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Oct 03, 2022 •104 likes •1.33k views

tt tr rt r r r s rst

  1. ◆♩t❡ ✿ ❈♊♠♣✉t✐♥❣ ✐s r❡❧❛t✐✈❡❧② ❡❛s②✱ s✐♥❝❡ ✐s ❛ ❝♊♥❝❛✈❡ ❢✉♥❝t✐♊♥ ♊❢ ✳ ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ■t ✐s ❡❛s② ✭✇❀②❄✮ t♩ s❡❡ t❀❛t R ( W ) ✐s ♊♣t✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ R ( W ⊗ W ) = 2 R ( W ) ∀ W. ❚❀❡ ❛❜♊✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♊❧❧♊✇s ✐❢ t❀❡ ❢♊❧❧♊✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❀♊❧❞s✿ I ( X 1 , X 2 ; Y 1 , Y 2 ) ≀ I ( X 1 ; Y 1 ) + I ( X 2 ; Y 2 ) . I ( X 1 , X 2 ; Y 1 , Y 2 ) = I ( X 1 , X 2 ; Y 1 ) + I ( X 1 , X 2 ; Y 2 | Y 1 ) = I ( X 1 , X 2 ; Y 1 ) + I ( Y 1 , X 1 , X 2 ; Y 2 ) − I ( Y 1 ; Y 2 ) = I ( X 1 ; Y 1 ) + I ( X 2 ; Y 2 ) − I ( Y 1 ; Y 2 ) ≀ I ( X 1 ; Y 1 ) + I ( X 2 ; Y 2 ) . ✹

  2. ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ■t ✐s ❡❛s② ✭✇❀②❄✮ t♩ s❡❡ t❀❛t R ( W ) ✐s ♊♣t✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ R ( W ⊗ W ) = 2 R ( W ) ∀ W. ❚❀❡ ❛❜♊✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♊❧❧♊✇s ✐❢ t❀❡ ❢♊❧❧♊✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❀♊❧❞s✿ I ( X 1 , X 2 ; Y 1 , Y 2 ) ≀ I ( X 1 ; Y 1 ) + I ( X 2 ; Y 2 ) . I ( X 1 , X 2 ; Y 1 , Y 2 ) = I ( X 1 , X 2 ; Y 1 ) + I ( X 1 , X 2 ; Y 2 | Y 1 ) = I ( X 1 , X 2 ; Y 1 ) + I ( Y 1 , X 1 , X 2 ; Y 2 ) − I ( Y 1 ; Y 2 ) = I ( X 1 ; Y 1 ) + I ( X 2 ; Y 2 ) − I ( Y 1 ; Y 2 ) ≀ I ( X 1 ; Y 1 ) + I ( X 2 ; Y 2 ) . ◆♩t❡ ✿ ❈♊♠♣✉t✐♥❣ R ( W ) = sup p ( x ) I ( X ; Y ) ✐s r❡❧❛t✐✈❡❧② ❡❛s②✱ s✐♥❝❡ I ( X ; Y ) ✐s ❛ ❝♊♥❝❛✈❡ ❢✉♥❝t✐♊♥ ♊❢ p ( x ) ✳ ✹

  3. ❲❡ ❛r❡ ♥♊✇ ✭❢✉❧❧② ✐♠♠❡rs❡❞✮ ✐♥ ❛ ✇✐r❡❧❡ss ✇♩r❧❞ ◆❡t✇♩r❊ ♊❢ ✉s❡rs s❀❛r✐♥❣ s❛♠❡ ♠❡❞✐✉♠ ❈❧❡❛r ♥❡❡❞ t♩ ♠❛①✐♠❛❧❧② ✉t✐❧✐③❡ t❀❡ ❧✐♠✐t❡❞ r❡s♩✉r❝❡s ✭♣♊✇❡r✱ ❜❛♥❞✇✐❞t❀✱ ❡♥❡r❣②✮ ❉❡✈❡❧♊♣ ❛ s✐♠✐❧❛r ✉♥❞❡rst❛♥❞✐♥❣ ✐♥ ♥❡t✇♩r❊ s❡tt✐♥❣s ❇✉t ✇❡ ✜rst ♥❡❡❞ t♩ ❢✉❧❧② ✉♥❞❡rst❛♥❞ t❀❡ ❜❛s✐❝ ❜✉✐❧❞✐♥❣ ❜❧♩❝❩s ❙✉❝❝❡ss❡s ❚❀❡ ✈❛r✐♩✉s ✐❞❡❛s ✐♥tr♊❞✉❝❡❞ ❜② ❙❀❛♥♥♊♥ ❀❛✈❡ ❧❡❞ t♩ ❛♥ ✐♥❢♊r♠❛t✐♊♥ r❡✈♊❧✉t✐♊♥ ❘❛♥❞♊♠ ❝♊❞✐♥❣ ❛♥❞ ✐ts ♊♣t✐♠❛❧✐t② ❀❛✈❡ ❞✐r❡❝t❧② ✐♥s♣✐r❡❞ • ▲♊✇ ❞❡♥s✐t② ♣❛r✐t② ❝❀❡❝❊ ❝♊❞❡s ✭▲❉P❈✮ • P♩❧❛r ❝♊❞❡s ⋆ ♣r♊♊❢ ♊❢ s✉❜✲❛❞❞✐t✐✈✐t② ✺

  4. ❙✉❝❝❡ss❡s ❚❀❡ ✈❛r✐♩✉s ✐❞❡❛s ✐♥tr♊❞✉❝❡❞ ❜② ❙❀❛♥♥♊♥ ❀❛✈❡ ❧❡❞ t♩ ❛♥ ✐♥❢♊r♠❛t✐♊♥ r❡✈♊❧✉t✐♊♥ ❘❛♥❞♊♠ ❝♊❞✐♥❣ ❛♥❞ ✐ts ♊♣t✐♠❛❧✐t② ❀❛✈❡ ❞✐r❡❝t❧② ✐♥s♣✐r❡❞ • ▲♊✇ ❞❡♥s✐t② ♣❛r✐t② ❝❀❡❝❊ ❝♊❞❡s ✭▲❉P❈✮ • P♩❧❛r ❝♊❞❡s ⋆ ♣r♊♊❢ ♊❢ s✉❜✲❛❞❞✐t✐✈✐t② ❲❡ ❛r❡ ♥♊✇ ✭❢✉❧❧② ✐♠♠❡rs❡❞✮ ✐♥ ❛ ✇✐r❡❧❡ss ✇♩r❧❞ • ◆❡t✇♩r❊ ♊❢ ✉s❡rs s❀❛r✐♥❣ s❛♠❡ ♠❡❞✐✉♠ • ❈❧❡❛r ♥❡❡❞ t♩ ♠❛①✐♠❛❧❧② ✉t✐❧✐③❡ t❀❡ ❧✐♠✐t❡❞ r❡s♩✉r❝❡s ✭♣♊✇❡r✱ ❜❛♥❞✇✐❞t❀✱ ❡♥❡r❣②✮ • ❉❡✈❡❧♊♣ ❛ s✐♠✐❧❛r ✉♥❞❡rst❛♥❞✐♥❣ ✐♥ ♥❡t✇♩r❊ s❡tt✐♥❣s ⋆ ❇✉t ✇❡ ✜rst ♥❡❡❞ t♩ ❢✉❧❧② ✉♥❞❡rst❛♥❞ t❀❡ ❜❛s✐❝ ❜✉✐❧❞✐♥❣ ❜❧♩❝❩s ✺

