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❍❛♣♣② ❆❇❈✿ ❊①♣❡❝t❛t✐♦♥✲Pr♦♣❛❣❛t✐♦♥ ❢♦r ❙✉♠♠❛r②✲▲❡ss✱ ▲✐❦❡❧✐❤♦♦❞✲❋r❡❡ ■♥❢❡r❡♥❝❡ ◆✐❝♦❧❛s ❈❤♦♣✐♥ ❈❘❊❙❚ ✭❊◆❙❆❊✮ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❙✐♠♦♥ ❇❛rt❤❡❧♠é ✭❚❯ ❇❡r❧✐♥✮ ✶ ✴ ✷✷

❇❛s✐❝ ❆❇❈ ❉❛t❛✿ ② ⋆ ✱ ♣r✐♦r ♣ ( θ ) ✱ ♠♦❞❡❧ ♣ ( ② | θ ) ✳ ▲✐❦❡❧✐❤♦♦❞ ♣ ( ② | θ ) ✐s ✐♥tr❛❝t❛❜❧❡✳ ✶ ❙❛♠♣❧❡ θ ∼ ♣ ( θ ) ✷ ❙❛♠♣❧❡ ② ∼ ♣ ( ② | θ ) ✸ ❆❝❝❡♣t θ ✐✛ � s ( ② ) − s ( ② ⋆ ) � ≤ ǫ ✷ ✴ ✷✷

❆❇❈ t❛r❣❡t ❚❤❡ ♣r❡✈✐♦✉s ❛❧❣♦r✐t❤♠ t❛r❣❡ts✿ ˆ ♣ ǫ ( θ | ② ⋆ ) ∝ ♣ ( θ ) ♣ ( ② | θ ) 1 {� s ( ② ) − s ( ② ⋆ ) �≤ ǫ } ❞ ② ✇❤✐❝❤ ❛♣♣r♦①✐♠❛t❡s t❤❡ tr✉❡ ♣♦st❡r✐♦r ♣ ( θ | ② ) ✳ ❚✇♦ ❧❡✈❡❧s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥✿ ✶ ◆♦♥✲♣❛r❛♠❡tr✐❝ ❡rr♦r✱ ❣♦✈❡r♥❡❞ ❜② ✏❜❛♥❞✇✐❞t❤✑ ǫ ❀ ♣ ǫ ( θ | ② ⋆ ) → ♣ ( θ | s ( ② ⋆ )) ❛s ǫ → ✵✳ ✷ ❇✐❛s ✐♥tr♦❞✉❝❡❞ ❜② s✉♠♠❛r② st❛t✳ s ✱ s✐♥❝❡ ♣ ( θ | s ( ② ⋆ )) � = ♣ ( θ | ② ⋆ ) ✳ ◆♦t❡ t❤❛t ♣ ( θ | s ( ② ⋆ )) ≈ ♣ ( θ | ② ⋆ ) ♠❛② ❜❡ ❛ r❡❛s♦♥❛❜❧❡ ❛♣♣r♦①✐♠❛t✐♦♥✱ ❜✉t ♣ ( ② ⋆ ) ❛♥❞ ♣ ( s ( ② ⋆ )) ❤❛✈❡ ♥♦ ❝❧❡❛r r❡❧❛t✐♦♥✿ ❤❡♥❝❡ st❛♥❞❛r❞ ❆❇❈ ❝❛♥♥♦t r❡❧✐❛❜❧② ❛♣♣r♦①✐♠❛t❡ t❤❡ ❡✈✐❞❡♥❝❡✳ ✸ ✴ ✷✷

