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sst rt s - - PowerPoint PPT Presentation

Sep 16, 2022 476 likes •1.09k views

sst rt s t t strts r Ps rtt

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SLIDE 1

❉✐ss✐♣❛t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✐♥ ❝❤❛♥♥❡❧s ✇✐t❤ ✐♥♣✉t ❝♦♥str❛✐♥ts

❨✉r② P♦❧②❛♥s❦✐②

❉❡♣❛rt♠❡♥t ♦❢ ❊❊❈❙ ▼■❚ ②♣❅♠✐t✳❡❞✉ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❨✐❤♦♥❣ ❲✉ ✭❯■❯❈✮

❆♣r ✷✽✱ ✷✵✶✹

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶

slide-2
SLIDE 2

❘❡❧❛②✐♥❣ ❞❛t❛ ❛❝r♦ss ❛ ❝❤❛✐♥ ♦❢ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s

f1 X0 X1

  • ❖r✐❣✐♥❛❧ ♠❡ss❛❣❡✿ X0
  • ❊♥❝♦❞❡rs fk s❛t✐s❢②✐♥❣ ♣♦✇❡r ❝♦♥str❛✐♥t

E[X2

k] ≤ P

◆♦✐s❡✿

✐✳✐✳❞✳

  • ♦❛❧✿ ❉❡s✐❣♥

✬s s♦ t❤❛t

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷

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SLIDE 3

❘❡❧❛②✐♥❣ ❞❛t❛ ❛❝r♦ss ❛ ❝❤❛✐♥ ♦❢ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s

f1 X0 X1 + Z1 Y1

  • ❖r✐❣✐♥❛❧ ♠❡ss❛❣❡✿ X0
  • ❊♥❝♦❞❡rs fk s❛t✐s❢②✐♥❣ ♣♦✇❡r ❝♦♥str❛✐♥t

E[X2

k] ≤ P

  • ◆♦✐s❡✿ Zk

✐✳✐✳❞✳

∼ N(0, 1)

  • ♦❛❧✿ ❉❡s✐❣♥

✬s s♦ t❤❛t

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷

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SLIDE 4

❘❡❧❛②✐♥❣ ❞❛t❛ ❛❝r♦ss ❛ ❝❤❛✐♥ ♦❢ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s

f1 X0 X1 + Z1 Y1 f2 X2 + Z2 Y2

  • ❖r✐❣✐♥❛❧ ♠❡ss❛❣❡✿ X0
  • ❊♥❝♦❞❡rs fk s❛t✐s❢②✐♥❣ ♣♦✇❡r ❝♦♥str❛✐♥t

E[X2

k] ≤ P

  • ◆♦✐s❡✿ Zk

✐✳✐✳❞✳

∼ N(0, 1)

  • ♦❛❧✿ ❉❡s✐❣♥

✬s s♦ t❤❛t

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷

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SLIDE 5

❘❡❧❛②✐♥❣ ❞❛t❛ ❛❝r♦ss ❛ ❝❤❛✐♥ ♦❢ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s

f1 X0 X1 + Z1 Y1 f2 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

  • ❖r✐❣✐♥❛❧ ♠❡ss❛❣❡✿ X0
  • ❊♥❝♦❞❡rs fk s❛t✐s❢②✐♥❣ ♣♦✇❡r ❝♦♥str❛✐♥t

E[X2

k] ≤ P

  • ◆♦✐s❡✿ Zk

✐✳✐✳❞✳

∼ N(0, 1)

  • ●♦❛❧✿ ❉❡s✐❣♥ f✬s s♦ t❤❛t X0 ≈ Xn

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷

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SLIDE 6

❈❛♥ ✇❡ ♣r❡s❡r✈❡ ❛♥② ✐♥❢♦r♠❛t✐♦♥❄

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

■♥t✉✐t✐♦♥ ❡❛❝❤ st❛❣❡ ❤❛s ✜♥✐t❡ ❡♥❡r❣② ❜✉❞❣❡t ❝❛♥♥♦t ❞❡♥♦✐s❡ ❝♦♠♣❧❡t❡❧② ♥♦✐s❡ ❛❝❝✉♠✉❧❛t❡s ❛♥❞ ❦✐❧❧s ❞❡♣❡♥❞❡♥❝② ❈♦♥❥❡❝t✉r❡✿ ❛s②♠♣t♦t✐❝ ❞❡❝♦✉♣❧✐♥❣ ❘❡❣❛r❞❧❡ss ♦❢ ♦r

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✸

slide-7
SLIDE 7

❈❛♥ ✇❡ ♣r❡s❡r✈❡ ❛♥② ✐♥❢♦r♠❛t✐♦♥❄

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

■♥t✉✐t✐♦♥ ❡❛❝❤ st❛❣❡ ❤❛s ✜♥✐t❡ ❡♥❡r❣② ❜✉❞❣❡t ⇒ ❝❛♥♥♦t ❞❡♥♦✐s❡ ❝♦♠♣❧❡t❡❧② ⇒ ♥♦✐s❡ ❛❝❝✉♠✉❧❛t❡s ❛♥❞ ❦✐❧❧s ❞❡♣❡♥❞❡♥❝② ❈♦♥❥❡❝t✉r❡✿ ❛s②♠♣t♦t✐❝ ❞❡❝♦✉♣❧✐♥❣ ❘❡❣❛r❞❧❡ss ♦❢ ♦r

