t r tt t - - PowerPoint PPT Presentation

▶
t r t t t r
SMART_READER_LITE
LIVE PREVIEW

t r tt t - - PowerPoint PPT Presentation

Jan 19, 2024 339 likes •576 views

t r tt t r rs rt sttt ttqs

slide-1
SLIDE 1

❲❤❡♥ ▼♦♥t❡ ❈❛r❧♦ ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥ ♠❡t ✐♥ ❛ ▼❛r❦♦✈✐❛♥ ❞❛♥❝❡

  • ❡rs❡♥❞❡ ❋♦rt

❈◆❘❙ ■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ❚♦✉❧♦✉s❡✱ ❋r❛♥❝❡

■❈❚❙ ✧❆❞✈❛♥❝❡s ✐♥ ❆♣♣❧✐❡❞ Pr♦❜❛❜✐❧✐t②✧✱ ❇❡♥❣❛❧✉r✉✱ ❆✉❣✉st ✷✵✶✾✳

slide-2
SLIDE 2

✳

❆ ❞❛♥❝❡✱ ✇❤② ❄

✳

slide-3
SLIDE 3

❚♦ ✐♠♣r♦✈❡ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞s t❛r❣❡tt✐♥❣✿ dπ = π dµ

  • ❚❤❡ ✧♥❛✐✈❡✧ ▼❈ s❛♠♣❧❡r ❞❡♣❡♥❞s ♦♥ ❞❡s✐❣♥ ♣❛r❛♠❡t❡rs ✐♥ Rp ♦r ✐♥ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥ θ
  • ❚❤❡♦r❡t✐❝❛❧ st✉❞✐❡s ❝❛r❛❝t❡r✐③❡ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ♦❢ t❤❡s❡s ♣❛r❛♠❡t❡rs θ⋆ ❜②

θ⋆ ∈ Θ s✳t✳

  • H(θ, x) dπ(x) = 0

♦r θ⋆ ∈ argminθ∈Θ

  • C(θ, x) dπ(x) = 0.
  • ❙tr❛t❡❣✐❡s✿

✲ ❙tr❛t❡❣② ✶✿ ❛ ♣r❡❧✐♠✐♥❛r② ✧♠❛❝❤✐♥❡r②✧ ❢♦r t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ θ⋆❀ t❤❡♥ r✉♥ t❤❡ ▼❈ s❛♠♣❧❡r ✇✐t❤ θ ← θ⋆ ✲ ❙tr❛t❡❣② ✷✿ ❧❡❛r♥ θ ❛♥❞ s❛♠♣❧❡ ❝♦♥❝♦♠✐t❛♥t❧②

slide-4
SLIDE 4

❚♦ ♠❛❦❡ ♦♣t✐♠✐③❛t✐♦♥ ♠❡t❤♦❞s tr❛❝t❛❜❧❡

  • ■♥tr❛❝t❛❜❧❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥

θ s✳t✳ h(θ) = 0 ✇❤❡♥ h ✐s ♥♦t ❡①♣❧✐❝✐t h(θ) =

  • X H(θ, x) dπθ(x)

♦r argminθ∈Θ

  • X C(θ, x) dπθ(x)
  • ■♥tr❛❝t❛❜❧❡ ❛✉①✐❧✐❛r② q✉❛♥t✐t✐❡s

❊①✲✶ ●r❛❞✐❡♥t✲❜❛s❡❞ ♠❡t❤♦❞s ∇f(θ) =

  • X H(θ, x) dπθ(x)

❊①✲✷ ▼❛❥♦r✐③❡✲▼✐♥✐♠✐③❛t✐♦♥ ♠❡t❤♦❞s ❛t ✐t❡r❛t✐♦♥ t✱ f(θ) ≤ Ft(θ) =

  • X Ht(θ, x) dπt,θ(x)
  • ❙tr❛t❡❣✐❡s✿ ❯s❡ ▼♦♥t❡ ❈❛r❧♦ t❡❝❤♥✐q✉❡s t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ✉♥❦♥♦✇♥ q✉❛♥t✐t✐❡s
slide-5
SLIDE 5

