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  1. ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s ❏❛♠❡s ❏✳ ❍❡❝❦♠❛♥ ❈❡♥t❡r ❢♦r t❤❡ ❊❝♦♥♦♠✐❝s ♦❢ ❍✉♠❛♥ ❉❡✈❡❧♦♣♠❡♥t ❯♥✐✈❡rs✐t② ♦❢ ❈❤✐❝❛❣♦ ❊❝♦♥ ✸✶✷✱ ❙♣r✐♥❣ ✷✵✶✾ ❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

  2. • ●❡♥❡r❛❧ ❞❡✜♥✐t✐♦♥s ♦❢ ✶ ❘❛♥❞♦♠ ❙❛♠♣❧✐♥❣ ✷ ❈❡♥s♦r❡❞ ❙❛♠♣❧✐♥❣ ✸ ❚r✉♥❝❛t❡❞ ❙❛♠♣❧❡ ✹ ❈❤♦✐❝❡ ❇❛s❡❞ ❙❛♠♣❧❡ ✺ ❆❧s♦ ❝♦♥s✐❞❡r tr✉♥❝❛t❡❞ ❛♥❞ ❝❡♥s♦r❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

  3. • ✭✶✮ ❘❛♥❞♦♠ s❛♠♣❧✐♥❣✿ ✭❘❡❛❧❧② s✐♠♣❧❡ r❛♥❞♦♠ s❛♠♣❧✐♥❣✮ • ✐✐❞✳ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐t❤ ❞❡♥s✐t② f ( X ) ✳ ❘❛♥❞♦♠ s❛♠♣❧✐♥❣ ✐♥ ❣❡♥❡r❛❧ ✐s ❞❡r✐✈❛t✐♦♥ ♦❢ ❛ s❛♠♣❧❡ ❜② ❛ ❝❛❧❝✉❧❛t❛❜❧❡ r✉❧❡✳ Pr♦❜✳ ♦❢ s❛♠♣❧❡ = f ( X ✶ ) f ( X ✷ ) f ( X ✸ ) ... f ( X N ) • Pr♦❜❧❡♠ ♦❢ ❣❡tt✐♥❣ ❛♥ X ✐s f ( X ) ✳ ❚❤✉s ✐♥ ❛ ♣♦♣✉❧❛t✐♦♥✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❣❡tt✐♥❣ ✐♥t♦ t❤❡ s❛♠♣❧❡ ✐s f ( X ) . ❚❤✐s ✐s s✐♠♣❧❡ r❛♥❞♦♠ s❛♠♣❧✐♥❣✳ • ✭✷✮ ❚r✉♥❝❛t❡❞ ❙❛♠♣❧❡ ❞❡♥s✐t② ♦❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ f ( X ) : a < X < b : b , a ♠❛② ❜❡ ✐♥✜♥✐t❡ ❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

  4. • ❲❡ ♦❜s❡r✈❡ X ✐❢ X < R ✭r✐❣❤t tr✉♥❝❛t✐♦♥✮ • ♦r ✐❢ ✐❢ X > L ✭❧❡❢t tr✉♥❝❛t✐♦♥✮✳ • ❑❡② ♣r♦♣❡rt②✿ ❧❛t❡♥t ✈❛r✐❛❜❧❡ X ✐♥ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✖ ✇❡ ❦♥♦✇ X ∗ = X ✇❤❡♥ L < X < R • ✭❆ss✉♠❡ s✐♠♣❧❡ r❛♥❞♦♠ s❛♠♣❧✐♥❣ ♦❢ ❛ ❧❛r❣❡r ♣♦♣✉❧❛t✐♦♥✮✳ ❲❡ ♦♥❧② ♦❜s❡r✈❡ X ∗ ❛♥❞ ✇❡ ❞♦ ♥♦t ❦♥♦✇ t❤❡ ♥✉♠❜❡r ♦❢ ♦❜s❡r✈❛t✐♦♥s ✐♥ ✭❧❛r❣❡r✮ r❛♥❞♦♠ s❛♠♣❧❡ ❢♦r ✇❤✐❝❤ X ✐s ♦✉ts✐❞❡ t❤❡ ✐♥t❡r✈❛❧✳ ❲❡ ♦♥❧② ❦♥♦✇ t❤❡ r❡❞✉❝❡❞ s❛♠♣❧❡ ✐❢ ❞❡♥s✐t② ✐♥ ♣♦♣✉❧❛t✐♦♥ ✭✉♥tr✉♥❝❛t❡❞✮ ✐s f ( X ) ✱ t❤❡♥ ❞❡♥s✐t② ♦❢ X ∗ ✐s f ( X ∗ ) L ≤ X ∗ ≤ R � R f ( z ) dz L ❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