  5. ❘❛♥❞♊♠ ❝♊❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♩ ❛❝❀✐❡✈❡ r❛t❡ ♣❛✐rs t❀❛t s❛t✐s❢② ❢♊r s♊♠❡ ❀ ✐t s✉✣❝❡s t♩ ❝♊♥s✐❞❡r ✳ ❈❛❧❧ t❀✐s r❡❣✐♊♥ ✳ ❆❀❧s✇❡❞❡ ◗✉❡st✐♊♥ ✿ ■s t❀✐s t❀❡ ❝❛♣❛❝✐t② ✭♊♣t✐♠❛❧✮ r❡❣✐♊♥❄ ✭❚❊❙✮ ✭❆❀❧s✇❡❞❡ ✬✌✷✮ ✶✳ ▌✉❧t✐♣❧❡ ❆❝❝❡ss ❈❀❛♥♥❡❧ ✭✉♣❧✐♥❊✮ ✭❙❀❛♥♥♊♥ ✬✻✶✮ X n 1 M 1 ❊♥❝♊❞❡r ✶ Y n ( ˆ M 1 , ˆ W ( y | x 1 , x 2 ) ❉❡❝♊❞❡r M 2 ) M 2 ❊♥❝♊❞❡r ✷ X n r❢✇✐r❡❧❡ss✲✇♊r❧❞ 2 ✻

  6. ◗✉❡st✐♊♥ ✿ ■s t❀✐s t❀❡ ❝❛♣❛❝✐t② ✭♊♣t✐♠❛❧✮ r❡❣✐♊♥❄ ✭❚❊❙✮ ✭❆❀❧s✇❡❞❡ ✬✌✷✮ ✶✳ ▌✉❧t✐♣❧❡ ❆❝❝❡ss ❈❀❛♥♥❡❧ ✭✉♣❧✐♥❊✮ ✭❙❀❛♥♥♊♥ ✬✻✶✮ X n 1 M 1 ❊♥❝♊❞❡r ✶ Y n ( ˆ M 1 , ˆ W ( y | x 1 , x 2 ) ❉❡❝♊❞❡r M 2 ) M 2 ❊♥❝♊❞❡r ✷ X n r❢✇✐r❡❧❡ss✲✇♊r❧❞ 2 ❘❛♥❞♊♠ ❝♊❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♩ ❛❝❀✐❡✈❡ r❛t❡ ♣❛✐rs ( R 1 , R 2 ) t❀❛t s❛t✐s❢② R 1 ≀ I ( X 1 ; Y | X 2 , Q ) R 2 ≀ I ( X 2 ; Y | X 1 , Q ) R 1 + R 2 ≀ I ( X 1 , X 2 ; Y | Q ) ❢♊r s♊♠❡ p ( q ) p ( x 1 | q ) p ( x 2 | q ) ❀ ✐t s✉✣❝❡s t♩ ❝♊♥s✐❞❡r |Q| ≀ 2 ✳ ❈❛❧❧ t❀✐s r❡❣✐♊♥ R ( W ) ✳ ❆❀❧s✇❡❞❡ ✻

  7. ✭❚❊❙✮ ✭❆❀❧s✇❡❞❡ ✬✌✷✮ ✶✳ ▌✉❧t✐♣❧❡ ❆❝❝❡ss ❈❀❛♥♥❡❧ ✭✉♣❧✐♥❊✮ ✭❙❀❛♥♥♊♥ ✬✻✶✮ X n 1 M 1 ❊♥❝♊❞❡r ✶ Y n ( ˆ M 1 , ˆ W ( y | x 1 , x 2 ) ❉❡❝♊❞❡r M 2 ) M 2 ❊♥❝♊❞❡r ✷ X n r❢✇✐r❡❧❡ss✲✇♊r❧❞ 2 ❘❛♥❞♊♠ ❝♊❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♩ ❛❝❀✐❡✈❡ r❛t❡ ♣❛✐rs ( R 1 , R 2 ) t❀❛t s❛t✐s❢② R 1 ≀ I ( X 1 ; Y | X 2 , Q ) R 2 ≀ I ( X 2 ; Y | X 1 , Q ) R 1 + R 2 ≀ I ( X 1 , X 2 ; Y | Q ) ❢♊r s♊♠❡ p ( q ) p ( x 1 | q ) p ( x 2 | q ) ❀ ✐t s✉✣❝❡s t♩ ❝♊♥s✐❞❡r |Q| ≀ 2 ✳ ❈❛❧❧ t❀✐s r❡❣✐♊♥ R ( W ) ✳ ❆❀❧s✇❡❞❡ ◗✉❡st✐♊♥ ✿ ■s t❀✐s t❀❡ ❝❛♣❛❝✐t② ✭♊♣t✐♠❛❧✮ r❡❣✐♊♥❄ ✻

  8. ✶✳ ▌✉❧t✐♣❧❡ ❆❝❝❡ss ❈❀❛♥♥❡❧ ✭✉♣❧✐♥❊✮ ✭❙❀❛♥♥♊♥ ✬✻✶✮ X n 1 M 1 ❊♥❝♊❞❡r ✶ Y n ( ˆ M 1 , ˆ W ( y | x 1 , x 2 ) ❉❡❝♊❞❡r M 2 ) M 2 ❊♥❝♊❞❡r ✷ X n r❢✇✐r❡❧❡ss✲✇♊r❧❞ 2 ❘❛♥❞♊♠ ❝♊❞✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♩ ❛❝❀✐❡✈❡ r❛t❡ ♣❛✐rs ( R 1 , R 2 ) t❀❛t s❛t✐s❢② R 1 ≀ I ( X 1 ; Y | X 2 , Q ) R 2 ≀ I ( X 2 ; Y | X 1 , Q ) R 1 + R 2 ≀ I ( X 1 , X 2 ; Y | Q ) ❢♊r s♊♠❡ p ( q ) p ( x 1 | q ) p ( x 2 | q ) ❀ ✐t s✉✣❝❡s t♩ ❝♊♥s✐❞❡r |Q| ≀ 2 ✳ ❈❛❧❧ t❀✐s r❡❣✐♊♥ R ( W ) ✳ ❆❀❧s✇❡❞❡ ◗✉❡st✐♊♥ ✿ ■s t❀✐s t❀❡ ❝❛♣❛❝✐t② ✭♊♣t✐♠❛❧✮ r❡❣✐♊♥❄ ✭❚❊❙✮ ✭❆❀❧s✇❡❞❡ ✬✌✷✮ ✻

  9. ❚❀❡ ❛❜♊✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♊❧❧♊✇s ✐❢ t❀❡ ❢♊❧❧♊✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❀♊❧❞s✿ ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ❉❡✜♥❡✱ ❢♊r λ ≥ 1 ✱ ï¿œ S λ ( W ) = max λR 1 + R 2 } ( R 1 ,R 2 ) ∈R ( W ) ï¿œ ï¿œ ( λ − 1) I ( X 1 ; Y | X 2 ) + I ( X 1 , X 2 ; Y ) = max p 1 ( x 1 ) p 2 ( x 2 ) ✌

  10. ❚❀❡ ❛❜♊✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♊❧❧♊✇s ✐❢ t❀❡ ❢♊❧❧♊✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❀♊❧❞s✿ ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ❉❡✜♥❡✱ ❢♊r λ ≥ 1 ✱ ï¿œ S λ ( W ) = max λR 1 + R 2 } ( R 1 ,R 2 ) ∈R ( W ) ï¿œ ï¿œ ( λ − 1) I ( X 1 ; Y | X 2 ) + I ( X 1 , X 2 ; Y ) = max p 1 ( x 1 ) p 2 ( x 2 ) 3 R 1 + R 2 R 1 2 R 1 + R 2 R 1 + R 2 R 2 ❙✉♣♣♊rt✐♥❣ ❀②♣❡r♣❧❛♥❡s ✌

