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❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

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

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ●❡♥❡r❛❧ ❞❡✜♥✐t✐♦♥s ♦❢

✶ ❘❛♥❞♦♠ ❙❛♠♣❧✐♥❣ ✷ ❈❡♥s♦r❡❞ ❙❛♠♣❧✐♥❣ ✸ ❚r✉♥❝❛t❡❞ ❙❛♠♣❧❡ ✹ ❈❤♦✐❝❡ ❇❛s❡❞ ❙❛♠♣❧❡ ✺ ❆❧s♦ ❝♦♥s✐❞❡r tr✉♥❝❛t❡❞ ❛♥❞ ❝❡♥s♦r❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ✭✶✮ ❘❛♥❞♦♠ 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 (XN)

• 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❡❞ ❙❛♠♣❧❡

f (X) : ❞❡♥s✐t② ♦❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ a < X < b : b, a ♠❛② ❜❡ ✐♥✜♥✐t❡

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ❲❡ ♦❜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 ∗) R

L

f (z)dz L ≤ X ∗ ≤ R

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ✭◆♦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

• ✭✹✮ ■❢ ✇❡ ❤❛✈❡ 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

L

f✷(Y✷)dY✷

• ❚❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ Y✶ ✐s ♥♦t tr✉♥❝❛t❡❞✳ ❲❡ ♦❜s❡r✈❡ Y✶ ♦♥❧②

✐❢ t❤❡ ❝♦♥❞✐t✐♦♥ ♦♥ Y✷ ✐s s❛t✐s✜❡❞✳

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ❚❤❡ 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✶✳ g(Y ∗

✶ ) =

R✷

L✷

f (Y ∗

✶ , Y✷)dY✷

R✶

L✶

R✷

L✷

f (Y✶, Y✷)dY✶dY✷

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ✭✻✮ ◆❡✇ 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✳ Yj ⊂ {✶, . . . , 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

❈❤♦✐❝❡ ❇❛s❡❞ ❙❛♠♣❧❡s

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• 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

ϕ(Z | Y )C(Y ) = g(Y | Z)h(Z) f (Y )

• C(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

• ◆♦t❡✱ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ✐♥ ❛♥ ❡①♦❣❡♥♦✉s s❛♠♣❧✐♥❣ s❝❤❡♠❡

✐s L =

I

• i=✶

f (Yi, Zi) =

I

• i=✶

f (Yi | Zi, θ)h(Zi) ln L =

I

• i=✶

ln f (Yi | Zi) +

• ln h(Zi).
• ❇② ❡①♦❣❡♥❡✐t②✱ ✇❡ ❣❡t t❤❡ ❧❛❝❦ ♦❢ ❞❡♣❡♥❞❡♥❝❡ ♦❢ ❞✐str✐❜✉t✐♦♥ ♦❢

Z ♦♥ θ.

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ▲✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ❢♦r ❛ ❝❤♦✐❝❡✲❜❛s❡❞ s❛♠♣❧✐♥❣ s❝❤❡♠❡ ✐s

ln L =

I

• i=✶

[ln g(Yi | Zi) + ln h(Zi) − ln f (Yi) + ln C(Yi)] .

• ■♥ s❡✈❡r❛❧✱

f (Y ) ❞❡♣❡♥❞s ♦♥ ♣❛r❛♠❡t❡rs θ. · .. ▼❛① ✇✐t❤ θ. ∂ ln L ∂θ =

I

• i=✶

∂ ln g(Yi | Zi) ∂θ −

I

• i=✶

∂ ln f (Yi) ∂θ .

• ❲❡ ♥❡❣❧❡❝t t❤❡ s❡❝♦♥❞ t❡r♠ ✐♥ ❢♦r♠✐♥❣ t❤❡ ✉s✉❛❧ ❡st✐♠❛t♦rs

✉s✐♥❣ ♦♥❧② t❤❡ ✜rst t❡r♠✳ ❚❤❛t ✐s t❤❡ s♦✉r❝❡ ♦❢ t❤❡ ✐♥❝♦♥s✐st❡♥❝②✳

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

❈❤♦✐❝❡ ❇❛s❡❞ ❙❛♠♣❧❡✿

• ❆♥ ❡①❛♠♣❧❡ ✐♥ ❞✐s❝r❡t❡ ❝❤♦✐❝❡✳
• ✭❝✮ ❉r❛✇ d ❜② ϕ(d).
• ✭❞✮ ❉r❛✇ X ❜② f (X | d = ✶).
• ❏♦✐♥t ❞❡♥s✐t② ♦❢ ❞❛t❛✿

ϕ(d = ✶)f (X | d = ✶, θ✵) = ϕ(d = ✶) Pr(d = ✶ | X, θ✵)f (X) Pr(d = ✶ | θ✵)

• ❍❡❝❦♠❛♥

❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ◆♦✇ ✐♥ ❛ ❝❤♦✐❝❡✲❜❛s❡❞ s❛♠♣❧❡

Pr∗(d = ✶ | X) = f (X | d = ✶, θ✵)ϕ(d = ✶) g ∗(X) ✇❤❡r❡ g ∗(X) ✐s t❤❡ s❛♠♣❧❡❞ X ❞❛t❛✳ ❏♦✐♥t ❞❡♥s✐t② ♦❢ ❞❛t❛ X ✐s ❣✐✈❡♥ ❜②✿ g ∗(X) = f (X | d = ✶, θ)ϕ(d = ✶) + f (X | d = ✵, θ)ϕ(d = ✶) ❛♥❞ Pr(d = ✶ | X) = f (X | d = ✶) Pr(d = ✶) f (X)

• ❆ss✉♠❡ f (X) > ✵✳ ❯s✐♥❣ ❇❛②❡s✬ t❤❡♦r❡♠ ❢♦r Y ✇r✐t❡✿

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• Pr∗(d = ✶ | X) =

Pr(d = ✶ | X, θ)f (X) Pr(d = ✶ | θ) ϕ(d = ✶) Pr(d = ✶ | X, θ)f (X) Pr(d = ✶ | θ) ϕ(d = ✶) + Pr(d = ✵ | X, θ)f (X) Pr(d = ✵ | θ) ϕ(d = ✵) = Pr(d = ✶ | X, θ)ϕ(d = ✶)/ Pr(d = ✶ | θ) Pr(d = ✶ | X, θ) ϕ(d = ✶) Pr(d = ✶ | θ) + Pr(d = ✵ | X, θ) ϕ(d = ✵) Pr(d = ✵ | θ) .