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✸

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SLIDE 8

❈❛♥ ✇❡ ♣r❡s❡r✈❡ ❛♥② ✐♥❢♦r♠❛t✐♦♥❄

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

■♥t✉✐t✐♦♥ ❡❛❝❤ st❛❣❡ ❤❛s ✜♥✐t❡ ❡♥❡r❣② ❜✉❞❣❡t ⇒ ❝❛♥♥♦t ❞❡♥♦✐s❡ ❝♦♠♣❧❡t❡❧② ⇒ ♥♦✐s❡ ❛❝❝✉♠✉❧❛t❡s ❛♥❞ ❦✐❧❧s ❞❡♣❡♥❞❡♥❝② ❈♦♥❥❡❝t✉r❡✿ ❛s②♠♣t♦t✐❝ ❞❡❝♦✉♣❧✐♥❣ ❘❡❣❛r❞❧❡ss ♦❢ X0 ♦r {f1, . . . , fn} PX0Xn ≈ PX0PXn n ≫ 1

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✸

slide-9
SLIDE 9

❍♦✇ t♦ ❣❛✉❣❡ ❞❡❝♦✉♣❧✐♥❣❄

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

PX0Xn ≈ PX0PXn ◗✉❛♥t✐t❛t✐✈❡❧②✿

  • ✭❚❱✮

TV(PX0Xn, PX0PXn) → 0

  • ✭❑▲✮

D(PX0XnPX0PXn) → 0

  • ✭❈♦rr❡❧❛t✐♦♥✮

ρ(X0, Xn) → 0

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✹

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SLIDE 10

❍♦✇ t♦ ❣❛✉❣❡ ❞❡❝♦✉♣❧✐♥❣❄

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

PX0Xn ≈ PX0PXn ◗✉❛♥t✐t❛t✐✈❡❧②✿

  • ✭❚❱✮

TV(PX0Xn, PX0PXn) → 0

  • ✭❑▲✮

D(PX0XnPX0PXn) → 0 ❊q✉✐✈✳✿ I(X0; Xn) → 0

  • ✭❈♦rr❡❧❛t✐♦♥✮

ρ(X0, Xn) → 0 ❊q✉✐✈✳✿ mmse(X0|Xn) → Var[X0]

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✹

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SLIDE 11

❍♦✇ t♦ ❣❛✉❣❡ ❞❡❝♦✉♣❧✐♥❣❄

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

PX0Xn ≈ PX0PXn ◗✉❛♥t✐t❛t✐✈❡❧②✿

  • ✭❚❱✮

TV(PX0Xn, PX0PXn) → 0

  • ✭❑▲✮

D(PX0XnPX0PXn) → 0 ❊q✉✐✈✳✿ I(X0; Xn) → 0

  • ✭❈♦rr❡❧❛t✐♦♥✮

ρ(X0, Xn) → 0 ❊q✉✐✈✳✿ mmse(X0|Xn) → Var[X0]

KL TV

Pinsker ineq. Rate-distortion

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✹

slide-12
SLIDE 12

❆tt❡♠♣t ✶✿ ❞❛t❛ ♣r♦❝❡ss✐♥❣

slide-13
SLIDE 13

❉❛t❛ ♣r♦❝❡ss✐♥❣ ✐♥❡q✉❛❧✐t②

  • ❑▲ ❞✐✈❡r❣❡♥❝❡

PX QX X Y PY QY PY |X

⇒ D(PY ||QY ) ≤ D(PX||QX) ✭❛♣♣❧✐❡s t♦ ❛♥② f✲❞✐✈❡r❣❡♥❝❡✱ ✐♥ ♣❛rt✐❝✉❧❛r ❚❱✮ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✻

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SLIDE 14

❉❛t❛ ♣r♦❝❡ss✐♥❣ ✐♥❡q✉❛❧✐t②

  • ❑▲ ❞✐✈❡r❣❡♥❝❡

PX QX X Y PY QY PY |X

⇒ D(PY ||QY ) ≤ D(PX||QX) ✭❛♣♣❧✐❡s t♦ ❛♥② f✲❞✐✈❡r❣❡♥❝❡✱ ✐♥ ♣❛rt✐❝✉❧❛r ❚❱✮

  • ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥

U → X → Y ⇒ I(U; Y ) ≤ I(U; X)

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✻

slide-15
SLIDE 15

❆♣♣❧②✐♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

I(X0; Xn) ≤ I(X0; Yn−1) ≤ I(X0; Xn−1) ≤ · · · ≤ I(X0; X1) ❖❑✿ ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣ ◆❡❡❞ ❛ q✉❛♥t✐t❛t✐✈❡ ❞❛t❛ ♣r♦❝❡ss✐♥❣ ✐♥❡q✉❛❧✐t②

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✼

slide-16
SLIDE 16

❆♣♣❧②✐♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

I(X0; Xn) ≤ I(X0; Yn−1) ≤ I(X0; Xn−1) ≤ · · · ≤ I(X0; X1)

  • ❖❑✿ I(X0; Xn) ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣
  • ◆❡❡❞ ❛ q✉❛♥t✐t❛t✐✈❡ ❞❛t❛ ♣r♦❝❡ss✐♥❣ ✐♥❡q✉❛❧✐t②