  11. ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ❉❡✜♥❡✱ ❢♊r λ ≥ 1 ✱ ï¿œ S λ ( W ) = max λR 1 + R 2 } ( R 1 ,R 2 ) ∈R ( W ) ï¿œ ï¿œ ( λ − 1) I ( X 1 ; Y | X 2 ) + I ( X 1 , X 2 ; Y ) = max p 1 ( x 1 ) p 2 ( x 2 ) ❆s ❜❡❢♊r❡✱ R ( W ) ✐s ♊♣t✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ S λ ( W ⊗ W ) = 2 S λ ( W ) ∀ W, λ ≥ 1 . ❚❀❡ ❛❜♊✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♊❧❧♊✇s ✐❢ t❀❡ ❢♊❧❧♊✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❀♊❧❞s✿ ( λ − 1) I ( X 11 , X 12 ; Y 1 , Y 2 | X 21 , X 22 ) + I ( X 11 , X 12 , X 21 , X 22 ; Y 1 , Y 2 ) ≀ ( λ − 1) I ( X 11 ; Y 1 | X 21 ) + I ( X 11 , X 21 ; Y 1 ) + ( λ − 1) I ( X 12 ; Y 2 | X 22 ) + I ( X 12 , X 22 ; Y 2 ) ✌

  12. ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ❉❡✜♥❡✱ ❢♊r λ ≥ 1 ✱ ï¿œ S λ ( W ) = max λR 1 + R 2 } ( R 1 ,R 2 ) ∈R ( W ) ï¿œ ï¿œ ( λ − 1) I ( X 1 ; Y | X 2 ) + I ( X 1 , X 2 ; Y ) = max p 1 ( x 1 ) p 2 ( x 2 ) ❆s ❜❡❢♊r❡✱ R ( W ) ✐s ♊♣t✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ S λ ( W ⊗ W ) = 2 S λ ( W ) ∀ W, λ ≥ 1 . ❚❀❡ ❛❜♊✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♊❧❧♊✇s ✐❢ t❀❡ ❢♊❧❧♊✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❀♊❧❞s✿ ( λ − 1) I ( X 11 , X 12 ; Y 1 , Y 2 | X 21 , X 22 ) + I ( X 11 , X 12 , X 21 , X 22 ; Y 1 , Y 2 ) ≀ ( λ − 1) I ( X 11 ; Y 1 | X 21 ) + I ( X 11 , X 21 ; Y 1 ) + ( λ − 1) I ( X 12 ; Y 2 | X 22 ) + I ( X 12 , X 22 ; Y 2 ) ❖♥❡ ❝❛♥ ❡st❛❜❧✐s❀ t❀✐s ✐♥ s❛♠❡ ✇❛② ❛s ♣♊✐♥t✲t♊✲♣♊✐♥t s❡tt✐♥❣✳ ✌

  13. ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ❉❡✜♥❡✱ ❢♊r λ ≥ 1 ✱ ï¿œ S λ ( W ) = max λR 1 + R 2 } ( R 1 ,R 2 ) ∈R ( W ) ï¿œ ï¿œ ( λ − 1) I ( X 1 ; Y | X 2 ) + I ( X 1 , X 2 ; Y ) = max p 1 ( x 1 ) p 2 ( x 2 ) ❆s ❜❡❢♊r❡✱ R ( W ) ✐s ♊♣t✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ S λ ( W ⊗ W ) = 2 S λ ( W ) ∀ W, λ ≥ 1 . ❚❀❡ ❛❜♊✈❡ ❡q✉❛❧✐t② ✭❛❞❞✐t✐✈✐t②✮ ❢♊❧❧♊✇s ✐❢ t❀❡ ❢♊❧❧♊✇✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ❀♊❧❞s✿ ( λ − 1) I ( X 11 , X 12 ; Y 1 , Y 2 | X 21 , X 22 ) + I ( X 11 , X 12 , X 21 , X 22 ; Y 1 , Y 2 ) ≀ ( λ − 1) I ( X 11 ; Y 1 | X 21 ) + I ( X 11 , X 21 ; Y 1 ) + ( λ − 1) I ( X 12 ; Y 2 | X 22 ) + I ( X 12 , X 22 ; Y 2 ) ◆♩t❡ ✿ ❈♊♠♣✉t✐♥❣ S λ ( W ) ✐s r❡❧❛t✐✈❡❧② ❡❛s② s✐♥❝❡ ï¿œ ï¿œ ( λ − 1) I ( X 1 ; Y | X 2 ) + I ( X 1 , X 2 ; Y ) ✐s ❝♊♥❝❛✈❡ ✐♥ p 1 ( x 1 ) , p 2 ( x 2 ) ✳ ✌

  14. ❙✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ❛♥❞ r❛♥❞♊♠ ❀❛s❀✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♩ ❛❝❀✐❡✈❡ r❛t❡ tr✐♣❧❡s t❀❛t s❛t✐s❢② ▌❛rt♊♥ ❢♊r s♊♠❡ ✳ ❈❛❧❧ t❀✐s r❡❣✐♊♥ ✳ ◗✉❡st✐♊♥ ✿ ■s t❀✐s t❀❡ ❝❛♣❛❝✐t② ✭♊♣t✐♠❛❧✮ r❡❣✐♊♥❄ ✭❖♣❡♥✮ ✭s✐♥❝❡ ▌❛rt♊♥ ✬✌✟✮ ✷✳ ❇r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✭❞♊✇♥❧✐♥❊✮ ✭❈♊✈❡r ✬✌✷✮ Y n 1 M 0 , ˆ ˆ W a ( y 1 | x ) ❉❡❝♊❞❡r ✶ M 1 X n ( M 0 , M 1 , M 2 ) ❊♥❝♊❞❡r Y n 2 M 0 , ˜ ˜ W b ( y 2 | x ) ❉❡❝♊❞❡r ✷ M 2 r❢✇✐r❡❧❡ss✲✇♊r❧❞ ✜

  15. ◗✉❡st✐♊♥ ✿ ■s t❀✐s t❀❡ ❝❛♣❛❝✐t② ✭♊♣t✐♠❛❧✮ r❡❣✐♊♥❄ ✭❖♣❡♥✮ ✭s✐♥❝❡ ▌❛rt♊♥ ✬✌✟✮ ✷✳ ❇r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✭❞♊✇♥❧✐♥❊✮ ✭❈♊✈❡r ✬✌✷✮ Y n 1 M 0 , ˆ ˆ W a ( y 1 | x ) ❉❡❝♊❞❡r ✶ M 1 X n ( M 0 , M 1 , M 2 ) ❊♥❝♊❞❡r Y n 2 M 0 , ˜ ˜ W b ( y 2 | x ) ❉❡❝♊❞❡r ✷ M 2 r❢✇✐r❡❧❡ss✲✇♊r❧❞ ❙✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ❛♥❞ r❛♥❞♊♠ ❀❛s❀✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♩ ❛❝❀✐❡✈❡ r❛t❡ tr✐♣❧❡s ( R 0 , R 1 , R 2 ) t❀❛t s❛t✐s❢② R 0 ≀ min { I ( Q ; Y 1 ) , I ( Q ; Y 2 ) } R 0 + R 1 ≀ I ( U, Q ; Y 1 ) R 0 + R 2 ≀ I ( V, Q ; Y 2 ) R 0 + R 1 + R 2 ≀ min { I ( Q ; Y 1 ) , I ( Q ; Y 2 ) } + I ( U ; Y 1 | Q ) + I ( V ; Y 2 | Q ) − I ( U ; V | Q ) ▌❛rt♊♥ ❢♊r s♊♠❡ p ( q, u, v, x ) ✳ ❈❛❧❧ t❀✐s r❡❣✐♊♥ R ( W a , W b ) ✳ ✜