■♥ t❤✐s t❛❧❦✱ ▼❛r❦♦✈ ✦

  • ❢r♦♠ t❤❡ ▼♦♥t❡ ❈❛r❧♦ ♣♦✐♥t ♦❢ ✈✐❡✇✿

✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ✉♣❞❛t✐♥❣ s❝❤❡♠❡ ❢♦r ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❛♠♣❧❡r ❄ ❈❛s❡✿ ▼❛r❦♦✈ ❝❤❛✐♥ ▼♦♥t❡ ❈❛r❧♦ s❛♠♣❧❡r

  • ❢r♦♠ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣♦✐♥t ♦❢ ✈✐❡✇✿

✇❤✐❝❤ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ▼♦♥t❡ ❈❛r❧♦ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ st♦❝❤❛st✐❝ ♦♣t✐♠✐③❛t✐♦♥ ❄ ❈❛s❡✿ ❙t♦❝❤❛st✐❝ ❆♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s ✇✐t❤ ▼❛r❦♦✈✐❛♥ ✐♥♣✉ts

  • ✭❚❛❧❦✮ ❆♣♣❧✐❝❛t✐♦♥ t♦ ❛ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❛❝❤✐♥❡ ▲❡❛r♥✐♥❣ ♣❜♠✿

♣❡♥❛❧✐③❡❞ ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ t❤r♦✉❣❤ ❙t♦❝❤❛st✐❝ Pr♦①✐♠❛❧✲●r❛❞✐❡♥t ❜❛s❡❞ ♠❡t❤♦❞s

slide-6
SLIDE 6

✳

P❛rt ■✿ ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s

✳

slide-7
SLIDE 7

✶st ❊①✳ ❆❞❛♣t✐✈❡ ■♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❜② ❲❛♥❣✲▲❛♥❞❛✉ ❛♣♣r♦❛❝❤❡s ✭✶✴✻✮ ❚❤❡ ♣r♦❜❧❡♠

  • ❆ ❤✐❣❤❧② ♠✉❧t✐♠♦❞❛❧ t❛r❣❡t ❞❡♥s✐t② dπ ♦♥ X ⊆ Rd✳

−2 −1 1 2 3 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −4 −2 2 4 1 2 3 4 5 6 7 8

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5

  • ❚✇♦ s❛♠♣❧❡rs ✇✐t❤ ❞✐✛❡r❡♥t ❜❡❤❛✈✐♦rs ✭♣❧♦t✿ t❤❡ x✲♣❛t❤ ♦❢ ❛ ❝❤❛✐♥ ✐♥ R2✮

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10

6

−2 −1.5 −1 −0.5 0.5 1 1.5 2 beta=4 2 4 6 8 10 12 x 10

4

−2 −1.5 −1 −0.5 0.5 1 1.5 2 beta=4

slide-8
SLIDE 8

✶st ❊①✳ ✭✷✴✻✮ ❚❤❡ str❛t❡❣② ❢♦r ❝❤♦♦s✐♥❣ t❤❡ ♣r♦♣♦s❛❧ ♠❡❝❛♥✐s♠

  • ❆ ❢❛♠✐❧② ♦❢ ♣r♦♣♦s❛❧ ♠❡❝❛♥✐s♠s ♦❜t❛✐♥❡❞ ❜② ❜✐❛s✐♥❣ ❧♦❝❛❧❧② t❤❡ t❛r❣❡t✿

✲ ❣✐✈❡♥ ❛ ♣❛rt✐t✐♦♥ X1, · · · , XI ♦❢ X✱ ✲ ❢♦r ❛♥② ✇❡✐❣❤t ✈❡❝t♦r θ = (θ(1), · · · , θ(I)) dπθ(x) = 1

I

i=1 θ⋆(i) θ(i) I

  • i=1

1Xi(x) dπ(x) θ(i) , ✇✐t❤ θ⋆(i) :=

  • Xi

dπ(u).