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✼

slide-17
SLIDE 17

❆tt❡♠♣t ✷✿ str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣

slide-18
SLIDE 18

❙tr♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ ✐♥❡q✉❛❧✐t②

  • ❑▲ ❞✐✈❡r❣❡♥❝❡

PX QX X Y PY QY PY |X

⇒ D(PY ||QY ) ≤ ηKLD(PX||QX) ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✾

slide-19
SLIDE 19

❙tr♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ ✐♥❡q✉❛❧✐t②

  • ❑▲ ❞✐✈❡r❣❡♥❝❡

PX QX X Y PY QY PY |X

⇒ D(PY ||QY ) ≤ ηKLD(PX||QX)

  • ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥

U → X → Y ⇒ I(U; Y ) ≤ ηKLI(U; X)

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✾

slide-20
SLIDE 20

❙tr♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ ✐♥❡q✉❛❧✐t②

PX QX X Y PY QY PY |X

  • ❋♦r ✜①❡❞ PY |X✱ ❝♦♥tr❛❝t✐♦♥ r❛t✐♦✿

ηKL = sup

PX=QX

D(PY ||QY ) D(PX||QX) = sup

U→X→Y

I(U; Y ) I(U; X)

  • ❆❤❧s✇❡❞❡✲●á❝s✿ ❢♦r ❞✐s❝r❡t❡ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❝❤❛♥♥❡❧s ηKL < 1

❤②♣❡r❝♦♥tr❛❝t✐✈✐t② r❛t✐♦ r❡❧❛t❡❞ t♦ ▲❙■ ❡t❝✿ ❬❲✐ts❡♥❤❛✉s❡♥❪✱ ❬❊r❦✐♣✲❈♦✈❡r❪✱ ❬❈♦❤❡♥✲❑❡♠♣❡r♠❛♥♥✲❩❜➔❣❛♥✉❪✱ ❬❉❡❧ ▼♦r❛❧✲▲❡❞♦✉①✲▼✐❝❧♦❪✱ ❬❆♥❛♥t❤❛r❛♠✲●♦❤❛r✐✲❑❛♠❛t❤✲◆❛✐r❪✱ ❬❘❛❣✐♥s❦②❪✱ ✳✳✳

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✵

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SLIDE 21

❙tr♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ ✐♥❡q✉❛❧✐t②

PX QX X Y PY QY PY |X

  • ❋♦r ✜①❡❞ PY |X✱ ❝♦♥tr❛❝t✐♦♥ r❛t✐♦✿

ηKL = sup

PX=QX

D(PY ||QY ) D(PX||QX) = sup

U→X→Y

I(U; Y ) I(U; X)

  • ❆❤❧s✇❡❞❡✲●á❝s✿ ❢♦r ❞✐s❝r❡t❡ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❝❤❛♥♥❡❧s ηKL < 1
  • . . .

ηKL = ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② r❛t✐♦

  • . . .

r❡❧❛t❡❞ t♦ ▲❙■ ❡t❝✿ ❬❲✐ts❡♥❤❛✉s❡♥❪✱ ❬❊r❦✐♣✲❈♦✈❡r❪✱ ❬❈♦❤❡♥✲❑❡♠♣❡r♠❛♥♥✲❩❜➔❣❛♥✉❪✱ ❬❉❡❧ ▼♦r❛❧✲▲❡❞♦✉①✲▼✐❝❧♦❪✱ ❬❆♥❛♥t❤❛r❛♠✲●♦❤❛r✐✲❑❛♠❛t❤✲◆❛✐r❪✱ ❬❘❛❣✐♥s❦②❪✱ ✳✳✳

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✵

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SLIDE 22

❙tr✐❝t ❝♦♥tr❛❝t✐♦♥

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

■❢ ηKL < 1✱ t❤❡♥ I(X0; Xn) ≤ I(X0; Yn−1) ≤ ηKLI(X0; Xn−1) ≤ · · · ≤ ηn−1

KL I(X0; X1)

→ 0 ❡①♣♦♥❡♥t✐❛❧❧② ❙❛❞✿ ❢♦r ✭❆❲●◆✮

  • ♦♦❞✿

❢♦r ✭❛♠♣❧✐t✉❞❡ ❝♦♥str✳✮

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✶

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SLIDE 23

❙tr✐❝t ❝♦♥tr❛❝t✐♦♥

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

■❢ ηKL < 1✱ t❤❡♥ I(X0; Xn) ≤ I(X0; Yn−1) ≤ ηKLI(X0; Xn−1) ≤ · · · ≤ ηn−1

KL I(X0; X1)

→ 0 ❡①♣♦♥❡♥t✐❛❧❧②

  • ❙❛❞✿

ηKL = 1 ❢♦r Y = X + Z, E[|X|2] ≤ P ✭❆❲●◆✮

  • ♦♦❞✿

❢♦r ✭❛♠♣❧✐t✉❞❡ ❝♦♥str✳✮

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✶

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SLIDE 24

❙tr✐❝t ❝♦♥tr❛❝t✐♦♥

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

■❢ ηKL < 1✱ t❤❡♥ I(X0; Xn) ≤ I(X0; Yn−1) ≤ ηKLI(X0; Xn−1) ≤ · · · ≤ ηn−1

KL I(X0; X1)

→ 0 ❡①♣♦♥❡♥t✐❛❧❧②

  • ❙❛❞✿

ηKL = 1 ❢♦r Y = X + Z, E[|X|2] ≤ P ✭❆❲●◆✮

  • ●♦♦❞✿ ηKL < 1 ❢♦r Y = X + Z,

|X| ≤ √ P ✭❛♠♣❧✐t✉❞❡ ❝♦♥str✳✮

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✶

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SLIDE 25

❉♦❜r✉s❤✐♥ ❝♦❡✣❝✐❡♥t

  • t♦t❛❧ ✈❛r✐❛t✐♦♥

TV(P, Q) = 1 2

  • |dP − dQ|
  • ❉♦❜r✉s❤✐♥ ❝♦❡✣❝✐❡♥t

ηTV = sup

PX=QX

TV(PY , QY ) TV(PX, QX) = sup

x,x′ TV(PY |X=x, PY |X=x′).