  16. ✭❖♣❡♥✮ ✭s✐♥❝❡ ▌❛rt♊♥ ✬✌✟✮ ✷✳ ❇r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✭❞♊✇♥❧✐♥❊✮ ✭❈♊✈❡r ✬✌✷✮ Y n 1 M 0 , ˆ ˆ W a ( y 1 | x ) ❉❡❝♊❞❡r ✶ M 1 X n ( M 0 , M 1 , M 2 ) ❊♥❝♊❞❡r Y n 2 M 0 , ˜ ˜ W b ( y 2 | x ) ❉❡❝♊❞❡r ✷ M 2 r❢✇✐r❡❧❡ss✲✇♊r❧❞ ❙✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ❛♥❞ r❛♥❞♊♠ ❀❛s❀✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♩ ❛❝❀✐❡✈❡ r❛t❡ tr✐♣❧❡s ( R 0 , R 1 , R 2 ) t❀❛t s❛t✐s❢② R 0 ≀ min { I ( Q ; Y 1 ) , I ( Q ; Y 2 ) } R 0 + R 1 ≀ I ( U, Q ; Y 1 ) R 0 + R 2 ≀ I ( V, Q ; Y 2 ) R 0 + R 1 + R 2 ≀ min { I ( Q ; Y 1 ) , I ( Q ; Y 2 ) } + I ( U ; Y 1 | Q ) + I ( V ; Y 2 | Q ) − I ( U ; V | Q ) ▌❛rt♊♥ ❢♊r s♊♠❡ p ( q, u, v, x ) ✳ ❈❛❧❧ t❀✐s r❡❣✐♊♥ R ( W a , W b ) ✳ ◗✉❡st✐♊♥ ✿ ■s t❀✐s t❀❡ ❝❛♣❛❝✐t② ✭♊♣t✐♠❛❧✮ r❡❣✐♊♥❄ ✜

  17. ✷✳ ❇r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✭❞♊✇♥❧✐♥❊✮ ✭❈♊✈❡r ✬✌✷✮ Y n 1 M 0 , ˆ ˆ W a ( y 1 | x ) ❉❡❝♊❞❡r ✶ M 1 X n ( M 0 , M 1 , M 2 ) ❊♥❝♊❞❡r Y n 2 M 0 , ˜ ˜ W b ( y 2 | x ) ❉❡❝♊❞❡r ✷ M 2 r❢✇✐r❡❧❡ss✲✇♊r❧❞ ❙✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ❛♥❞ r❛♥❞♊♠ ❀❛s❀✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♩ ❛❝❀✐❡✈❡ r❛t❡ tr✐♣❧❡s ( R 0 , R 1 , R 2 ) t❀❛t s❛t✐s❢② R 0 ≀ min { I ( Q ; Y 1 ) , I ( Q ; Y 2 ) } R 0 + R 1 ≀ I ( U, Q ; Y 1 ) R 0 + R 2 ≀ I ( V, Q ; Y 2 ) R 0 + R 1 + R 2 ≀ min { I ( Q ; Y 1 ) , I ( Q ; Y 2 ) } + I ( U ; Y 1 | Q ) + I ( V ; Y 2 | Q ) − I ( U ; V | Q ) ▌❛rt♊♥ ❢♊r s♊♠❡ p ( q, u, v, x ) ✳ ❈❛❧❧ t❀✐s r❡❣✐♊♥ R ( W a , W b ) ✳ ◗✉❡st✐♊♥ ✿ ■s t❀✐s t❀❡ ❝❛♣❛❝✐t② ✭♊♣t✐♠❛❧✮ r❡❣✐♊♥❄ ✭❖♣❡♥✮ ✭s✐♥❝❡ ▌❛rt♊♥ ✬✌✟✮ ✜

  18. ◆♩t❡ ✿ ❈♊♠♣✉t✐♥❣ ✐s ❛ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠✳ ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ✭ R 0 = 0 ✮ ❉❡✜♥❡✱ ❢♊r λ ≥ 1 ✱ S λ ( W ) = ( R 1 ,R 2 ) ∈R ( W a ,W b ) { λR 1 + R 2 } max ï¿œ = max ( λ − 1) I ( U, Q ; Y 1 ) + min { I ( Q ; Y 1 ) , I ( Q ; Y 2 ) } + I ( U ; Y 1 | Q ) p ( u,v,w,x ) ï¿œ + I ( V ; Y 2 | Q ) − I ( U ; V | Q ) ï¿œ = min max ( λ − α ) I ( Q ; Y 1 ) + αI ( Q ; Y 2 ) + λI ( U ; Y 1 | Q ) α ∈ [0 , 1] p ( u,v,w,x ) ï¿œ + I ( V ; Y 2 | Q ) − I ( U ; V | Q ) ❆s ❜❡❢♊r❡✱ R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ S λ ( W a ⊗ W a , W b ⊗ W b ) = 2 S λ ( W a , W b ) ∀ W a , W b , λ ≥ 1 . ✟

  19. ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ✭ R 0 = 0 ✮ ❉❡✜♥❡✱ ❢♊r λ ≥ 1 ✱ S λ ( W ) = ( R 1 ,R 2 ) ∈R ( W a ,W b ) { λR 1 + R 2 } max ï¿œ = max ( λ − 1) I ( U, Q ; Y 1 ) + min { I ( Q ; Y 1 ) , I ( Q ; Y 2 ) } + I ( U ; Y 1 | Q ) p ( u,v,w,x ) ï¿œ + I ( V ; Y 2 | Q ) − I ( U ; V | Q ) ï¿œ = min max ( λ − α ) I ( Q ; Y 1 ) + αI ( Q ; Y 2 ) + λI ( U ; Y 1 | Q ) α ∈ [0 , 1] p ( u,v,w,x ) ï¿œ + I ( V ; Y 2 | Q ) − I ( U ; V | Q ) ❆s ❜❡❢♊r❡✱ R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ✐❢ ❛♥❞ ♊♥❧② ✐❢ S λ ( W a ⊗ W a , W b ⊗ W b ) = 2 S λ ( W a , W b ) ∀ W a , W b , λ ≥ 1 . ◆♩t❡ ✿ ❈♊♠♣✉t✐♥❣ S λ ( W a , W b ) ✐s ❛ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠✳ ✟

  20. ❈♊✈❡r ✬✌✷✿ ❞❡✈❡❧♊♣♠❡♥t ♊❢ s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ str❛t❡❣② ●❛❧❧❛❣❡r ✬✌✹✿ ❝♊♥✈❡rs❡ t♩ t❀❡ ❞❡❣r❛❞❡❞ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✭ s✉❜✲❛❞❞✐t✐✈✐t② ✮ ❲❡✐♥❣❛rt❡♥✲❙t❡✐♥❜❡r❣✲❙❀❛♠❛✐ ✬✵✻✿ ❖♣t✐♠❛❧✐t② ♊❢ ✭♊♥ ✮ ❢♊r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧❀ ❞❡✈❡❧♊♣✐♥❣ ❛ ❢❛♠✐❧② ♊❢ t✐❣❀t ❝♊♥✈❡① r❡❧❛①❛t✐♊♥s t♩ ❝♊♠♣✉t❡ t❀❡ ♊♣t✐♠❛❧ ✈❛❧✉❡ ♊❢ ❛ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠ ●❡♥❣✲◆❛✐r ✬✶✹✿ ❖♣t✐♠❛❧✐t② ♊❢ ❢♊r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧✿ ❚❡❝❀♥✐q✉❡ ❢♊r ❡st❛❜❧✐s❀✐♥❣ ❡①tr❡♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♊♥s ✉s✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♊❢ ❢✉♥❝t✐♊♥❛❧s ❙✉❝❝❡ss❡s ■♥ s♣✐t❡ ♊❢ t❀❡ ✉♥❞❡r❧②✐♥❣ ♣r♊❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♊♥✲❝♊♥✈❡① • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ♊♥ R 1 = 0 ✭♩r R 2 = 0 ✮ ⋆ ❉❡❣r❛❞❡❞ ♠❡ss❛❣❡ s❡ts✿ ❑♩r♥❡r ❛♥❞ ▌❛rt♊♥ ✭✬✌✌✮ • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ❢♊r s♊♠❡ ❝❧❛ss❡s ♊❢ ❝❀❛♥♥❡❧s ⋆ ●❛❧❧❛❣❡r ✬✌✹✱ ❑♩r♥❡r ❛♥❞ ▌❛rt♊♥ ✭✬✌✺✱ ✬✌✌✱ ✬✌✟✮✱ ●❡❧❢❛♥❞ ❛♥❞ P✐♥s❊❡r ✭✬✌✜✮✱ P♩❧t②r❡✈ ✭✬✌✜✮✱ ❊❧ ●❛♠❛❧ ✭✬✌✟✱ ✬✜✵✮ ⋆ ❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❀❛♠❛✐ ✬✵✻✱ ◆❛✐r ✬✶✵✱ ●❡♥❣ ❛♥❞ ●♊❀❛r✐ ❛♥❞ ◆❛✐r ❛♥❞ ❹✉ ✬✶✹✱ ●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹ • ◆♊✈❡❧ ✐❞❡❛s ❛♥❞ t❡❝❀♥✐q✉❡s ✇❡r❡ ♥❡❡❞❡❞ t♩ ❡st❛❜❧✐s❀ t❀❡s❡ ❝❛♣❛❝✐t② r❡❣✐♊♥s ✶✵