  • ❖♣t✐♠❛❧ ♣r♦♣♦s❛❧✿ dπθ⋆ ❁♣r♦♦❢❃
  • ❯♥❢♦rt✉♥❛t❡❧②✱ θ⋆ ✉♥❛✈❛✐❧❛❜❧❡✳
slide-9
SLIDE 9

✶st ❊①✳ ✭✸✴✻✮ ■❢ πθ⋆ ✇❡r❡ ❛✈❛✐❧❛❜❧❡

  • ❚❤❡ ❛❧❣♦r✐t❤♠ ✇♦✉❧❞ ❜❡✿

✲ ❙❛♠♣❧❡ X1, · · · , Xn, · · · ✐✳✐✳❞✳ ✇✐t❤ ❞✐str✐❜✉t✐♦♥ dπθ⋆ ✭♦r ❛ ▼❈▼❈ ✇✐t❤ t❛r❣❡t dπθ⋆✮ ✲ ❈♦♠♣✉t❡ t❤❡ ✐♠♣♦rt❛♥❝❡ r❛t✐♦ dπ dπθ⋆ (Xk) = I

I

  • i=1

1Xi(Xk) θ⋆(i)

  • ❲❤❡♥ ❛♣♣r♦①✐♠❛t✐♥❣ ❛♥ ❡①♣❡❝t❛t✐♦♥✱ s❡t
  • φ dπ ≈ I

T

T

  • t=1

 

I

  • i=1

1Xi(Xt) θ⋆(i)

  φ(Xt).

slide-10
SLIDE 10

✶st ❊①✳ ✭✹✴✻✮ θ⋆ ❛♥❞ t❤❡r❡❢♦r❡ dπθ⋆ ❛r❡ ✉♥❦♥♦✇♥✱ s♦ ❄

  • θ⋆ ∈ RI ❝♦❧❧❡❝ts
  • Xi dπ ❢♦r ❛❧❧ i ∈ {1, · · · , I}✱
  • θ⋆ t❤❡ ✉♥✐q✉❡ r♦♦t ♦❢ θ →
  • X H(θ, x) dπθ(x) ∈ RI ✇❤❡r❡ ❢♦r ❛❧❧ i ∈ {1, · · · , I}

Hi(θ, x) := θ(i)1X(i)(x) − θ(i)

I

  • j=1

1Xj(x)θ(j). t❤✉s s✉❣❣❡st✐♥❣ t❤❡ ✉s❡ ♦❢ ❛ ❙t♦❝❤❛st✐❝ ❆♣♣r♦①✐♠❛t✐♦♥ ♣r♦❝❡❞✉r❡✿ θ⋆ ≈ limt θt θt+1 = θt + γt+1H(θt, Xt+1) Xt+1 ∼ d πθt

  • ❚❤✐s ✉♣❞❛t❡ s❝❤❡♠❡ ✐s ❛ ♥♦r♠❛❧✐③❡❞ ❝♦✉♥t❡r ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ✈✐s✐ts t♦ Xi

❁♣r♦♦❢❃

slide-11
SLIDE 11

✶st ❊①✳ ✭✺✴✻✮ ❚❤❡ ❛❧❣♦r✐t❤♠✿ ❲❛♥❣✲▲❛♥❞❛✉ ❜❛s❡❞ ♣r♦❝❡❞✉r❡s

  • ■♥✐t✐❛❧✐s❛t✐♦♥✿ ❛ ✇❡✐❣❤t ✈❡❝t♦r θ0

❘❡♣❡❛t ❢♦r t = 1, · · · , T ✲ s❛♠♣❧❡ ❛ ♣♦✐♥t Xt+1 ∼ dπθt ✲ ✉♣❞❛t❡ t❤❡ ❡st✐♠❛t❡ ♦❢ θ⋆ θt+1 = θt + γt+1 H(θt, Xt+1) . ✇❤❡r❡ Xt+1 ∼ Pθt(Xt, ·) ❛♥❞ Pθ ✐♥✈✳ ✇rt dπθ✳

  • ❊①♣❡❝t❡❞✿

✲ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ θt t♦ θ⋆✿ ❙❆ s❝❤❡♠❡✱ ❢❡❞ ✇✐t❤ ❛❞❛♣t✐✈❡ ✭❝♦♥tr♦❧❧❡❞✮ ▼❈▼❈ s❛♠♣❧❡r✱ ✲ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ Xt t♦ d πθ⋆

slide-12
SLIDE 12

✶st ❊①✳ ✭✻✴✻✮ ❉♦❡s ✐t ✇♦r❦ ❄ P❧♦t✿ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ θt ❛♥❞ ✜rst ❡①✐t t✐♠❡s ❢r♦♠ ♦♥❡ ♠♦❞❡