❊P✲❆❇❈ t❛r❣❡t ❆ss✉♠❡ t❤❛t t❤❡ ❞❛t❛ ② ❞❡❝♦♠♣♦s❡s ✐♥t♦ ( ② ✶ , . . . , ② ♥ ) ✱ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❆❇❈ ❛♣♣r♦①✐♠❛t✐♦♥✿ � ˆ � ♥ � ♣ ǫ ( θ | ② ⋆ ) ∝ ♣ ( θ ) ♣ ( ② ✐ | ② ⋆ ✶ : ✐ − ✶ , θ ) 1 { � ② ✐ − ② ⋆ ✐ �≤ ε } ❞② ✐ ✭✶✮ ✐ = ✶ ❙t❛♥❞❛r❞ ❆❇❈ ❝❛♥♥♦t t❛r❣❡t t❤✐s ❛♣♣r♦①✐♠❛t❡ ♣♦st❡r✐♦r✱ ❜❡❝❛✉s❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t � ② ✐ − ② ⋆ ✐ � ≤ ε ❢♦r ❛❧❧ ✐ s✐♠✉❧t❛♥❡♦✉s❧② ✐s ❡①♣♦♥❡♥t✐❛❧❧② s♠❛❧❧ ✇✳r✳t✳ ♥ ✳ ❇✉t ✐t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ s♦♠❡ s✉♠♠❛r② st❛ts s ✱ ❛♥❞ ♣ ǫ ( θ | ② ⋆ ) → ♣ ( θ | ② ⋆ ) ❛s ǫ → ✵ ✭♦♥❡ ❧❡✈❡❧ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥✮✳ ❚❤❡ ❊P✲❆❇❈ ❛❧❣♦r✐t❤♠ ❝♦♠♣✉t❡s ❛ ●❛✉ss✐❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ✭✶✮✳ ✹ ✴ ✷✷

◆♦✐s② ❆❇❈ ✐♥t❡r♣r❡t❛t✐♦♥ ◆♦t❡ t❤❛t t❤❡ ❊P✲❆❇❈ t❛r❣❡t ♦❢ t❤❡ ♣r❡✈✐♦✉s s❧✐❞❡ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ ❝♦rr❡❝t ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ♦❢ ❛ ♠♦❞❡❧ ✇❤❡r❡ t❤❡ ❞❛t❛♣♦✐♥ts ❛r❡ ❝♦rr✉♣t❡❞ ✇✐t❤ ❛ ❯ [ − ǫ, ǫ ] ♥♦✐s❡✱ ❢♦❧❧♦✇✐♥❣ ❲✐❧❦✐♥s♦♥ ✭✷✵✵✽✮✳ ✺ ✴ ✷✷

❊P✿ ❛♥ ✐♥tr♦❞✉❝t✐♦♥ ■♥tr♦❞✉❝❡❞ ✐♥ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ ❜② ▼✐♥❦❛ ✭✷✵✵✶✮✳ ❈♦♥s✐❞❡r ❛ ❣❡♥❡r✐❝ ♣♦st❡r✐♦r✿ ♥ � π ( θ ) = ♣ ( θ | ② ) ∝ ♣ ( θ ) ❧ ✐ ( θ ) ✭✷✮ ✐ = ✶ ✇❤❡r❡ t❤❡ ❧ ✐ ❛r❡ ♥ ❝♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ ❧✐❦❡❧✐❤♦♦❞✳ ❆✐♠ ✐s t♦ ❛♣♣r♦①✐♠❛t❡ π ✇✐t❤ ♥ � q ( θ ) ∝ ❢ ✐ ( θ ) ✭✸✮ ✐ = ✵ ✇❤❡r❡ t❤❡ ❢ ✐ ✬s ❛r❡ t❤❡ ✏s✐t❡s✑✳ ❚♦ ♦❜t❛✐♥ ❛ ●❛✉ss✐❛♥ ❛♣♣r♦①✐♠❛t✐♦♥✱ � � − ✶ ✷ θ t ◗ ✐ θ + r t t❛❦❡ ❢ ✐ ( θ ) ∝ ❡①♣ ✱ s♦ t❤❛t✿ ✐ θ � � ♥ � � ♥ � t � � � − ✶ ✷ θ t q ( θ ) ∝ ❡①♣ θ + ✭✹✮ ◗ ✐ r ✐ θ ✐ = ✵ ✐ = ✵ ✇❤❡r❡ ◗ ✐ ❛♥❞ r ✐ ❛r❡ t❤❡ s✐t❡ ♣❛r❛♠❡t❡rs✳ ✻ ✴ ✷✷