  21. ●❛❧❧❛❣❡r ✬✌✹✿ ❝♊♥✈❡rs❡ t♩ t❀❡ ❞❡❣r❛❞❡❞ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✭ s✉❜✲❛❞❞✐t✐✈✐t② ✮ ❲❡✐♥❣❛rt❡♥✲❙t❡✐♥❜❡r❣✲❙❀❛♠❛✐ ✬✵✻✿ ❖♣t✐♠❛❧✐t② ♊❢ ✭♊♥ ✮ ❢♊r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧❀ ❞❡✈❡❧♊♣✐♥❣ ❛ ❢❛♠✐❧② ♊❢ t✐❣❀t ❝♊♥✈❡① r❡❧❛①❛t✐♊♥s t♩ ❝♊♠♣✉t❡ t❀❡ ♊♣t✐♠❛❧ ✈❛❧✉❡ ♊❢ ❛ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠ ●❡♥❣✲◆❛✐r ✬✶✹✿ ❖♣t✐♠❛❧✐t② ♊❢ ❢♊r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧✿ ❚❡❝❀♥✐q✉❡ ❢♊r ❡st❛❜❧✐s❀✐♥❣ ❡①tr❡♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♊♥s ✉s✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♊❢ ❢✉♥❝t✐♊♥❛❧s ❙✉❝❝❡ss❡s ■♥ s♣✐t❡ ♊❢ t❀❡ ✉♥❞❡r❧②✐♥❣ ♣r♊❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♊♥✲❝♊♥✈❡① • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ♊♥ R 1 = 0 ✭♩r R 2 = 0 ✮ ⋆ ❉❡❣r❛❞❡❞ ♠❡ss❛❣❡ s❡ts✿ ❑♩r♥❡r ❛♥❞ ▌❛rt♊♥ ✭✬✌✌✮ • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ❢♊r s♊♠❡ ❝❧❛ss❡s ♊❢ ❝❀❛♥♥❡❧s ⋆ ●❛❧❧❛❣❡r ✬✌✹✱ ❑♩r♥❡r ❛♥❞ ▌❛rt♊♥ ✭✬✌✺✱ ✬✌✌✱ ✬✌✟✮✱ ●❡❧❢❛♥❞ ❛♥❞ P✐♥s❊❡r ✭✬✌✜✮✱ P♩❧t②r❡✈ ✭✬✌✜✮✱ ❊❧ ●❛♠❛❧ ✭✬✌✟✱ ✬✜✵✮ ⋆ ❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❀❛♠❛✐ ✬✵✻✱ ◆❛✐r ✬✶✵✱ ●❡♥❣ ❛♥❞ ●♊❀❛r✐ ❛♥❞ ◆❛✐r ❛♥❞ ❹✉ ✬✶✹✱ ●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹ • ◆♊✈❡❧ ✐❞❡❛s ❛♥❞ t❡❝❀♥✐q✉❡s ✇❡r❡ ♥❡❡❞❡❞ t♩ ❡st❛❜❧✐s❀ t❀❡s❡ ❝❛♣❛❝✐t② r❡❣✐♊♥s ⋆ ❈♊✈❡r ✬✌✷✿ ❞❡✈❡❧♊♣♠❡♥t ♊❢ s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ str❛t❡❣② ✶✵

  22. ❲❡✐♥❣❛rt❡♥✲❙t❡✐♥❜❡r❣✲❙❀❛♠❛✐ ✬✵✻✿ ❖♣t✐♠❛❧✐t② ♊❢ ✭♊♥ ✮ ❢♊r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧❀ ❞❡✈❡❧♊♣✐♥❣ ❛ ❢❛♠✐❧② ♊❢ t✐❣❀t ❝♊♥✈❡① r❡❧❛①❛t✐♊♥s t♩ ❝♊♠♣✉t❡ t❀❡ ♊♣t✐♠❛❧ ✈❛❧✉❡ ♊❢ ❛ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠ ●❡♥❣✲◆❛✐r ✬✶✹✿ ❖♣t✐♠❛❧✐t② ♊❢ ❢♊r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧✿ ❚❡❝❀♥✐q✉❡ ❢♊r ❡st❛❜❧✐s❀✐♥❣ ❡①tr❡♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♊♥s ✉s✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♊❢ ❢✉♥❝t✐♊♥❛❧s ❙✉❝❝❡ss❡s ■♥ s♣✐t❡ ♊❢ t❀❡ ✉♥❞❡r❧②✐♥❣ ♣r♊❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♊♥✲❝♊♥✈❡① • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ♊♥ R 1 = 0 ✭♩r R 2 = 0 ✮ ⋆ ❉❡❣r❛❞❡❞ ♠❡ss❛❣❡ s❡ts✿ ❑♩r♥❡r ❛♥❞ ▌❛rt♊♥ ✭✬✌✌✮ • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ❢♊r s♊♠❡ ❝❧❛ss❡s ♊❢ ❝❀❛♥♥❡❧s ⋆ ●❛❧❧❛❣❡r ✬✌✹✱ ❑♩r♥❡r ❛♥❞ ▌❛rt♊♥ ✭✬✌✺✱ ✬✌✌✱ ✬✌✟✮✱ ●❡❧❢❛♥❞ ❛♥❞ P✐♥s❊❡r ✭✬✌✜✮✱ P♩❧t②r❡✈ ✭✬✌✜✮✱ ❊❧ ●❛♠❛❧ ✭✬✌✟✱ ✬✜✵✮ ⋆ ❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❀❛♠❛✐ ✬✵✻✱ ◆❛✐r ✬✶✵✱ ●❡♥❣ ❛♥❞ ●♊❀❛r✐ ❛♥❞ ◆❛✐r ❛♥❞ ❹✉ ✬✶✹✱ ●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹ • ◆♊✈❡❧ ✐❞❡❛s ❛♥❞ t❡❝❀♥✐q✉❡s ✇❡r❡ ♥❡❡❞❡❞ t♩ ❡st❛❜❧✐s❀ t❀❡s❡ ❝❛♣❛❝✐t② r❡❣✐♊♥s ⋆ ❈♊✈❡r ✬✌✷✿ ❞❡✈❡❧♊♣♠❡♥t ♊❢ s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ str❛t❡❣② ⋆ ●❛❧❧❛❣❡r ✬✌✹✿ ❝♊♥✈❡rs❡ t♩ t❀❡ ❞❡❣r❛❞❡❞ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✭ s✉❜✲❛❞❞✐t✐✈✐t② ✮ ✶✵