  • ▼❛r❦♦✈ ♣r♦❝❡ss✱ st❛t ♣❤②s✐❝s✱ ✉♥✐q✉❡♥❡ss ♦❢ ●✐❜❜s ♠❡❛s✉r❡
  • ❬❈♦❤❡♥✲❑❡♠♣❡r♠❛♥♥✲❩❜➔❣❛♥✉✬✾✽❪

ηKL ≤ ηTV

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✷

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SLIDE 26

❉♦❜r✉s❤✐♥ ❝♦❡✣❝✐❡♥t

  • ■❢ |X| ≤ A ❛✳s✳✱

ηKL ≤ ηTV = sup

x,x′∈[−A,A]

TV(N(0, x), N(0, x′)) = TV(N(−A, 1), N(A, 1)) < 1.

  • ❆♠♣❧✐t✉❞❡ ❝♦♥str❛✐♥t ⇒ ❡①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♠✉t✉❛❧

✐♥❢♦r♠❛t✐♦♥✱ ❡t❝✳

  • ❙❤♦rt ♣r♦♦❢ ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ str♦♥❣❡st r❡s✉❧ts ♦♥ ●❛✉ss✐❛♥

❧✐♥❡✲♥❡t✇♦r❦s ❬❙✉❜r❛♠❛♥✐❛♥ ❡t ❛❧✳ ✬✶✶✱ ✬✶✷❪ ❲❤❛t ❛❜♦✉t ❛✈❡r❛❣❡ ♣♦✇❡r ❝♦♥str❛✐♥t❄

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✸

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SLIDE 27

❉♦❜r✉s❤✐♥ ❝♦❡✣❝✐❡♥t

  • ■❢ |X| ≤ A ❛✳s✳✱

ηKL ≤ ηTV = sup

x,x′∈[−A,A]

TV(N(0, x), N(0, x′)) = TV(N(−A, 1), N(A, 1)) < 1.

  • ❆♠♣❧✐t✉❞❡ ❝♦♥str❛✐♥t ⇒ ❡①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♠✉t✉❛❧

✐♥❢♦r♠❛t✐♦♥✱ ❡t❝✳

  • ❙❤♦rt ♣r♦♦❢ ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ str♦♥❣❡st r❡s✉❧ts ♦♥ ●❛✉ss✐❛♥

❧✐♥❡✲♥❡t✇♦r❦s ❬❙✉❜r❛♠❛♥✐❛♥ ❡t ❛❧✳ ✬✶✶✱ ✬✶✷❪

  • ❲❤❛t ❛❜♦✉t ❛✈❡r❛❣❡ ♣♦✇❡r ❝♦♥str❛✐♥t❄

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✸

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SLIDE 28

❆tt❡♠♣t ✸✿ tr✉♥❝❛t✐♦♥ ❛r❣✉♠❡♥ts

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SLIDE 29

❋❆■▲✳

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SLIDE 30

◆♦ ❝♦♥tr❛❝t✐♦♥✦

  • ◆♦ ❝♦♥tr❛❝t✐♦♥ ❡✈❡♥ ✇✐t❤ ♣♦✇❡r ❝♦♥str❛✐♥t

   PX = (1 − t)δ0 + tδ√

P/t

QX = (1 − t)δ0 + tδ−√

P/t

⇒ TV(PY , QY ) TV(PX, QX) → 1

  • s❛♠❡ ❢♦r ❞✐✈❡r❣❡♥❝❡ ❛♥❞ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥
  • ▼❛②❜❡ s✉❜❡①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡❄

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✻

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SLIDE 31

▼❛✐♥ r❡s✉❧ts✿ ■♥❢♦r♠❛t✐♦♥ ♥❡✈❡rt❤❡❧❡ss ❞✐ss✐♣❛t❡s

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

❚❤❡♦r❡♠ ❋♦r ❛♥② ♣r♦❝❡ss♦rs {fn}✱ TV(PX0Xn, PX0PXn) ≤ CP log n ρ(X0, Xn) ≤

  • CP log log n

log n I(X0; Xn) ≤ CP log log n log n ❚❤❡♦r❡♠ ❚❤❡r❡ ❡①✐st ✱ ✇❤❡r❡ ❛r❡ ✉♥✐✈❡rs❛❧ ❝♦♥st❛♥ts✳ ❊✈❡r②t❤✐♥❣ ❝♦♥✈❡r❣❡s ❜✉t ü❜❡r s❧♦✇❧②✦

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✼

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SLIDE 32

▼❛✐♥ r❡s✉❧ts✿ ■♥❢♦r♠❛t✐♦♥ ♥❡✈❡rt❤❡❧❡ss ❞✐ss✐♣❛t❡s

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

❚❤❡♦r❡♠ ❋♦r ❛♥② ♣r♦❝❡ss♦rs {fn}✱ TV(PX0Xn, PX0PXn) ≤ CP log n ρ(X0, Xn) ≤

  • CP log log n

log n I(X0; Xn) ≤ CP log log n log n ❚❤❡♦r❡♠ ❚❤❡r❡ ❡①✐st {fn}✱ TV(PX0Xn, PX0PXn) ≥ cP log n ρ(X0, Xn) ≥