❙✐t❡ ✉♣❞❛t❡ ❲❡ ✇✐s❤ t♦ ♠✐♥✐♠✐s❡ ❑▲ ( π � q ) ✳ ❚♦ t❤❛t ❛✐♠✱ ✇❡ ✉♣❞❛t❡ ❡❛❝❤ s✐t❡ ( ◗ ✐ , r ✐ ) ✐♥ t✉r♥✱ ❛s ❢♦❧❧♦✇s✳ ❈♦♥s✐❞❡r t❤❡ ❤②❜r✐❞✿ � ❤ ✐ ( θ ) ∝ q − ✐ ( θ ) ❧ ✐ ( θ ) , q − ✐ ( θ ) = ❢ ❥ ( θ ) ❥ � = ✐ ❛♥❞ ❛❞❥✉st ( ◗ ✐ , r ✐ ) s♦ t❤❛t ❑▲ ( ❤ ✐ � q ) ✐s ♠✐♥✐♠❛❧✳ ❖♥❡ ♠❛② ❡❛s✐❧② ♣r♦✈❡ t❤❛t t❤✐s ♠❛② ❜❡ ❞♦♥❡ ❜② ♠♦♠❡♥t ♠❛t❝❤✐♥❣✱ ✐✳❡✳ ❝❛❧❝✉❧❛t❡✿ � θθ ❚ � µ ❤ = E ❤ ✐ [ θ ] , Σ ❤ = E ❤ ✐ − µ ✐ µ ❚ ✐ s❡t ◗ ❤ = Σ − ✶ ❤ , r ❤ = Σ − ✶ ❤ µ ❤ ✱ t❤❡♥ ❛❞❥✉st ( ◗ ✐ , r ✐ ) s♦ t❤❛t ( ◗ ❤ , r ❤ ) ❛♥❞ ( ◗ , r ) = ( � ♥ ✐ = ✵ ◗ ✐ , � ♥ ✐ = ✵ r ✐ ) ✭t❤❡ ♠♦♠❡♥ts ♦❢ q ✮ ♠❛t❝❤✳ ◗ ✐ ← Σ − ✶ r ✐ ← Σ − ✶ − ◗ − ✐ , ❤ µ ❤ − r − ✐ . ❤ ✼ ✴ ✷✷

❊P q✉✐❝❦ s✉♠♠❛r② • ❈♦♥✈❡r❣❡♥❝❡ ✐s ✉s✉❛❧❧② ♦❜t❛✐♥❡❞ ❛❢t❡r ❛ ❢❡✇ ❝♦♠♣❧❡t❡ ❝②❝❧❡s ♦✈❡r ❛❧❧ t❤❡ s✐t❡s✳ • ❖✉t♣✉t ✐s ❛ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥ ✇❤✐❝❤ ✐s ✏❝❧♦s❡st✑ t♦ t❛r❣❡t π ✱ ✐♥ ❑▲ s❡♥s❡✳ • ❲❡ ✉s❡ t❤❡ ●❛✉ss✐❛♥ ❢❛♠✐❧② ❢♦r q ✱ ❜✉t ♦♥❡ ♠❛② t❛❦❡ ❛♥♦t❤❡r ❡①♣♦♥❡♥t✐❛❧ ❢❛♠✐❧②✳ • ❋❡❛s✐❜❧✐t② ♦❢ ❊P ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❤♦✇ ❡❛s② ✐t ✐s t♦ ❝♦♠♣✉t❡ t❤❡ ♠♦♠❡♥ts ♦❢ ♦r❞❡r ✶ ❛♥❞ ✷ ♦❢ t❤❡ ❤②❜r✐❞ ❞✐str✐❜✉t✐♦♥ ✭✐✳❡✳ ❛ ●❛✉ss✐❛♥ ❞❡♥s✐t② q − ✐ t✐♠❡s ❛ s✐♥❣❧❡ ❧✐❦❡❧✐❤♦♦❞ ❝♦♥tr✐❜✉t✐♦♥ ❧ ✐ ✮✳ ✽ ✴ ✷✷

