s ❛♥❞ ♣r♦❝❡ss❡s✱ ❝❛♣t✉r✐♥❣ ✭✐♥✲✮❞❡♣❡♥❞❡♥❝✐❡s ❜❡t✇❡❡♥ t❤❡ ✈❛r✐❛❜❧❡s ♦❢ ✐♥t❡r❡st ◮ ❛❧❧♦✇ ✐♥❝❧✉s✐♦♥ ♦❢ ❞♦♠❛✐♥ ❦♥♦✇❧❡❞❣❡ ◮ ❝❛♥ ❜❡ ✉♣❞❛t❡❞ ❢♦r ♥❡✇ ♦❜s❡r✈❛t✐♦♥s ◮ ❛❧❧♦✇ r❡❛s♦♥✐♥❣ ✉♥❞❡r ❛♥❞ ♣r♦♣❛❣❛t✐♦♥ ♦❢ ✉♥❝❡rt❛✐♥t② ❑✳ ❱♦❣❡❧✱ ❈✳ ❘✐❣❣❡❧s❡♥✱ ❇✳ ▼❡r③✱ ❍✳ ❑r❡✐❜✐❝❤✱ ❋✳ ❙❝❤❡r❜❛✉♠ ❯♥✐✈❡rs✐t② ♦❢ P♦ts❞❛♠✱ ●❋❩ P♦ts❞❛♠ P●▼ ✷✵✶✷ ✲ ❆♣♣❧✐❝❛t✐♦♥s

❋❧♦♦❞ ❉❛♠❛❣❡ ❛♥❞ ■♥✢✉❡♥❝✐♥❣ ❋❛❝t♦rs✿ ❆ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦ P❡rs♣❡❝t✐✈❡ Pr♦❜❧❡♠ ♦❢t❡♥ ❢❛❝❡❞ ♣r♦❜❧❡♠s ❞✐s❝r❡t✐③❛t✐♦♥✿ ♠❛♥② ✈❛r✐❛❜❧❡s ❛r❡ ❝♦♥t✐♥✉♦✉s ✭❢✉♥❝t✐♦♥❛❧ ❢♦r♠ s♦♠❡t✐♠❡s ✉♥❦♥♦✇♥✮ ♦r ❞✐s❝r❡t❡ ✇✐t❤ ❤✐❣❤ ♥✉♠❜❡r ♦❢ st❛t❡s❀ ❢♦r ❞✐str✐❜✉t✐♦♥ ❢r❡❡ ❧❡❛r♥✐♥❣ t❤❡ ✈❛r✐❛❜❧❡s ❛r❡ ❞✐s❝r❡t✐③❡❞ ❤✐❣❤ r❡s♦❧✉t✐♦♥ ♦❢ t❛r❣❡t ✈❛r✐❛❜❧❡✿ ♠❛✐♥ ✐♥t❡r❡st ✐s ♦❢t❡♥ ✐♥ t❤❡ ♣r❡❞✐❝t✐♦♥ ♦❢ t❛r❣❡t ✈❛r✐❛❜❧❡❀ t❤❡ ❞✐s❝r❡t✐③❛t✐♦♥ ✉s❡❞ ❢♦r ♥❡t✇♦r❦ ❧❡❛r♥✐♥❣ ✐s q✉✐t❡ ❝♦❛rs❡ ❑✳ ❱♦❣❡❧✱ ❈✳ ❘✐❣❣❡❧s❡♥✱ ❇✳ ▼❡r③✱ ❍✳ ❑r❡✐❜✐❝❤✱ ❋✳ ❙❝❤❡r❜❛✉♠ ❯♥✐✈❡rs✐t② ♦❢ P♦ts❞❛♠✱ ●❋❩ P♦ts❞❛♠ P●▼ ✷✵✶✷ ✲ ❆♣♣❧✐❝❛t✐♦♥s