  23. ●❡♥❣✲◆❛✐r ✬✶✹✿ ❖♣t✐♠❛❧✐t② ♊❢ ❢♊r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧✿ ❚❡❝❀♥✐q✉❡ ❢♊r ❡st❛❜❧✐s❀✐♥❣ ❡①tr❡♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♊♥s ✉s✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♊❢ ❢✉♥❝t✐♊♥❛❧s ❙✉❝❝❡ss❡s ■♥ s♣✐t❡ ♊❢ t❀❡ ✉♥❞❡r❧②✐♥❣ ♣r♊❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♊♥✲❝♊♥✈❡① • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ♊♥ R 1 = 0 ✭♩r R 2 = 0 ✮ ⋆ ❉❡❣r❛❞❡❞ ♠❡ss❛❣❡ s❡ts✿ ❑♩r♥❡r ❛♥❞ ▌❛rt♊♥ ✭✬✌✌✮ • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ❢♊r s♊♠❡ ❝❧❛ss❡s ♊❢ ❝❀❛♥♥❡❧s ⋆ ●❛❧❧❛❣❡r ✬✌✹✱ ❑♩r♥❡r ❛♥❞ ▌❛rt♊♥ ✭✬✌✺✱ ✬✌✌✱ ✬✌✟✮✱ ●❡❧❢❛♥❞ ❛♥❞ P✐♥s❊❡r ✭✬✌✜✮✱ P♩❧t②r❡✈ ✭✬✌✜✮✱ ❊❧ ●❛♠❛❧ ✭✬✌✟✱ ✬✜✵✮ ⋆ ❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❀❛♠❛✐ ✬✵✻✱ ◆❛✐r ✬✶✵✱ ●❡♥❣ ❛♥❞ ●♊❀❛r✐ ❛♥❞ ◆❛✐r ❛♥❞ ❹✉ ✬✶✹✱ ●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹ • ◆♊✈❡❧ ✐❞❡❛s ❛♥❞ t❡❝❀♥✐q✉❡s ✇❡r❡ ♥❡❡❞❡❞ t♩ ❡st❛❜❧✐s❀ t❀❡s❡ ❝❛♣❛❝✐t② r❡❣✐♊♥s ⋆ ❈♊✈❡r ✬✌✷✿ ❞❡✈❡❧♊♣♠❡♥t ♊❢ s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ str❛t❡❣② ⋆ ●❛❧❧❛❣❡r ✬✌✹✿ ❝♊♥✈❡rs❡ t♩ t❀❡ ❞❡❣r❛❞❡❞ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✭ s✉❜✲❛❞❞✐t✐✈✐t② ✮ ⋆ ❲❡✐♥❣❛rt❡♥✲❙t❡✐♥❜❡r❣✲❙❀❛♠❛✐ ✬✵✻✿ ❖♣t✐♠❛❧✐t② ♊❢ R ( W a , W b ) ✭♊♥ R 0 = 0 ✮ ❢♊r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧❀ ❞❡✈❡❧♊♣✐♥❣ ❛ ❢❛♠✐❧② ♊❢ t✐❣❀t ❝♊♥✈❡① r❡❧❛①❛t✐♊♥s t♩ ❝♊♠♣✉t❡ t❀❡ ♊♣t✐♠❛❧ ✈❛❧✉❡ ♊❢ ❛ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠ ✶✵

  24. ❙✉❝❝❡ss❡s ■♥ s♣✐t❡ ♊❢ t❀❡ ✉♥❞❡r❧②✐♥❣ ♣r♊❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♊♥✲❝♊♥✈❡① • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ♊♥ R 1 = 0 ✭♩r R 2 = 0 ✮ ⋆ ❉❡❣r❛❞❡❞ ♠❡ss❛❣❡ s❡ts✿ ❑♩r♥❡r ❛♥❞ ▌❛rt♊♥ ✭✬✌✌✮ • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ❢♊r s♊♠❡ ❝❧❛ss❡s ♊❢ ❝❀❛♥♥❡❧s ⋆ ●❛❧❧❛❣❡r ✬✌✹✱ ❑♩r♥❡r ❛♥❞ ▌❛rt♊♥ ✭✬✌✺✱ ✬✌✌✱ ✬✌✟✮✱ ●❡❧❢❛♥❞ ❛♥❞ P✐♥s❊❡r ✭✬✌✜✮✱ P♩❧t②r❡✈ ✭✬✌✜✮✱ ❊❧ ●❛♠❛❧ ✭✬✌✟✱ ✬✜✵✮ ⋆ ❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❀❛♠❛✐ ✬✵✻✱ ◆❛✐r ✬✶✵✱ ●❡♥❣ ❛♥❞ ●♊❀❛r✐ ❛♥❞ ◆❛✐r ❛♥❞ ❹✉ ✬✶✹✱ ●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹ • ◆♊✈❡❧ ✐❞❡❛s ❛♥❞ t❡❝❀♥✐q✉❡s ✇❡r❡ ♥❡❡❞❡❞ t♩ ❡st❛❜❧✐s❀ t❀❡s❡ ❝❛♣❛❝✐t② r❡❣✐♊♥s ⋆ ❈♊✈❡r ✬✌✷✿ ❞❡✈❡❧♊♣♠❡♥t ♊❢ s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ str❛t❡❣② ⋆ ●❛❧❧❛❣❡r ✬✌✹✿ ❝♊♥✈❡rs❡ t♩ t❀❡ ❞❡❣r❛❞❡❞ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✭ s✉❜✲❛❞❞✐t✐✈✐t② ✮ ⋆ ❲❡✐♥❣❛rt❡♥✲❙t❡✐♥❜❡r❣✲❙❀❛♠❛✐ ✬✵✻✿ ❖♣t✐♠❛❧✐t② ♊❢ R ( W a , W b ) ✭♊♥ R 0 = 0 ✮ ❢♊r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧❀ ❞❡✈❡❧♊♣✐♥❣ ❛ ❢❛♠✐❧② ♊❢ t✐❣❀t ❝♊♥✈❡① r❡❧❛①❛t✐♊♥s t♩ ❝♊♠♣✉t❡ t❀❡ ♊♣t✐♠❛❧ ✈❛❧✉❡ ♊❢ ❛ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠ ⋆ ●❡♥❣✲◆❛✐r ✬✶✹✿ ❖♣t✐♠❛❧✐t② ♊❢ R ( W a , W b ) ❢♊r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧✿ ❚❡❝❀♥✐q✉❡ ❢♊r ❡st❛❜❧✐s❀✐♥❣ ❡①tr❡♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♊♥s ✉s✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♊❢ ❢✉♥❝t✐♊♥❛❧s ✶✵

  25. ✞✳ ■♥t❡r❢❡r❡♥❝❡ ❈❀❛♥♥❡❧ ✭❆❀❧s✇❡❞❡ ✬✌✹✮ ❈r❡❞✐t✿ ✇✇✇✳♣❡rs♊♥❛❧✳♣s✉✳❡❞✉✎❜①❣✷✶✺✎r❡s❡❛r❝❀✳❀t♠❧ X n Y n 1 1 ˆ M 1 W a ( y 1 | x 1 , x 2 ) ❊♥❝♊❞❡r ✶ ❉❡❝♊❞❡r ✶ M 1 Y n 2 ˆ W b ( y 2 | x 1 , x 2 ) M 2 ❊♥❝♊❞❡r ✷ ❉❡❝♊❞❡r ✷ M 2 X n 2 ✶✶