❊P✲❆❇❈ ●♦✐♥❣ ❜❛❝❦ t♦ t❤❡ ❊P✲❆❇❈ t❛r❣❡t✿ � ˆ � ♥ � ♣ ǫ ( θ | ② ⋆ ) ∝ ♣ ( θ ) ♣ ( ② ✐ | ② ⋆ ✶ : ✐ − ✶ , θ ) 1 { � ② ✐ − ② ⋆ ✐ �≤ ε } ❞② ✐ ✭✺✮ ✐ = ✶ ✇❡ t❛❦❡ ˆ ❧ ✐ ( θ ) = ♣ ( ② ✐ | ② ⋆ ✶ : ✐ − ✶ , θ ) 1 { � ② ✐ − ② ⋆ ✐ �≤ ε } ❞② ✐ . ■♥ t❤❛t ❝❛s❡✱ t❤❡ ❤②❜r✐❞ ❞✐str✐❜✉t✐♦♥ ✐s ❛ ●❛✉ss✐❛♥ t✐♠❡s ❧ ✐ ✳ ❚❤❡ ♠♦♠❡♥ts ❛r❡ ♥♦t ❛✈❛✐❧❛❜❧❡ ✐♥ ❝❧♦s❡✲❢♦r♠ ✭♦❜✈✐♦✉s❧②✮✱ ❜✉t t❤❡② ❛r❡ ❡❛s✐❧② ♦❜t❛✐♥❡❞✱ ✉s✐♥❣ s♦♠❡ ❢♦r♠ ♦❢ ❆❇❈ ❢♦r ❛ s✐♥❣❧❡ ♦❜s❡r✈❛t✐♦♥✳ ✾ ✴ ✷✷

❊P✲❆❇❈ s✐t❡ ✉♣❞❛t❡ ■♥♣✉ts✿ ǫ ✱ ② ⋆ ✱ ✐ ✱ ❛♥❞ t❤❡ ♠♦♠❡♥t ♣❛r❛♠❡t❡rs µ − ✐ ✱ Σ − ✐ ♦❢ t❤❡ ●❛✉ss✐❛♥ ♣s❡✉❞♦✲♣r✐♦r q − ✐ ✳ ✶ ❉r❛✇ ▼ ✈❛r✐❛t❡s θ [ ♠ ] ❢r♦♠ ❛ ◆ ( µ − ✐ , Σ − ✐ ) ❞✐str✐❜✉t✐♦♥✳ ✷ ❋♦r ❡❛❝❤ θ [ ♠ ] ✱ ❞r❛✇ ② [ ♠ ] ✶ : ✐ − ✶ , θ [ ♠ ] ) ✳ ∼ ♣ ( ② ✐ | ② ⋆ ✐ ✸ ❈♦♠♣✉t❡ t❤❡ ❡♠♣✐r✐❝❛❧ ♠♦♠❡♥ts � ▼ ♠ = ✶ θ [ ♠ ] 1 � ▼ � � � ② [ ♠ ] − ② ⋆ ✐ �≤ ε ✐ � ▼ ❛❝❝ = 1 � � , µ ❤ = � ② [ ♠ ] − ② ⋆ ✐ �≤ ε ▼ ❛❝❝ ✐ ♠ = ✶ ✭✻✮ ♠ = ✶ θ [ ♠ ] � θ [ ♠ ] � t 1 � � ▼ � � ② [ ♠ ] − ② ⋆ ✐ �≤ ε � ✐ µ ( ❤ ✐ ) t . ✭✼✮ − � Σ ❤ = µ ( ❤ ✐ ) � ▼ ❛❝❝ ❘❡t✉r♥ � µ ( ❤ ✐ ) ❛♥❞ � ❩ ( ❤ ✐ ) = ▼ ❛❝❝ / ▼ ✱ � Σ ( ❤ ✐ ) ✳ ✶✵ ✴ ✷✷