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5

◮ s❡❡ ❋✱ ❑✉❤♥✱ ❏♦✉r❞❛✐♥✱ ▲❡❧✐è✈r❡✱ ❙t♦❧t③ ✭✷✵✶✹✮❀ ❋✱ ❏♦✉r❞❛✐♥✱ ▲❡❧✐è✈r❡✱ ❙t♦❧t③ ✭✷✵✶✺✱✷✵✶✼✱✷✵✶✽✮ ❢♦r st✉❞✐❡s ♦❢ t❤❡s❡ ❲❛♥❣✲▲❛♥❞❛✉ ❜❛s❡s ❛❧❣♦r✐t❤♠s❀ ✐♥❝❧✉❞✐♥❣ s❡❧❢✲t✉♥❡❞ ❙❆ ✉♣❞❛t❡ r✉❧❡s ✭γt ✐s r❛♥❞♦♠✮✳

0.5 e6 1 e6 1.5 e6 2 e6 2.5 e6 3 e6 0.02 0.04 0.06 0.08 0.1 0.12 0.14 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 0.02 0.04 0.06 0.08 0.1 0.12

  • 1
  • 0.5

0.5 1 2e+08 4e+08 6e+08 8e+08 1e+09 1.2e+09 1.4e+09

Xn,1 Iterations

100 1000 10000 100000 1e+06 1e+07 1e+08 1e+09 2 4 6 8 10 12 14

Exit time β

d = 96 d = 48 d = 24 d = 12 d = 6 d = 3 slope 1.25

slide-13
SLIDE 13

❈♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ✶st ❡①❛♠♣❧❡

  • ■t❡r❛t✐✈❡ s❛♠♣❧❡r
  • ❊❛❝❤ ✐t❡r❛t✐♦♥ ❝♦♠❜✐♥❡s ✿

✭✐✮ ❛ s❛♠♣❧✐♥❣ st❡♣ Xt+1 ∼ Pθt(Xt, ·)❀ ❛♥❞ ✭✐✐✮ ❛♥ ♦♣t✐♠✐③❛t✐♦♥ st❡♣ t♦ ✉♣❞❛t❡ t❤❡ ❦♥♦✇♥❧❡❞❣❡ ♦❢ s♦♠❡ ♦♣t✐♠❛❧ ♣❛r❛♠❡t❡r✳

  • ❚❤❡ ♣♦✐♥ts {X1, · · · , Xt, · · · } ❝❛♥ ❜❡ s❡❡♥ ❛s t❤❡ ♦✉t♣✉t ♦❢ ❛ ❝♦♥tr♦❧❧❡❞ ▼❛r❦♦✈

❝❤❛✐♥ E

  • f(Xt+1)|Ft
  • = Pθt(Xt, ·)

Ft := σ(X0:t, θ0) ✇❤❡r❡ Pθ ❤❛s dπθ ❛s ✐ts ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ❞✐str✐❜✉t✐♦♥✳

  • ❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♣❛r❛♠❡t❡r θt ✐s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ ❙❆ s❝❤❡♠❡ ✇✐t❤

✧❝♦♥tr♦❧❧❡❞ ▼❛r❦♦✈✐❛♥✧ ❞②♥❛♠✐❝s θt+1 = θt + γt+1 H(θt, Xt+1)

slide-14
SLIDE 14

✷♥❞ ❊①❛♠♣❧❡✿ ♣❡♥❛❧✐③❡❞ ▼▲ ✐♥ ❧❛t❡♥t ✈❛r✐❛❜❧❡ ♠♦❞❡❧s ✭✶✴✻✮

  • ❆♥ ❡①❛♠♣❧❡ ❢r♦♠ P❤❛r♠❛❝♦❦✐♥❡t✐❝✿

✲ N ♣❛t✐❡♥ts✳ ✲ ❆t t✐♠❡ 0✿ ❞♦s❡ D ♦❢ ❛ ❞r✉❣✳ ✲ ❋♦r ♣❛t✐❡♥t #i✱ ♦❜s❡r✈❛t✐♦♥s Yi1, · · · , YiJi ❣✐✈✐♥❣ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❝♦♥❝❡♥tr❛✲ t✐♦♥ ❛t t✐♠❡s ti1, · · · , tiJi✳

  • ❚❤❡ ♠♦❞❡❧✿

Yij = F

  • tij, Xi
  • + ǫij

ǫij

i.i.d.