  • cP

log n I(X0; Xn) ≥ cP log n ✇❤❡r❡ c, C ❛r❡ ✉♥✐✈❡rs❛❧ ❝♦♥st❛♥ts✳ ❊✈❡r②t❤✐♥❣ ❝♦♥✈❡r❣❡s ❜✉t ü❜❡r s❧♦✇❧②✦

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✼

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SLIDE 33

▼❛✐♥ r❡s✉❧ts✿ ■♥❢♦r♠❛t✐♦♥ ♥❡✈❡rt❤❡❧❡ss ❞✐ss✐♣❛t❡s

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

❚❤❡♦r❡♠ ❋♦r ❛♥② ♣r♦❝❡ss♦rs {fn}✱ TV(PX0Xn, PX0PXn) ≤ CP log n ρ(X0, Xn) ≤

  • CP log log n

log n I(X0; Xn) ≤ CP log log n log n ❚❤❡♦r❡♠ ❚❤❡r❡ ❡①✐st {fn}✱ TV(PX0Xn, PX0PXn) ≥ cP log n ρ(X0, Xn) ≥

  • cP

log n I(X0; Xn) ≥ cP log n ✇❤❡r❡ c, C ❛r❡ ✉♥✐✈❡rs❛❧ ❝♦♥st❛♥ts✳ ❊✈❡r②t❤✐♥❣ ❝♦♥✈❡r❣❡s ❜✉t ü❜❡r s❧♦✇❧②✦

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✼

slide-34
SLIDE 34

Pr♦♦❢ ✐❞❡❛s

slide-35
SLIDE 35

❘❡✈✐s✐t str♦♥❣ ❞❛t❛✲♣r♦❝❡ss✐♥❣

PX QX X Y PY QY PY |X

  • ❙tr♦♥❣ ❉✳P✳✿

D(PY QY ) ≤ ηKLD(PXQX)

  • ▼♦r❡ ♣r❡❝✐s❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✿ ❥♦✐♥t r❛♥❣❡

(PX, QX) → (D(PX||QX), D(PY ||QY )) ∈ R2

+

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✶✾

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SLIDE 36

❏♦✐♥t r❛♥❣❡

❞❛t❛ ♣r♦❝❡ss✐♥❣ str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ s❧♦♣❡ ❂

♦✉t♣✉t ❞✐✈❡r❣❡♥❝❡ ✐♥♣✉t ❞✐✈❡r❣❡♥❝❡

❜♦✉♥❞❛r②

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✵

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SLIDE 37

❏♦✐♥t r❛♥❣❡

❞❛t❛ ♣r♦❝❡ss✐♥❣ str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ s❧♦♣❡ ❂

♦✉t♣✉t ❞✐✈❡r❣❡♥❝❡ ✐♥♣✉t ❞✐✈❡r❣❡♥❝❡

❜♦✉♥❞❛r②

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✵

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SLIDE 38

❏♦✐♥t r❛♥❣❡

❞❛t❛ ♣r♦❝❡ss✐♥❣ str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ s❧♦♣❡ ❂ η < 1

♦✉t♣✉t ❞✐✈❡r❣❡♥❝❡ ✐♥♣✉t ❞✐✈❡r❣❡♥❝❡

❜♦✉♥❞❛r②

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✵

slide-39
SLIDE 39

❏♦✐♥t r❛♥❣❡

❞❛t❛ ♣r♦❝❡ss✐♥❣ str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ s❧♦♣❡ ❂

♦✉t♣✉t ❞✐✈❡r❣❡♥❝❡ ✐♥♣✉t ❞✐✈❡r❣❡♥❝❡

❜♦✉♥❞❛r②

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✵

slide-40
SLIDE 40

❏♦✐♥t r❛♥❣❡

❞❛t❛ ♣r♦❝❡ss✐♥❣ str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ s❧♦♣❡ ❂

♦✉t♣✉t ❞✐✈❡r❣❡♥❝❡ ✐♥♣✉t ❞✐✈❡r❣❡♥❝❡

❜♦✉♥❞❛r②

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✵

slide-41
SLIDE 41

❏♦✐♥t r❛♥❣❡

❞❛t❛ ♣r♦❝❡ss✐♥❣ str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ s❧♦♣❡ ❂

♦✉t♣✉t ❞✐✈❡r❣❡♥❝❡ ✐♥♣✉t ❞✐✈❡r❣❡♥❝❡

❜♦✉♥❞❛r②

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✵

slide-42
SLIDE 42

❏♦✐♥t r❛♥❣❡

❞❛t❛ ♣r♦❝❡ss✐♥❣ str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ s❧♦♣❡ ❂

♦✉t♣✉t ❞✐✈❡r❣❡♥❝❡ ✐♥♣✉t ❞✐✈❡r❣❡♥❝❡

❜♦✉♥❞❛r②

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✵

slide-43
SLIDE 43

❏♦✐♥t r❛♥❣❡

❞❛t❛ ♣r♦❝❡ss✐♥❣ str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ s❧♦♣❡ ❂

♦✉t♣✉t ❞✐✈❡r❣❡♥❝❡ ✐♥♣✉t ❞✐✈❡r❣❡♥❝❡

❜♦✉♥❞❛r②

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✵

slide-44
SLIDE 44

❏♦✐♥t r❛♥❣❡

❞❛t❛ ♣r♦❝❡ss✐♥❣ str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ s❧♦♣❡ ❂

♦✉t♣✉t ❞✐✈❡r❣❡♥❝❡ ✐♥♣✉t ❞✐✈❡r❣❡♥❝❡

❜♦✉♥❞❛r② FKL(t)