  26. ◗✉❡st✐♊♥ ✿ ■s t❀✐s t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥❄ ✞✳ ■♥t❡r❢❡r❡♥❝❡ ❈❀❛♥♥❡❧ ✭❆❀❧s✇❡❞❡ ✬✌✹✮ ❙✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣✱ ♠❡ss❛❣❡ s♣❧✐tt✐♥❣✱ ❝♊❞❡❞ t✐♠❡✲s❀❛r✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♩ ❛❝❀✐❡✈❡ r❛t❡ ♣❛✐rs ( R 1 , R 2 ) t❀❛t s❛t✐s❢② R 1 < I ( X 1 ; Y 1 | U 2 , Q ) , R 2 < I ( X 2 ; Y 2 | U 1 , Q ) , R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | Q ) + I ( X 2 ; Y 2 | U 1 , U 2 , Q ) , R 1 + R 2 < I ( X 2 , U 1 ; Y 2 | Q ) + I ( X 1 ; Y 1 | U 1 , U 2 , Q ) , R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | U 1 , Q ) + I ( X 2 , U 1 ; Y 2 | U 2 , Q ) , ❍❛♥ 2 R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | Q ) + I ( X 1 ; Y 1 | U 1 , U 2 , Q ) + I ( X 2 , U 1 ; Y 2 | U 2 , Q ) , R 1 + 2 R 2 < I ( X 2 , U 1 ; Y 2 | Q ) + I ( X 2 ; Y 2 | U 1 , U 2 , Q ) + I ( X 1 , U 2 ; Y 1 | U 1 , Q ) ❢♊r s♊♠❡ ♣♠❢ p ( q ) p ( u 1 , x 1 | q ) p ( u 2 , x 2 | q ) ✱ ✇❀❡r❡ | U 1 | ≀ | X 1 | + 4 ✱ | U 2 | ≀ | X 2 | + 4 ✱ ❛♥❞ | Q | ≀ 7 ✳ ❈❛❧❧ t❀✐s r❡❣✐♊♥ R ( W a , W b ) ✳ ❑♊❜❛②❛s❀✐ ✶✶

  27. ✞✳ ■♥t❡r❢❡r❡♥❝❡ ❈❀❛♥♥❡❧ ✭❆❀❧s✇❡❞❡ ✬✌✹✮ ❙✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣✱ ♠❡ss❛❣❡ s♣❧✐tt✐♥❣✱ ❝♊❞❡❞ t✐♠❡✲s❀❛r✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♩ ❛❝❀✐❡✈❡ r❛t❡ ♣❛✐rs ( R 1 , R 2 ) t❀❛t s❛t✐s❢② R 1 < I ( X 1 ; Y 1 | U 2 , Q ) , R 2 < I ( X 2 ; Y 2 | U 1 , Q ) , R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | Q ) + I ( X 2 ; Y 2 | U 1 , U 2 , Q ) , R 1 + R 2 < I ( X 2 , U 1 ; Y 2 | Q ) + I ( X 1 ; Y 1 | U 1 , U 2 , Q ) , R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | U 1 , Q ) + I ( X 2 , U 1 ; Y 2 | U 2 , Q ) , ❍❛♥ 2 R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | Q ) + I ( X 1 ; Y 1 | U 1 , U 2 , Q ) + I ( X 2 , U 1 ; Y 2 | U 2 , Q ) , R 1 + 2 R 2 < I ( X 2 , U 1 ; Y 2 | Q ) + I ( X 2 ; Y 2 | U 1 , U 2 , Q ) + I ( X 1 , U 2 ; Y 1 | U 1 , Q ) ❢♊r s♊♠❡ ♣♠❢ p ( q ) p ( u 1 , x 1 | q ) p ( u 2 , x 2 | q ) ✱ ✇❀❡r❡ | U 1 | ≀ | X 1 | + 4 ✱ | U 2 | ≀ | X 2 | + 4 ✱ ❛♥❞ | Q | ≀ 7 ✳ ❈❛❧❧ t❀✐s r❡❣✐♊♥ R ( W a , W b ) ✳ ◗✉❡st✐♊♥ ✿ ■s t❀✐s t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥❄ ❑♊❜❛②❛s❀✐ ✶✶

  28. ✞✳ ■♥t❡r❢❡r❡♥❝❡ ❈❀❛♥♥❡❧ ✭❆❀❧s✇❡❞❡ ✬✌✹✮ ❙✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣✱ ♠❡ss❛❣❡ s♣❧✐tt✐♥❣✱ ❝♊❞❡❞ t✐♠❡✲s❀❛r✐♥❣ ❝❛♥ ❜❡ ✉s❡❞ t♩ ❛❝❀✐❡✈❡ r❛t❡ ♣❛✐rs ( R 1 , R 2 ) t❀❛t s❛t✐s❢② R 1 < I ( X 1 ; Y 1 | U 2 , Q ) , R 2 < I ( X 2 ; Y 2 | U 1 , Q ) , R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | Q ) + I ( X 2 ; Y 2 | U 1 , U 2 , Q ) , R 1 + R 2 < I ( X 2 , U 1 ; Y 2 | Q ) + I ( X 1 ; Y 1 | U 1 , U 2 , Q ) , R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | U 1 , Q ) + I ( X 2 , U 1 ; Y 2 | U 2 , Q ) , ❍❛♥ 2 R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | Q ) + I ( X 1 ; Y 1 | U 1 , U 2 , Q ) + I ( X 2 , U 1 ; Y 2 | U 2 , Q ) , R 1 + 2 R 2 < I ( X 2 , U 1 ; Y 2 | Q ) + I ( X 2 ; Y 2 | U 1 , U 2 , Q ) + I ( X 1 , U 2 ; Y 1 | U 1 , Q ) ❢♊r s♊♠❡ ♣♠❢ p ( q ) p ( u 1 , x 1 | q ) p ( u 2 , x 2 | q ) ✱ ✇❀❡r❡ | U 1 | ≀ | X 1 | + 4 ✱ | U 2 | ≀ | X 2 | + 4 ✱ ❛♥❞ | Q | ≀ 7 ✳ ❈❛❧❧ t❀✐s r❡❣✐♊♥ R ( W a , W b ) ✳ ◗✉❡st✐♊♥ ✿ ■s t❀✐s t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥❄ ❍❛❞ ❜❡❡♥ ♊♣❡♥ ✭s✐♥❝❡ ❍❛♥ ❛♥❞ ❑♊❜❛②❛s❀✐ ✬✜✶✮ ❑♊❜❛②❛s❀✐ ✶✶

  29. ✐s ♥♊t ♊♣t✐♠❛❧ ✐♥ ❣❡♥❡r❛❧ ✭◆❛✐r✱ ❳✐❛✱ ❚❛③❞❛♥♣❛♥❛❀ ✬✶✺✮ ❇r♩❛❞❝❛st ❛♥❞ ✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧s ❛r❡ ❢❛r t♩♩ ✐♠♣♊rt❛♥t ❚♩ ❧❡t ♥♊♥✲❝♊♥✈❡①✐t② ❞✐ss✉❛❞❡ ✉s ❚♩ ♥♊t ✐♥✈❡st✐❣❛t❡ s✐♠♣❧❡ ❝❧❛ss❡s t❀❛t r❡q✉✐r❡ ♥❡✇ ✐❞❡❛s ❙✉❝❝❡ss❡s ■♥ s♣✐t❡ ♊❢ t❀❡ ✉♥❞❡r❧②✐♥❣ ♣r♊❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♊♥✲❝♊♥✈❡① • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ❢♊r s♊♠❡ ❝❧❛ss❡s ♊❢ ❝❀❛♥♥❡❧s ⋆ ❈❛r❧❡✐❛❧ ✬✌✺✱ ❙❛t♩ ✬✜✶✱ ❊❧ ●❛♠❛❧ ❛♥❞ ❈♩st❛ ✭✬✜✶ ❛♥❞ ✬✜✻✮ • R ( W a , W b ) ✐s ❝❧♩s❡ t♩ ♊♣t✐♠❛❧ ❢♊r ●❛✉ss✐❛♥ ■♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧ ⋆ ❊t❊✐♥ ❛♥❞ ❚s❡ ❛♥❞ ❲❛♥❣ ✭✬✵✟✮ • ◆♊✈❡❧ ✐❞❡❛s ❛♥❞ ♠❛t❀❡♠❛t✐❝❛❧ r❡s✉❧ts ❝❛♠❡ ♩✉t ❢r♩♠ t❀❡ ✐♥✈❡st✐❣❛t✐♊♥ ♊❢ ♊♣t✐♠❛❧✐t② ⋆ ❈♊♥❝❛✈✐t② ♊❢ ❡♥tr♊♣② ♣♊✇❡r ✭❈♩st❛ ✬✜✺✮ ⋆ ●❡♥✐❡ ❜❛s❡❞ ❛♣♣r♊❛❝❀ t♩ ♣r♊✈❡ s✉❜✲❛❞❞✐t✐✈✐t② ✭❊❧ ●❛♠❛❧ ❛♥❞ ❈♩st❛ ✬✜✶✱ ❑r❛♠❡r ✬✵✷✮ ✶✷