∼ N(0, σ2) ✇❤❡r❡ Xi ∈ RL ✐s ♠♦❞❡❧❡❞ ❛s Xi = Ziβ + di ∈ RL di

i.i.d.

∼ NL(0, Ω) ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ǫ• ❛♥❞ Zi ❦♥♦✇♥ ♠❛tr✐① s✳t✳ ❡❛❝❤ r♦✇ ♦❢ Xi ❤❛s ✐♥ ✐♥t❡r❝❡♣t ✭✜①❡❞ ❡✛❡❝t✮ ❛♥❞ ❝♦✈❛r✐❛t❡s✳

  • ❙t❛t✐st✐❝❛❧ ❛♥❛❧②s✐s✿ ✭✐✮ ❡st✐♠❛t✐♦♥ ♦❢ θ = (β, σ2, Ω)✱ ✉♥❞❡r s♣❛rs✐t② ❝♦♥str❛✐♥ts

♦♥ β❀ ✭✐✐✮ s❡❧❡❝t✐♦♥ ♦❢ t❤❡ ❝♦✈❛r✐❛t❡s ❜❛s❡❞ ♦♥ ˆ β✳

slide-15
SLIDE 15

✷♥❞ ❊①✳ ✭✷✴✻✮ P❡♥❛❧✐③❡❞ ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞

  • ❚❤❡ ❧✐❦❡❧✐❤♦♦❞ ♦❢ Y := {Yij, 1 ≤ i ≤ N, 1 ≤ j ≤ Ji} ✐s ♥♦t ❡①♣❧✐❝✐t✿

✲ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ Yi,j ❣✐✈❡♥ Xi ✐s s✐♠♣❧❡❀ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ Xi ✐s s✐♠♣❧❡✳ ✲ ❚❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ❤❛s ❛♥ ❡①♣❧✐❝✐t ❡①♣r❡ss✐♦♥ ✲ ■t ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❧❛t❡♥t ✈❛r✐❛❜❧❡ ♠♦❞❡❧✿ log L(Y ; θ) = log

  • p(Y, x1:N; θ) dν(x1:N)
  • ❙♣❛rs✐t② ❝♦♥str❛✐♥ts ♦♥ t❤❡ ♣❛r❛♠❡t❡r θ✿ t❤r♦✉❣❤ ❛ ♣❡♥❛❧t② t❡r♠ g(θ)
  • ❚❤❡ ♣❡♥❛❧✐③❡❞ ▼▲ ✐s ♦❢ t❤❡ ❢♦r♠

argminΘ (− log L(Y ; θ) + g(θ)) ✇✐t❤ ❛♥ ✐♥tr❛❝t❛❜❧❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✳

slide-16
SLIDE 16

✷♥❞ ❊①✳ ✭✸✴✻✮ ❲❤❛t ❛❜♦✉t ✜rst✲♦r❞❡r ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ❄

  • ❖♥ t❤❡ ❧✐❦❡❧✐❤♦♦❞ t❡r♠✿

✲ ❯s✉❛❧❧② r❡❣✉❧❛r ❡♥♦✉❣❤ s♦ t❤❛t t❤❡ ●r❛❞✐❡♥t ❡①✐sts ❛♥❞ ❁♣r♦♦❢❃ ∇θ log L(Y ; θ) =

∂θ p(Y, x; θ)

p(Y, x; θ) p(Y, x; θ) dµ(x)

p(Y, z; θ) dµ(z)

=

  • ∂θ (log p(Y, x; θ))

dπθ(x)

  • t❤❡ ❛ ♣♦st❡r✐♦r✐ ❞✐str✐❜✉t✐♦♥ ♦❢ x ❣✐✈❡♥ Y

t❤❡ ❞❡♣ ✉♣♦♥ Y ✐s ♦♠✐tt❡❞

✲ t❤❡ ❛ ♣♦st❡r✐♦r✐ ❞✐str✐❜✉t✐♦♥ ✐s ❦♥♦✇♥ ✉♣ t♦ ❛ ♥♦r♠❛❧✐③✐♥❣ ❝♦♥st❛♥t✳

  • ❖♥ t❤❡ ♣❡♥❛❧t② t❡r♠

✲ ▼❛② ❜❡ ♥♦♥ s♠♦♦t❤✱ ❜✉t✿ ❝♦♥✈❡① ❛♥❞ ❧♦✇❡r s❡♠✐✲❝♦♥t✐♥✉♦✉s ✲ ❍❡♥❝❡ ❛ Pr♦①✐♠❛❧ ♦♣❡r❛t♦r ✭✐♠♣❧✐❝✐t ❣r❛❞✐❡♥t✮ ✐s ❛ss♦❝✐❛t❡❞ ✲ ❁❙❡❡ t❤❡ t❛❧❦✱ ♦♥ t✉❡s❞❛② ❛❢t❡r♥♦♦♥❃✳