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✵

slide-45
SLIDE 45

❈♦♥✈❡r❣❡♥❝❡ ✇✐t❤♦✉t ❝♦♥tr❛❝t✐♦♥

♦✉t♣✉t ❞✐✈❡r❣❡♥❝❡ ✐♥♣✉t ❞✐✈❡r❣❡♥❝❡

P✉♥❝❤❧✐♥❡✿ ■❢ D(PXQX) ✈s D(PX ∗ NQX ∗ N) ❝✉r✈❡❞ ⇒ ❞♦♥❡ ✭❑▲→ 0✮✳ ❙❛❞ ♥❡✇s✿ ❋♦r ❑▲ t❤❡ ❜♦✉♥❞❛r②

  • ♦♦❞ ♥❡✇s✿ ❋♦r ❚❱ t❤❡ ❜♦✉♥❞❛r②

✭✦✮ ✕ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ♦❢ t❤❡ ❝❤❛♥♥❡❧

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✶

slide-46
SLIDE 46

❈♦♥✈❡r❣❡♥❝❡ ✇✐t❤♦✉t ❝♦♥tr❛❝t✐♦♥

♦✉t♣✉t ❞✐✈❡r❣❡♥❝❡ ✐♥♣✉t ❞✐✈❡r❣❡♥❝❡

P✉♥❝❤❧✐♥❡✿ ■❢ D(PXQX) ✈s D(PX ∗ NQX ∗ N) ❝✉r✈❡❞ ⇒ ❞♦♥❡ ✭❑▲→ 0✮✳

  • ❙❛❞ ♥❡✇s✿

❋♦r ❑▲ t❤❡ ❜♦✉♥❞❛r② FKL(t) = t

  • ●♦♦❞ ♥❡✇s✿ ❋♦r ❚❱ t❤❡ ❜♦✉♥❞❛r② FTV(t) < t

✭✦✮ FTV(t), t ∈ [0, 1] ✕ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ♦❢ t❤❡ ❝❤❛♥♥❡❧

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✶

slide-47
SLIDE 47

❙tr❛t❡❣②

♣♦✇❡r ❝♦♥str❛✐♥t ❝❛♥♥♦t tr❛♥s♠✐t ✶ ❜✐t ❞❡❝♦✉♣❧✐♥❣ ✐♥ ❚❱ ❞❡❝♦✉♣❧✐♥❣ ✐♥ ❑▲ ✭▼■ → 0✮ ❞❡❝♦rr❡❧❛t✐♦♥ ρ(X0, Xn) → 0

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✷

slide-48
SLIDE 48

❙tr❛t❡❣②

♣♦✇❡r ❝♦♥str❛✐♥t ❝❛♥♥♦t tr❛♥s♠✐t ✶ ❜✐t ❞❡❝♦✉♣❧✐♥❣ ✐♥ ❚❱ ❞❡❝♦✉♣❧✐♥❣ ✐♥ ❑▲ ✭▼■ → 0✮ ❞❡❝♦rr❡❧❛t✐♦♥ ρ(X0, Xn) → 0

  • ❛✉ss✐❛♥ ♥♦✐s❡
  • ❡♥❡r❛❧

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✷

slide-49
SLIDE 49

❚r❛♥s♠✐t ♦♥❡ ❜✐t

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

  • X0 = ±1 ❡q✉✐♣r♦❜❛❜❧②
  • ❈♦♥❞✐t✐♦♥❛❧ ❧❛✇✿ P(·|X0 = 1) = P ❛♥❞ P(·|X0 = −1) = Q
  • ❘❡❞✉❝❡ t♦ t❡st✐♥❣ PXn ✈s✳ QXn

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✸

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SLIDE 50

❉♦❜r✉s❤✐♥ ❝✉r✈❡

PX QX X Y

+

Z ∼ N (0, 1)

PY QY

FTV(t)

t = TV(PX, QX) TV(PY , QY )

  • ❯♣♣❡r ❜♦✉♥❞❛r②✿ FTV : [0, 1] → [0, 1]

FTV(t) = sup

  • TV(PY , QY ) : TV(PX, QX) ≤ t,

EPX|X|2 + EQX|X|2 ≤ 2P

  • .
  • ❉♦❜r✉s❤✐♥ ❝♦❡✣❝✐❡♥t ηTV ❂ ♠❛①✐♠❛❧ s❧♦♣❡ ♦❢ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ FTV

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✹

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SLIDE 51

❉♦❜r✉s❤✐♥ ❝✉r✈❡ ♦❢ ❆❲●◆

❚❤❡♦r❡♠ ❯♥❞❡r ♣♦✇❡r ❝♦♥str❛✐♥t E|X|2 ≤ P✱ FTV(t) = t

  • 1 − 2Q
  • P

t

  • ✇❤❡r❡ Q = ❝♦♠♣❧❡♠❡♥t❛r② ♥♦r♠❛❧ ❈❉❋✳
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