  5. • ✭◆♦t❡ ❢✉rt❤❡r t❤❛t t❤❡r❡ ❛r❡ ❛♥ ✐♥✜♥✐t② ♦❢ ✉♥❞❡r✐❞❡♥t✐✜❡❞ ❞✐str✐❜✉t✐♦♥s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ tr✉♥❝❛t❡❞ ♦♥❡✳✮ • ✭✸✮ ❈❡♥s♦r❡❞ ❙❛♠♣❧❡✿ ❲❡ ♦❜s❡r✈❡ X ∗ ❛s ❜❡❢♦r❡ ❜✉t ✇❡ ❦♥♦✇ t❤❡ ♥✉♠❜❡r ♦❢ ♦❜s❡r✈❛t✐♦♥s ♦✉ts✐❞❡ ✐♥t❡r✈❛❧✳ • ❲❡ ❡♥❝♦✉♥t❡r t✇♦ t②♣❡s ♦❢ ❝❡♥s♦r✐♥❣✿ • ✭❛✮ ❚②♣❡ ♦♥❡ ❝❡♥s♦r✐♥❣ ✿ ✇❡ ♦♥❧② ♦❜s❡r✈❡ ❛ ✈❛r✐❛❜❧❡ ✐❢ ✐t ❧✐❡s ✐♥ ❛ r❛♥❣❡✱ ♥✉♠❜❡r ♦❢ ✈❛❧✉❡s ♦❢ Y ♦✉ts✐❞❡ t❤❡ r❛♥❣❡ ✐s ❦♥♦✇♥✳ • ✭❜✮ ❚②♣❡ ❚✇♦ ❈❡♥s♦r✐♥❣✿ ❋✐①❡❞ ♣r♦♣♦rt✐♦♥ ♦❢ t❤❡ s❛♠♣❧❡ ✐s ❝❡♥s♦r❡❞ ✐♥ ❛❞✈❛♥❝❡ ✭ ❡✳❣✳ st♦♣ ♦❜s❡r✈✐♥❣ ❧✐❣❤t ❜✉❧❜ ❜✉r♥♦✉t ✇❤❡♥ ✇❡ ❤❛✈❡ ❛ ♣r♦♣♦rt✐♦♥ ✲ s❛② ♠✮✳ ❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

  6. • ✭✹✮ ■❢ ✇❡ ❤❛✈❡ t❤❛t ✐♥ ❜♦t❤ ✭✸✮ ❛♥❞ ✭✷✮✱ X ✐s ❛ tr✉♥❝❛t❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✭t❤❡ r❛♥❣❡ ♦❢ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✐s tr✉♥❝❛t❡❞✮✳ • ✭✺✮ ◆❡✇ t❡r♠✿ ❝♦✐♥❡❞ ✐♥ r❡❝❡♥t ❡❝♦♥♦♠❡tr✐❝ ✇♦r❦ ✖ ❝❡♥s♦r❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✳ ■t ✐s ✐♥❤❡r❡♥t❧② ❛ ❜✐✈❛r✐❛t❡ ❝♦♥❝❡♣t✳ ❏♦✐♥t pdf − f ( Y ✶ , Y ✷ ) ✳ ❚❤❡♥ ✇❡ ❤❛✈❡ t❤❛t ✇❡ ♦❜s❡r✈❡ Y ✶ ♦♥❧② ✐❢ Y ✷ ❡①❝❡❡❞s s♦♠❡ ✈❛❧✉❡ ♦r ❧✐❡s ✐♥ s♦♠❡ r❛♥❣❡✱ ❡✳❣✳ ✭✶✮ L < Y ✷ < R Pr♦❜✳ ♦❢ t❤✐s ❡✈❡♥t ✐s � R f ✷ ( Y ✷ ) dY ✷ L • ❚❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ Y ✶ ✐s ♥♦t tr✉♥❝❛t❡❞ ✳ ❲❡ ♦❜s❡r✈❡ Y ✶ ♦♥❧② ✐❢ t❤❡ ❝♦♥❞✐t✐♦♥ ♦♥ Y ✷ ✐s s❛t✐s✜❡❞✳ ❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

  7. • ❚❤❡ s❛♠♣❧❡ ♠❛② ♦r ♠❛② ♥♦t ❜❡ tr✉♥❝❛t❡❞✳ ❚❤✉s✱ ✐t ✐s t❤❡ ❝❛s❡ t❤❛t ✐❢ ✇❡ ♦❜s❡r✈❡ Y ✶ , ❣✐✈❡♥ s❡❧❡❝t✐♦♥ ❝r✐t❡r✐♦♥ ✭✯✮✱ ❜✉t ✇❡ ❞♦ ♥♦t ❦♥♦✇ t❤❡ ♥✉♠❜❡r ♦❢ ♦❜s❡r✈❛t✐♦♥s ✐♥ t❤❡ ❧❛r❣❡r r❛♥❞♦♠ s❛♠♣❧❡ ✈❛r✐❛❜❧❡ ❢♦r ✇❤✐❝❤ t❤❡ Y ✷ r❡str✐❝t✐♦♥ ✐s ✈✐♦❧❛t❡❞✱ ✇❡ ❤❛✈❡ ❛ tr✉♥❝❛t❡❞ s❛♠♣❧❡ ❛♥❞ ❛ ❝❡♥s♦r❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳ ◆♦✇ ❝❧❡❛r❧② ✇❡ ♠❛② ♣✉t ❛ r❡str✐❝t✐♦♥ ♦♥ Y ✶ ❡✳❣✳ ✇❡ ♦❜s❡r✈❡ Y ✶ ♦♥❧② ✐❢ L ✷ < Y ✷ < R ✷ ❛♥❞ L ✶ < Y ✶ < R ✶ . ❚❤✉s ❞❡✜♥❡ Y ∗ ✶ = Y ✶ ❢♦r L ✶ < Y ✶ < R ✶ ✳ � R ✷ f ( Y ∗ ✶ , Y ✷ ) dY ✷ L ✷ g ( Y ∗ ✶ ) = � R ✶ � R ✷ f ( Y ✶ , Y ✷ ) dY ✶ dY ✷ L ✶ L ✷ ❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