slide-17
SLIDE 17

✷♥❞ ❊①✳ ✭✹✴✻✮ ❲❤❛t ❛❜♦✉t ❊▼✲❧✐❦❡ ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ❄

  • ❊①♣❡❝t❛t✐♦♥✲▼❛①✐♠✐③❛t✐♦♥ ✐♥tr♦❞✉❝❡❞ t♦ s♦❧✈❡ ❜❡❧♦✇✿

♠♦❞✐❢✐❡❞ ❢♦r ❛ ♠✐♥✐♠✐③❛t✐♦♥ argminθ∈Θ

  • log
  • X p(x; θ)dµ(x) − g(θ)
  • ✇❤❡r❡ t❤❡ ✜rst ♣❛rt ✐s ✉♥tr❛❝t❛❜❧❡❀ ❜② ✐t❡r❛t✐♥❣ t✇♦ st❡♣s

✲ ❊①♣❡❝t❛t✐♦♥ st❡♣ Q(θ, θt) :=

  • log p(x; θ) p(x; θt) dµ(x)

p(z; θt) dµ(z) =

  • log p(x; θ) dπθt(x)

✲ ▼✐♥✐♠✐③❛t✐♦♥ st❡♣ θt+1 := argminθ (−Q(θ, θt) + g(θ)) .

  • θ → Q(θ, θt) ✐s ❛♥ ✐♥t❡❣r❛❧ ✇❤✐❝❤ ✐s ✉♥tr❛❝t❛❜❧❡❀ dπθ ✐s ❦♥♦✇♥ ✉♣ t♦ ❛ ♥♦r♠❛❧✐③✐♥❣

❝♦♥st❛♥t✳

s❡❡ ❋✱▼♦✉❧✐♥❡s ✭✷✵✵✸✮❀ ❋✱❖❧❧✐❡r✱❙❛♠s♦♥ ✭✷✵✶✽✮

slide-18
SLIDE 18

✷♥❞ ❊①✳ ✭✺✴✻✮

  • ❇♦t❤ ✐♥ ❊▼✲❧✐❦❡ ❛♣♣r♦❛❝❤❡s ❛♥❞ ✐♥ ❣r❛❞✐❡♥t✲❜❛s❡❞ ❛♣♣r♦❛❝❤❡s✱

✲ ❢❛❝❡❞ ✇✐t❤ ✉♥tr❛❝t❛❜❧❡ ❛✉①✐❧✐❛r② q✉❛♥t✐t✐❡s ♦❢ t❤❡ ❢♦r♠

  • X H(θ, x) dπθt(x)

✭✶✮ ❛t ✐tr❡r❛t✐♦♥ t ♦❢ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠✳ ✲ ✉♥tr❛❝t❛❜❧❡ ✐♥t❡❣r❛❧❀ dπθ ✐s ♦❢t❡♥ ❦♥♦✇♥ ✉♣ t♦ ❛ ♥♦r♠❛❧✐③✐♥❣ ❝♦♥st❛♥t✳

  • ❲❤❛t ❦✐♥❞ ♦❢ st♦❝❤❛st✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ✭✶✮ ❛t ✐t❡r❛t✐♦♥ t ❄