❋❧♦♦❞ ❉❛♠❛❣❡ ❛♥❞ ■♥✢✉❡♥❝✐♥❣ ❋❛❝t♦rs✿ ❆ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦ P❡rs♣❡❝t✐✈❡ Pr♦❜❧❡♠ ❋❧♦♦❞ ❞❛♠❛❣❡ ❛ss❡ss♠❡♥t ✲ ❞❛t❛s❡t ◮ ✷✵✵✷ ❛♥❞ ✷✵✵✺✴✷✵✵✻ ✢♦♦❞s ✐♥ ❊❧❜❡ ❛♥❞ ❉❛♥✉❜❡ ❝❛t❝❤♠❡♥ts ✭●❡r♠❛♥②✮ ◮ ✷✽ ✈❛r✐❛❜❧❡s ❞❡s❝r✐❜✐♥❣ ✢♦♦❞ ❡✈❡♥t✱ ❛✛❡❝t❡❞ ❜✉✐❧❞✐♥❣s✱ ♣r❡❝❛✉t✐♦♥✱ ✇❛r♥✐♥❣✱ s♦❝✐♦✲❡❝♦♥♦♠✐❝ ❢❛❝t♦rs ◮ t❛r❣❡t ✈❛r✐❛❜❧❡✿ r❡❧❛t✐✈❡ ❜✉✐❧❞✐♥❣ ❧♦ss ◮ ✶✶✸✺ ♣❛rt❧② ♠✐ss✐♥❣ ♦❜s❡r✈❛t✐♦♥s ❑✳ ❱♦❣❡❧✱ ❈✳ ❘✐❣❣❡❧s❡♥✱ ❇✳ ▼❡r③✱ ❍✳ ❑r❡✐❜✐❝❤✱ ❋✳ ❙❝❤❡r❜❛✉♠ ❯♥✐✈❡rs✐t② ♦❢ P♦ts❞❛♠✱ ●❋❩ P♦ts❞❛♠ P●▼ ✷✵✶✷ ✲ ❆♣♣❧✐❝❛t✐♦♥s

❋❧♦♦❞ ❉❛♠❛❣❡ ❛♥❞ ■♥✢✉❡♥❝✐♥❣ ❋❛❝t♦rs✿ ❆ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦ P❡rs♣❡❝t✐✈❡ ▼❡t❤♦❞s ❙❝♦r✐♥❣ ♠❡tr✐❝ ❇◆ ▼❆P s❝♦r❡ ❢♦r ❇◆ ❧❡❛r♥✐♥❣✿ s❡❛r❝❤✐♥❣ ❢♦r t❤❡ ♣❛✐r ♦❢ ♥❡t✇♦r❦ str✉❝t✉r❡ ✭❉❆●✮ ❛♥❞ ♣❛r❛♠❡t❡r ✭ Θ ✮✱ t❤❛t ♠❛①✐♠✐③❡ t❤❡ ❥♦✐♥t ♣♦st❡r✐♦r P ( ❉❆● , Θ | ❞ ) ∝ P ( ❞ | ❉❆● , Θ) P ( ❉❆● , Θ) ∝ P ( ❞ | ❉❆● , Θ) P (Θ | ❉❆● ) P ( ❉❆● ) P ( ❞ | ❉❆● , Θ) ♠✉❧t✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ ❢♦r ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s P (Θ | ❉❆● ) ❞❡✜♥❡❞ t♦ ❜❡ ❛ ♣r♦❞✉❝t ❉✐r✐❝❤❧❡t ❞✐str✐❜✉t✐♦♥ P ( ❉❆● ) ❞❡✜♥❡❞ t♦ ❜❡ ✉♥✐❢♦r♠ ♦✈❡r ❉❆●s → ✐❣♥♦r❡❞ ❑✳ ❱♦❣❡❧✱ ❈✳ ❘✐❣❣❡❧s❡♥✱ ❇✳ ▼❡r③✱ ❍✳ ❑r❡✐❜✐❝❤✱ ❋✳ ❙❝❤❡r❜❛✉♠ ❯♥✐✈❡rs✐t② ♦❢ P♦ts❞❛♠✱ ●❋❩ P♦ts❞❛♠ P●▼ ✷✵✶✷ ✲ ❆♣♣❧✐❝❛t✐♦♥s