FTV ◆♦t❡✿

  • FTV s♠♦♦t❤ ❜✉t ♥♦t ❛♥❛❧②t✐❝✿ FTV′(0) = 1✱ FTV(k)(0) = 0
  • ✐t❡r❛t✐✈❡ ♠❛♣♣✐♥❣✿

TV(PXn, QXn) ≤ FTV ◦ FTV · · · ◦ FTV(1) = O

  • 1

log n

  • → 0

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✺

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SLIDE 52

Pr♦♦❢ ✐❞❡❛s

FTV(t) = t

  • 1 − 2Q
  • P

t

  • Pr♦♦❢✳
  • ●✐✈❡♥ PX, QX ✇✐t❤ TV(PX, QX) = t
  • ◆❡❡❞✿ ✉♣♣❡r✲❜♦✉♥❞ TV(PX ∗ N, QX ∗ N)
  • ❋✐rst ❝♦✉♣❧❡ PX t♦ QX✿

π[X = X′] = TV(PX, QX) = t

  • ■❞❡❛✿ ✇❤❡♥ |X − X′| ≪ 1

⇒ Y ≈ Y ′ ✇❤❡♥ |X − X′| ≫ 1 ⇒ ❄❄❄ ❜✉t✿ s✐♥❝❡ E|X|2 ≤ P t❤✐s ✐s r❛r❡✦

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✻

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SLIDE 53

❉❡t❛✐❧s✳

  • ■♥t❡r♣❧❛② ❜❡t✇❡❡♥ ❊✉❝❧✐❞❡❛♥ ❞✐st❛♥❝❡ ❛♥❞ TV✿

θ(x) TV(N(0, 1), N(x, 1))

  • ◆❡①t ♥♦t✐❝❡✿

TV(PX ∗ N, PY ∗ N) ≤ E[θ(X − X′)]

❬❚❱ ✕ ❲❛ss❡rst❡✐♥ ❞✐st❛♥❝❡✦❪

= E[θ(X − X′)|X = X′] · t ≤ tθ

  • 2
  • P

t

  • ❬θ ✐s ❝♦♥❝❛✈❡✦❪
  • ▲✉❝❦② t✇✐❝❡✿ ❝❤♦✐❝❡ ♦❢ TV✱ ✉♥✐♠♦❞❛❧ ♥♦✐s❡✳
  • ❋♦r t❤❡ ❧♦✇❡r ❜♦✉♥❞✿ t❛❦❡ (1 − t)δ0 + tδ±√

P/T ✳

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✼

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SLIDE 54
  • ❡♥❡r❛❧✐③❛t✐♦♥s
  • ❝♦st✿ E|X|2 ≤ P

→ ❛♥② ❝♦♥✈❡① ❝♦st

  • ♥♦✐s❡✿ ●❛✉ss✐❛♥

→ ❛♥② ✉♥✐♠♦❞❛❧ ❞❡♥s✐t② ❛♥❞ ♠♦r❡

  • ❝❤❛♥♥❡❧s✿ s❝❛❧❛r✲✐♥♣✉t

→ ✈❡❝t♦r✲✐♥♣✉t ✭❡✈❡♥ ∞✲❞✐♠✦✮ ▼♦r❡ t❤❛♥ ②♦✉ ✇❛♥t t♦ ❦♥♦✇ ❤❡r❡✿

P✳ ✫ ❨✐❤♦♥❣ ❲✉ ✭✷✵✶✹✮✳ ❉✐ss✐♣❛t✐♦♥ ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✐♥ ❝❤❛♥♥❡❧s ✇✐t❤ ✐♥♣✉t ❝♦♥str❛✐♥ts✳ Pr❡♣r✐♥t✳ ❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✽

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SLIDE 55

❖♥❡ s✉❝❤ ❣❡♥❡r❛❧✐③❛t✐♦♥

f1 X0 X1 + Z1 f2 Y1 X2 + Z2 Y2 · · · + Zn−1 Xn−1 fn Yn−1 Xn

❚❤❡♦r❡♠ ▲❡t Xj, Zj ❜❡ d✲❞✐♠❡♥s✐♦♥❛❧✱ d ∈ N ∪ {+∞}✳ ■❢ Zj ∼ N(0, Id) ❛♥❞ EXj2

2 ≤ E < ∞

t❤❡♥ I(X0; Xn) ≤ const · E log log n log n

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✷✾

slide-56
SLIDE 56

■s ❞✐❣✐t❛❧ ✐♥❢♦r♠❛t✐♦♥ ✐♠♠♦rt❛❧❄

▲✐❢❡ ♦❢ ❛ ❜✐t Copy Copy

...

❈♦♣✐❡r ❤❛s ✜♥✐t❡ ❡♥❡r❣② ⇒ ✐♥❢♦r♠❛t✐♦♥ ❞✐ss✐♣❛t❡s ✭s❧♦✇❧②✦✮ ❉❘❆▼ ❝♦♥tr♦❧❧❡r ❉♦❡s r❡❢r❡s❤ ❡✈❡r② ✻✹♠s✦

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✸✵

slide-57
SLIDE 57

■s ❞✐❣✐t❛❧ ✐♥❢♦r♠❛t✐♦♥ ✐♠♠♦rt❛❧❄

▲✐❢❡ ♦❢ ❛ ❜✐t Copy Copy

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❈♦♣✐❡r ❤❛s ✜♥✐t❡ ❡♥❡r❣② ⇒ ✐♥❢♦r♠❛t✐♦♥ ❞✐ss✐♣❛t❡s ✭s❧♦✇❧②✦✮ ❉❘❆▼ ❝♦♥tr♦❧❧❡r