  8. • ✭✻✮ ◆❡✇ t❡r♠ ✐♥ ❞✐s❝r❡t❡ ❝❤♦✐❝❡ ❧✐t❡r❛t✉r❡ ✕ ❝❤♦✐❝❡ ❜❛s❡❞ s❛♠♣❧✐♥❣ ✳ ❈♦♥s✐❞❡r t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ Y t♦ ❜❡ ❞✐s❝r❡t❡✳ Z ❛r❡ ❡①♦❣❡♥♦✉s ❡①♣❧❛♥❛t♦r② ✈❛r✐❛❜❧❡s✳ ❚❤❡ t❤❡♦r② ♣r♦❞✉❝❡s ❛ g ( Y | Z , θ ) : ❞✐s❝r❡t❡ ❝❤♦✐❝❡ ♠♦❞❡❧ h ( Z ) ✐♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ ❡①♦❣❡♥♦✉s ✈❛r✐❛❜❧❡s✳ Y j ⊂ { ✶ , . . . , J } ❡❧❡♠❡♥ts ♦❢ ❝❤♦✐❝❡ s❡t✳ ❊①♦❣❡♥♦✉s ❙❛♠♣❧✐♥❣✿ ✇❡ ♣✐❝❦ Z ✱ t❤❡♥ ♦❜s❡r✈❡ Y ✳ ❙❛♠♣❧❡ Z ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡♥s✐t② k ( Z ) ❛♥❞ ♦❜s❡r✈❡ t❤❡ ✈❛❧✉❡ ♦❢ Y ✱ t❤❡ ❝❤♦✐❝❡✳ ▲✐❦❡❧✐❤♦♦❞ ♦❢ ❛♥ ♦❜s❡r✈❛t✐♦♥ ( Y , Z ) ✐s g ( Y | Z , θ ) k ( Z ) ✇❤❡♥ k ( Z ) = h ( Z ) ✱ ✇❡ ❤❛✈❡ r❛♥❞♦♠ s❛♠♣❧✐♥❣✳ ❖t❤❡r✇✐s❡ ✇❡ ❤❛✈❡ str❛t✐✜❡❞ s❛♠♣❧✐♥❣✳ ❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

  9. ❈❤♦✐❝❡ ❇❛s❡❞ ❙❛♠♣❧❡s ❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

  10. • P✐❝❦ Y ✜rst ✭ ❡✳❣✳ tr❛✈❡❧ ♠♦❞❡✮✳ Pr♦❜❛❜✐❧✐t② ♦❢ s❡❧❡❝t✐♥❣ Y ✐s C ( Y ) . • f ( Y , Z ) ✐s t❤❡ ❥♦✐♥t ❞❡♥s✐t② ♦❢ Y ❛♥❞ Z ✐♥ t❤❡ ♣♦♣✉❧❛t✐♦♥✳ f ( Y , Z | θ ) = g ( Y | Z , θ ) h ( Z ) = ϕ ( Z | Y ) f ( Y | θ ) � f ( Y | θ ) = g ( Y | Z , θ ) h ( Z ) dZ • ●✐✈❡♥ Y ✇❡ ♦❜s❡r✈❡ Z ✭t❤❡ ✐♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥ ✐s t❤❛t ✇❡ ❛r❡ s❛♠♣❧✐♥❣ ♦♥❧② ♦♥ Y , ♥♦t ♦♥ Y ❛♥❞ Z ) ✳ Pr♦❜❛❜✐❧✐t② ♦❢ s❛♠♣❧❡❞ Z , Y ✐s ϕ ( Z | Y ) C ( Y ) . • ❆ ❢❛❝t ✇❡ ✉s❡ ❧❛t❡r ✐s ❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

  11. � g ( Y | Z ) h ( Z ) � ϕ ( Z | Y ) C ( Y ) = C ( Y ) f ( Y ) g ( Y | Z ) h ( Z ) C ( Y ) = � . �� g ( Y | Z ) h ( Z ) dZ � ❲❤❡♥ C ( Y ) = f ( Y ) = g ( Y | Z ) h ( Z ) dZ ✱ ❝❤♦✐❝❡ ❜❛s❡❞ s❛♠♣❧✐♥❣ ✐s r❛♥❞♦♠ s❛♠♣❧✐♥❣✳ ❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

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