✲ ◗✉❛❞r❛t✉r❡ t❡❝❤♥✐q✉❡s✿ ♣♦♦r ❜❡❤❛✈✐♦r ✇✳r✳t✳ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ X ✲ ■✳✐✳❞✳ s❛♠♣❧❡s ❢r♦♠ πθt t♦ ❞❡✜♥❡ ❛ ▼♦♥t❡ ❈❛r❧♦ ❛♣♣r♦①✐♠❛t✐♦♥✿ ♥♦t ♣♦ss✐❜❧❡✱ ✐♥ ❣❡♥❡r❛❧✳ ✲ ✉s❡ T s❛♠♣❧❡s ❢r♦♠ ❛ ▼❈▼❈ s❛♠♣❧❡r {Xj,t+1, j ≥ 0} ✇✐t❤ ✉♥✐q✉❡ ✐♥✈✳ ❞✐st✳ dπθt✳

slide-19
SLIDE 19

✷♥❞ ❊①✳ ✭✻✴✻✮ ❉♦❡s ✐t ✇♦r❦ ❄ s❡❡ ❋✱▼♦✉❧✐♥❡s ✭✷✵✵✸✮ ❢♦r ❊▼✲❧✐❦❡ ❛♣♣r♦❛❝❤❡s❀ s❡❡ ❆t❝❤❛❞é✱❋✱▼♦✉❧✐♥❡s ✭✷✵✶✼✮ ❛♥❞ ❋✱❖❧❧✐❡r✱❙❛♠s♦ ✭✷✵✶✽✮ ❢♦r ❣r❛❞✐❡♥t✲❜❛s❡❞ ❛♣♣r♦❛❝❤❡s❀ s❡❡ ❋✱❖❧❧✐❡r✱❙❛♠s♦♥ ✭✷✵✶✽✮ ❢♦r t❤❡ ♣❛r❛❧❧❡❧ ❜❡t✇❡❡♥ ❊▼✲❧✐❦❡ ❛♥❞ ●r❛❞✐❡♥t✲❜❛s❡❞ t❡❝❤♥✐q✉❡s

0.0 0.1 0.2 0.3 0.4 0.5 −3 −2 −1 1 0.50 0.75 1.00 1.25 1.50 0.00 0.25 0.50 0.75 1.00 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 Iteration Parameter value

Proximal MCEM Decreasing Step Size

0.0 0.1 0.2 0.3 0.4 0.5 −3 −2 −1 1 0.5 0.6 0.7 0.8 0.9 1.0 0.00 0.25 0.50 0.75 1.00 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 Iteration Parameter value

Proximal MCEM Adaptive Step Size

0.0 0.1 0.2 0.3 0.4 0.5 −3 −2 −1 1 0.50 0.75 1.00 1.25 0.00 0.25 0.50 0.75 1.00 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 Iteration Parameter value

Proximal SAEM Decreasing Step Size

−0.1 0.0 0.1 0.2 0.3 0.4 0.5 −3 −2 −1 1 0.5 0.6 0.7 0.8 0.9 1.0 0.00 0.25 0.50 0.75 1.00 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 Iteration Parameter value

Proximal SAEM Adaptive Step Size

slide-20
SLIDE 20

❈♦♥❝❧✉s✐♦♥ ♦❢ t❤❡ ✷♥❞ ❡①❛♠♣❧❡

  • ■t❡r❛t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥ t❡❝❤♥✐q✉❡
  • ❊❛❝❤ ✐t❡r❛t✐♦♥ ❝♦♠❜✐♥❡s ✿ ✭✐✮ ❛♥ ✉♣❞❛t❡ ♦❢ t❤❡ ♣❛r❛♠❡t❡r❀ ✭✐✐✮ ❛ s❛♠♣❧✐♥❣ st❡♣

Xj+1,t+1 ∼ Pθt(Xj,t+1, ·) t♦ ❛♣♣r♦①✐♠❛t❡ ❛✉①✐❧✐❛r② q✉❛♥t✐t✐❡s✳

  • ❚❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ {θt}t ✐s t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ st♦❝❤❛st✐❝❛❧❧② ♣❡rt✉r❜❡❞ ✐t❡r❛✲

t✐✈❡ ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠✳ ❆t ❡❛❝❤ ✐t❡r❛t✐♦♥✿ ❛♥ ❡①❛❝t q✉❛♥t✐t②

H(θ, x) dπθt(x)

✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛ ▼♦♥t❡ ❈❛r❧♦ s✉♠

  • H(θ, x) dπθt(x) ≈

1 mt+1

mt+1

  • j=1

H(θ, Xj,t+1)

  • ❚❤❡ ♣♦✐♥ts {Xj,t+1}j s❛t✐s❢②

E

  • f(Xj,t+1)|Ft
  • = P j

θt(X0,t+1, ·)