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❉♦❡s r❡❢r❡s❤ ❡✈❡r② ✻✹♠s✦

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✸✵

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SLIDE 58

❆♣♣❧✐❝❛t✐♦♥✿ ❯♥✐q✉❡♥❡ss ♦❢ ●✐❜❜s ♠❡❛s✉r❡s

  • ❛♠❡✿

· · · − X−2 − X−1 − X0 − X1 − X2 − · · ·

  • ●✐✈❡♥✿ ❝♦♥❞✐t✐♦♥❛❧s PXj|Xj−1,Xj+1
  • ❋✐♥❞✿ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ PX+∞

−∞

❙♦♠❡ ❜❛❝❦❣r♦✉♥❞✿

  • ❙❤♦✇s ❤♦✇ ❧♦❝❛❧ ✐♥t❡r❛❝t✐♦♥s ❧❡❛❞ t♦ ❢✉♥♥② ❣❧♦❜❛❧ ❡✛❡❝ts✳
  • ❊①❛♠♣❧❡✿ ♠✉❧t✐♣❧❡ s♦❧✉t✐♦♥s ❝♦rr❡s♣♦♥❞ t♦ ♣❤❛s❡✲tr❛♥s✐t✐♦♥ ✭❡✳❣✳

✷❉✲■s✐♥❣✮

  • ❘✉❧❡ ♦❢ t❤✉♠❜✿ ❧✐♥❦s ❛r❡ ✇❡❛❦ ✭❤✐❣❤ t❡♠♣✳✮ ⇒ ♥♦ ♣❤❛s❡ tr✳

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✸✶

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SLIDE 59

❆♣♣❧✐❝❛t✐♦♥✿ ❯♥✐q✉❡♥❡ss ♦❢ ●✐❜❜s ♠❡❛s✉r❡s

❉♦❜r✉s❤✐♥✬s ♠❡t❤♦❞✿ ❍♦✇ t♦ s❤♦✇ ✉♥✐q✉❡♥❡ss❄

  • ❈♦♥tr❛♣♦s✐t✐✈❡✿

· · · − X−2 − X−1 − X0 − X+1 − X+2 − · · · · · · − ˜ X−2 − ˜ X−1 − ˜ X0 − ˜ X+1 − ˜ X+2 − · · ·

  • ❈♦✉♣❧❡ X±1 t♦ ˜

X±1

  • ■❢ ❝❤❛♥♥❡❧ (X−1, X+1) → X0 ✐s ❚❱✲❝♦♥tr❛❝t✐✈❡ ✭ηTV < 1✮ t❤❡♥

✐♠♣r♦✈❡ ❝♦✉♣❧✐♥❣ ♦❢ X0 t♦ ˜ X0✳ ❘❡♣❡❛t✳ ❖✉r ❝♦♥tr✐❜✉t✐♦♥✿ ❚❤❡♦r❡♠ ▲❡t ♣❛✐r✇✐s❡ ♣♦t❡♥t✐❛❧s ❜❡ ✉♥✐❢♦r♠❧② ✏❧♦✇❡r✲❜♦✉♥❞❡❞✑✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛t ♠♦st ♦♥❡ ●✐❜❜s ♠❡❛s✉r❡ s✳t✳

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✸✷

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SLIDE 60

❆♣♣❧✐❝❛t✐♦♥✿ ❯♥✐q✉❡♥❡ss ♦❢ ●✐❜❜s ♠❡❛s✉r❡s

❉♦❜r✉s❤✐♥✬s ♠❡t❤♦❞✿ ❍♦✇ t♦ s❤♦✇ ✉♥✐q✉❡♥❡ss❄

  • ❈♦♥tr❛♣♦s✐t✐✈❡✿

· · · − X−2 − X−1 − X0 − X+1 − X+2 − · · · · · · − ˜ X−2 − ˜ X−1 − ˜ X0 − ˜ X+1 − ˜ X+2 − · · ·

  • ❈♦✉♣❧❡ X±1 t♦ ˜

X±1

  • ■❢ ❝❤❛♥♥❡❧ (X−1, X+1) → X0 ✐s ❚❱✲❝♦♥tr❛❝t✐✈❡ ✭ηTV < 1✮ t❤❡♥

✐♠♣r♦✈❡ ❝♦✉♣❧✐♥❣ ♦❢ X0 t♦ ˜ X0✳ ❘❡♣❡❛t✳ ❖✉r ❝♦♥tr✐❜✉t✐♦♥✿ ❚❤❡♦r❡♠ ▲❡t ♣❛✐r✇✐s❡ ♣♦t❡♥t✐❛❧s Φj(xj, xj+1) ❜❡ ✉♥✐❢♦r♠❧② ✏❧♦✇❡r✲❜♦✉♥❞❡❞✑✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛t ♠♦st ♦♥❡ ●✐❜❜s ♠❡❛s✉r❡ s✳t✳ sup

k

E|Xk|2 < ∞

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✸✷

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SLIDE 61

❚❛❦❡✲❛✇❛② ♠❡ss❛❣❡

❧✐♥❡❛r str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ ✐♥❡q✉❛❧✐t② ♥♦♥❧✐♥❡❛r str♦♥❣ ❞❛t❛ ♣r♦❝❡ss✐♥❣ ✐♥❡q✉❛❧✐t②

❨✉r② P♦❧②❛♥s❦✐② ❛♥❞ ❨✐❤♦♥❣ ❲✉ ❉♦❜r✉s❤✐♥ ❝✉r✈❡ ✸✸