❋❧♦♦❞ ❉❛♠❛❣❡ ❛♥❞ ■♥✢✉❡♥❝✐♥❣ ❋❛❝t♦rs✿ ❆ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦ P❡rs♣❡❝t✐✈❡ ▼❡t❤♦❞s ❙❝♦r✐♥❣ ♠❡tr✐❝ ❇◆ ▼❆P s❝♦r❡ ❢♦r ❇◆ ❧❡❛r♥✐♥❣✿ s❡❛r❝❤✐♥❣ ❢♦r t❤❡ ♣❛✐r ♦❢ ♥❡t✇♦r❦ str✉❝t✉r❡ ✭❉❆●✮ ❛♥❞ ♣❛r❛♠❡t❡r ✭ Θ ✮✱ t❤❛t ♠❛①✐♠✐③❡ t❤❡ ❥♦✐♥t ♣♦st❡r✐♦r P ( ❉❆● , Θ | ❞ ) ∝ P ( ❞ | ❉❆● , Θ) P ( ❉❆● , Θ) ❇◆ ▼❆P s❝♦r❡ ❢♦r ❇◆ ❧❡❛r♥✐♥❣ ❛♥❞ ✈❛r✐❛❜❧❡ ❞✐s❝r❡t✐③❛t✐♦♥ ✿ s❡❛r❝❤✐♥❣ ❢♦r t❤❡ tr✐♣❧❡ ♦❢ ♥❡t✇♦r❦ str✉❝t✉r❡ ✭❉❆●✮ ❛♥❞ ♣❛r❛♠❡t❡r ✭ Θ ✮ ❛♥❞ ❞✐s❝r❡t✐③❛t✐♦♥ ✭ Λ ✮✱ t❤❛t ♠❛①✐♠✐③❡ t❤❡ ❥♦✐♥t ♣♦st❡r✐♦r P ( ❉❆● , Θ , Λ | ❞ ❝ ) ∝ P ( ❞ ❝ | ❉❆● , Θ , Λ) P ( ❉❆● , Θ , Λ) ❑✳ ❱♦❣❡❧✱ ❈✳ ❘✐❣❣❡❧s❡♥✱ ❇✳ ▼❡r③✱ ❍✳ ❑r❡✐❜✐❝❤✱ ❋✳ ❙❝❤❡r❜❛✉♠ ❯♥✐✈❡rs✐t② ♦❢ P♦ts❞❛♠✱ ●❋❩ P♦ts❞❛♠ P●▼ ✷✵✶✷ ✲ ❆♣♣❧✐❝❛t✐♦♥s

❋❧♦♦❞ ❉❛♠❛❣❡ ❛♥❞ ■♥✢✉❡♥❝✐♥❣ ❋❛❝t♦rs✿ ❆ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦ P❡rs♣❡❝t✐✈❡ ▼❡t❤♦❞s P ( ❉❆● , Θ , Λ | ❞ ❝ ) P ( ❞ ❝ | ❉❆● , Θ , Λ) P ( ❉❆● , Θ , Λ) ∝ P ( ❞ | ❉❆● , Θ , Λ) P ( ❞ ❝ | ❞ , Λ) × = P (Θ | ❉❆● , Λ) P (Λ | ❉❆● ) P ( ❉❆● ) ❑✳ ❱♦❣❡❧✱ ❈✳ ❘✐❣❣❡❧s❡♥✱ ❇✳ ▼❡r③✱ ❍✳ ❑r❡✐❜✐❝❤✱ ❋✳ ❙❝❤❡r❜❛✉♠ ❯♥✐✈❡rs✐t② ♦❢ P♦ts❞❛♠✱ ●❋❩ P♦ts❞❛♠ P●▼ ✷✵✶✷ ✲ ❆♣♣❧✐❝❛t✐♦♥s

❋❧♦♦❞ ❉❛♠❛❣❡ ❛♥❞ ■♥✢✉❡♥❝✐♥❣ ❋❛❝t♦rs✿ ❆ ❇❛②❡s✐❛♥ ◆❡t✇♦r❦ P❡rs♣❡❝t✐✈❡ ▼❡t❤♦❞s P ( ❉❆● , Θ , Λ | ❞ ❝ ) P ( ❞ ❝ | ❉❆● , Θ , Λ) P ( ❉❆● , Θ , Λ) ∝ P ( ❞ | ❉❆● , Θ , Λ) P ( ❞ ❝ | ❞ , Λ) × = P (Θ | ❉❆● , Λ) P (Λ | ❉❆● ) P ( ❉❆● ) ▼♦♥t✐✱ ❈♦♦♣❡r❀ ✶✾✾✽ ❑✳ ❱♦❣❡❧✱ ❈✳ ❘✐❣❣❡❧s❡♥✱ ❇✳ ▼❡r③✱ ❍✳ ❑r❡✐❜✐❝❤✱ ❋✳ ❙❝❤❡r❜❛✉♠ ❯♥✐✈❡rs✐t② ♦❢ P♦ts❞❛♠✱ ●❋❩ P♦ts❞❛♠ P●▼ ✷✵✶✷ ✲ ❆♣♣❧✐❝❛t✐♦♥s