Ft := σ(X:,0:t, θ0), X0,t+1 = Xmt,t ✇❤❡r❡ Pθ ❤❛s dπθ ❛s ✐ts ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ❞✐str✐❜✉t✐♦♥✳

slide-21
SLIDE 21

❈♦♥❝❧✉s✐♦♥ ♦❢ t❤✐s ✜rst ♣❛rt ✭✶✴✸✮✿ ✐s ❛ t❤❡♦r② r❡q✉✐r❡❞ ❄

slide-22
SLIDE 22

❈♦♥❝❧✉s✐♦♥ ♦❢ t❤✐s ✜rst ♣❛rt ✭✷✴✸✮✿ ✐s ❛ t❤❡♦r② r❡q✉✐r❡❞ ✇❤❡♥ s❛♠♣❧✐♥❣ ❄ ❨❊❙ ✦ ❝♦♥✈❡r❣❡♥❝❡ ❝❛♥ ❜❡ ❧♦st ❜② t❤❡ ❛❞❛♣t✐♦♥ ♠❡❝❛♥✐s♠ ❊✈❡♥ ✐♥ ❛ s✐♠♣❧❡ ❝❛s❡ ✇❤❡♥ ∀θ ∈ Θ, Pθ ✐♥✈❛r✐❛♥t ✇rt dπ, ♦♥❡ ❝❛♥ ❞❡✜♥❡ ❛ s✐♠♣❧❡ ❛❞❛♣t✐♦♥ ♠❡❝❛♥✐s♠ Xt+1|♣❛st1:t ∼ Pθt(Xt, ·) θt ∈ σ(X1:t) s✉❝❤ t❤❛t lim

t E [f(Xt)] =

  • f dπ.

❁♣r♦♦❢❃ ❆ {0, 1}✲✈❛❧✉❡❞ ❝❤❛✐♥ {Xt}t ❞❡✜♥❡❞ ❜② Xt+1 ∼ PXt(Xt, ·) ✇❤❡r❡ t❤❡ tr❛♥s✐t✐♦♥ ♠❛tr✐❝❡s ❛r❡ P0 =

  • t0

(1 − t0) (1 − t0) t0

  • P1 =
  • t1

(1 − t1) (1 − t1) t1

  • ❚❤❡♥ P0 ❛♥❞ P1 ❛r❡ ✐♥✈❛r✐❛♥t ✇✳r✳t [1/2, 1/2] ❜✉t {Xt} ✐s ❛ ▼❛r❦♦✈ ❝❤❛✐♥ ✐♥✈❛r✐❛♥t ✇✳r✳t✳ [t1, t0]
slide-23
SLIDE 23

❈♦♥❝❧✉s✐♦♥ ♦❢ t❤✐s ✜rst ♣❛rt ✭✸✴✸✮✿ ✐s ❛ t❤❡♦r② r❡q✉✐r❡❞ ✇❤❡♥ ♦♣t✐♠✐③✐♥❣ ❄ ❨❊❙ ✦ ❯♥❢♦rt✉♥❛t❡❧② ✱

  • ❛ ❜✐❛s❡❞ ❛♣♣r♦①✐♠❛t✐♦♥ ❁♣r♦♦❢❃

E

 

1 mt+1

mt+1

  • j=1

H(θ, Xj,t+1)

  • Ft

  = ? =

  • X H(θ, x) dπθt(x)
  • ❋♦r ❛ r❡❞✉❝❡❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st✿ ❛ ❜✐❛s ✇❤✐❝❤ ✇❡ ✇♦✉❧❞ ❧✐❦❡ ◆❖❚ ✈❛♥✐s❤✐♥❣

✐✳❡✳ mt = m(= 1)✳ ❊①✳ ❙t♦❝❤❛st✐❝ ❆♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤ ❝♦♥tr♦❧❧❡❞ ▼❛r❦♦✈✐❛♥ ❞②♥❛♠✐❝s θt+1 = θt + γt+1 H(θt, Xt+1) Xt+1 ∼ Pθt(Xt, ·) = θt + γt+1

  • H(θt, x)dπθt(x)
  • h(θt)

+γt+1

  • H(Xt+1, θt) − h(θt)
  • ♥♦♥ ❝❡♥t❡r❡❞