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ◆♦✇ ✇❡ ♠✐ss❛♠♣❧❡ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✇✐t❤ ❞❡♥s✐t②

f (X | d = ✶) ✐♥ ❛ ❝❤♦✐❝❡ ❜❛s❡❞ s❛♠♣❧❡✿

Pr∗(d = ✶ | X) = f (X | d = ✶, θ✵)ϕ(d = ✶) f (X | d = ✶, θ)ϕ(d = ✶) + f (X | d = ✵, θ✵)ϕ(d = ✵) = f (X) Pr(d = ✶ | X) Pr(d = ✶) ϕ(d = ✶) f (X) Pr(d = ✶ | X) Pr(d = ✶) ϕ(d = ✶) + f (X) Pr(d = ✵ | X) Pr(d = ✵) ϕ(d = ✵) = Pr(d = ✶ | X) Pr(d = ✶ | X) + Pr(d = ✵ | X)ϕ(d = ✵) ϕ(d = ✶) · Pr(d = ✶) Pr(d = ✵) = ✶ ✶ + Pr(d = ✵ | X) Pr(d = ✶ | X)

• · ϕ(d = ✵)

ϕ(d = ✶) · Pr(d = ✶) Pr(d = ✵)

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ❲✐t❤ ❧♦❣✐t ✇❡ ❣❡t

Pr∗(d = ✶ | X) = ✶ ✶ + e

−(α✵+Xβ)+ln  ϕ(d = ✵)

ϕ(d = ✶)

·Pr(d = ✶)

Pr(d = ✵)

 

. ❚❤✐s ❣♦❡s ✐♥t♦ ❛♥ ✐♥t❡r❝❡♣t t❡r♠✿ = eα∗+Xβ ✶ + eα∗+Xβ α∗ = α✵ − ln ϕ(d = ✵) ϕ(d = ✶) · Pr(d = ✶) Pr(d = ✵)

• .

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ❍♦✇ t♦ s♦❧✈❡ ♣r♦❜❧❡♠✿ ❘❡✇❡✐❣❤t ❞❛t❛ ❜② r❡❧❛t✐✈❡ ❢r❡q✉❡♥❝② ✐♥

♣♦♣✉❧❛t✐♦♥✳

• ✭■❞❡❛ ❞✉❡ t♦ ❈✳❘✳ ❘❛♦✱ ✶✾✻✺✱ ✶✾✽✻✳✮
• ❏♦✐♥t ❞❡♥s✐t② ♦❢ t❤❡ ❞❛t❛ ✐s

f (X | d = ✶)ϕ(d = ✶). ❯s❡ ❇❛②❡s✬ r✉❧❡ t♦ ♦❜t❛✐♥ P(d = ✶ | X)f (X) P(d = ✶) ϕ(d = ✶).

• ◆♦✇ ✇❡✐❣❤t ❜②

P(d = ✶) ϕ(d = ✶).

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ❙♦❧✉t✐♦♥✿ ❘❡✇❡✐❣❤t t❤❡ ❞❛t❛ t♦ ❢♦r♠ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❡✐❣❤t❡❞

❧✐❦❡❧✐❤♦♦❞✿

✶ N

N

• i=✶

Pr(di = ✶) ϕ(di = ✶) (d∗

i ) ln Pr(di = ✶ | X, θ) + Pr(di = ✵)

ϕ(di = ✵) (✶ − d∗

i ) ln Pr(di

✵ P

• {[Pr(d = ✶ | X, θ✵)f (X | θ✵)] ln Pr(d = ✶ | X, θ)+
• [Pr(d = ✵ | X, θ✵)f (X | θ✵)] ln Pr(d = ✵ | X, θ)} f (X | d)dX

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s

• ❚❤✐s st❡♣ ✉s❡s t❤❡ r❡s✉❧t t❤❛t r❡✇❡✐❣❤t✐♥❣ t❤❡ ❞❛t❛ ❣✐✈❡s ✉s t❤❡

tr✉❡ ❞❡♥s✐t②.

• ❇❡tt❡r ✇❛② t♦ s❡❡ ✇❤❛t ✐s ❣✐✈✐♥❣ ♦♥✿

f (X | d = ✶)ϕ(d = ✶) g ∗(X) = Pr(d = ✶ | X)f (X) g ∗(X) ϕ(d = ✶) Pr(d = ✶).

• ❘❡✇❡✐❣❤t t❤❡ ❞❛t❛✿ ✇❤❡♥ ✇❡ r❡✇❡✐❣❤t t❤❡ ❞❛t❛✱ g ∗ ✐s r❡st♦r❡❞

t♦ f ✳

f (X) = f (X | d = ✶)ϕ(d = ✶) P(d = ✶) ϕ(d = ✶)

• +f (X | d = ✵)ϕ(d = ✵)Pr(d = ✵)

ϕ(d = ✵) .

❍❡❝❦♠❛♥ ❉❡✜♥✐t✐♦♥ ♦❢ ❙❛♠♣❧❡s