  30. ❙✉❝❝❡ss❡s ■♥ s♣✐t❡ ♊❢ t❀❡ ✉♥❞❡r❧②✐♥❣ ♣r♊❜❧❡♠ ❜❡✐♥❣ ✐♥tr✐♥s✐❝❛❧❧② ♥♊♥✲❝♊♥✈❡① • R ( W a , W b ) ✐s ♊♣t✐♠❛❧ ❢♊r s♊♠❡ ❝❧❛ss❡s ♊❢ ❝❀❛♥♥❡❧s ⋆ ❈❛r❧❡✐❛❧ ✬✌✺✱ ❙❛t♩ ✬✜✶✱ ❊❧ ●❛♠❛❧ ❛♥❞ ❈♩st❛ ✭✬✜✶ ❛♥❞ ✬✜✻✮ • R ( W a , W b ) ✐s ❝❧♩s❡ t♩ ♊♣t✐♠❛❧ ❢♊r ●❛✉ss✐❛♥ ■♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧ ⋆ ❊t❊✐♥ ❛♥❞ ❚s❡ ❛♥❞ ❲❛♥❣ ✭✬✵✟✮ • ◆♊✈❡❧ ✐❞❡❛s ❛♥❞ ♠❛t❀❡♠❛t✐❝❛❧ r❡s✉❧ts ❝❛♠❡ ♩✉t ❢r♩♠ t❀❡ ✐♥✈❡st✐❣❛t✐♊♥ ♊❢ ♊♣t✐♠❛❧✐t② ⋆ ❈♊♥❝❛✈✐t② ♊❢ ❡♥tr♊♣② ♣♊✇❡r ✭❈♩st❛ ✬✜✺✮ ⋆ ●❡♥✐❡ ❜❛s❡❞ ❛♣♣r♊❛❝❀ t♩ ♣r♊✈❡ s✉❜✲❛❞❞✐t✐✈✐t② ✭❊❧ ●❛♠❛❧ ❛♥❞ ❈♩st❛ ✬✜✶✱ ❑r❛♠❡r ✬✵✷✮ • R ( W a , W b ) ✐s ♥♊t ♊♣t✐♠❛❧ ✐♥ ❣❡♥❡r❛❧ ✭◆❛✐r✱ ❳✐❛✱ ❚❛③❞❛♥♣❛♥❛❀ ✬✶✺✮ ❇r♩❛❞❝❛st ❛♥❞ ✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧s ❛r❡ ❢❛r t♩♩ ✐♠♣♊rt❛♥t • ❚♩ ❧❡t ♥♊♥✲❝♊♥✈❡①✐t② ❞✐ss✉❛❞❡ ✉s • ❚♩ ♥♊t ✐♥✈❡st✐❣❛t❡ s✐♠♣❧❡ ❝❧❛ss❡s t❀❛t r❡q✉✐r❡ ♥❡✇ ✐❞❡❛s ✶✷

  31. ❆ ❝❧❛ss ♊❢ ♊♣❡♥ ♣r♊❜❧❡♠s ❆ s✉❜✲❝♊❧❧❡❝t✐♊♥ ♊❢ t❀❡ ✶✺ ♊♣❡♥ ♣r♊❜❧❡♠s ❧✐st❡❞ ✐♥ ❈❀❛♣s✳ ✺✲✟✳ ✺✳✶ ❲❀❛t ✐s t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥ ♊❢ ❧❡ss ♥♊✐s② ❜r♩❛❞❝❛st✲❝❀❛♥♥❡❧s ✇✐t❀ ❢♊✉r ♩r ♠♩r❡ r❡❝❡✐✈❡rs❄ ✭t✇♊✲r❡❝❡✐✈❡r✿ ❑♩r♥❡r✲▌❛rt♊♥ ✬✌✻✱ t❀r❡❡✲r❡❝❡✐✈❡r✿ ◆❛✐r✲❲❛♥❣ ✬✶✵✮ ✺✳✷ ❲❀❛t ✐s t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥ ♊❢ ♠♩r❡ ❝❛♣❛❜❧❡ ❜r♩❛❞❝❛st✲❝❀❛♥♥❡❧s ✇✐t❀ t❀r❡❡ ♩r ♠♩r❡ r❡❝❡✐✈❡rs❄ ✭t✇♊✲r❡❝❡✐✈❡r✿ ❊❧ ●❛♠❛❧ ✬✌✟✮ ✻✳✶ ❲❀❛t ✐s t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥ ♊❢ t❀❡ ●❛✉ss✐❛♥ ■♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧ ✇✐t❀ ✇❡❛❊ ✐♥t❡r❢❡r❡♥❝❡❄ ✭str♊♥❣✲✐♥t❡r❢❡r❡♥❝❡✿ ❙❛t♩ ✬✌✟❀ ♠✐①❡❞✲✐♥t❡r❢❡r❡♥❝❡ ❝♩r♥❡r✲♣♊✐♥ts✿ ❙❛t♊✬ ✜✶✱ ❈♩st❛✬✜✺❀ ✇❡❛❊✲✐♥t❡r❢❡r❡♥❝❡ ❝♩r♥❡r✲♣♊✐♥ts✿ r❛t❡✲s✉♠ ✭♣❛rt✐❛❧✮✿ t❀r❡❡✲❣r♊✉♣s ✬✵✟ ✮ ✻✳✹ ■s t❀❡ ❍❛♥✲❑♊❜❛②❛s❀✐ ✐♥♥❡r ❜♊✉♥❞ t✐❣❀t ✐♥ ❣❡♥❡r❛❧ ❢♊r ✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧s❄ ✜✳✷ ■s s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ♊♣t✐♠❛❧ ❢♊r t❀❡ ❣❡♥❡r❛❧ ✞✲r❡❝❡✐✈❡r ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✇✐t❀ ♊♥❡ ♠❡ss❛❣❡ t♩ ❛❧❧ t❀r❡❡ r❡❝❡✐✈❡rs ❛♥❞ ❛♥♊t❀❡r ♠❡ss❛❣❡ t♩ t✇♩ r❡❝❡✐✈❡rs❄ ✜✳✞ ❲❀❛t ✐s t❀❡ s✉♠✲❝❛♣❛❝✐t② ♊❢ t❀❡ ❜✐♥❛r② s❊❡✇✲s②♠♠❡tr✐❝ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧❄ ✜✳✹ ■s t❀❡ ▌❛rt♊♥ ✐♥♥❡r ❜♊✉♥❞ t✐❣❀t ✐♥ ❣❡♥❡r❛❧ ❢♊r ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧s❄ ✟✳✷ ❈❛♥ t❀❡ ❝♊♥✈❡rs❡ ❢♊r t❀❡ ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ❜❡ ♣r♊✈❡❞ ❞✐r❡❝t❧② ❜② ♊♣t✐♠✐③✐♥❣ t❀❡ ◆❛✐r✲❊❧ ●❛♠❛❧ ♩✉t❡r ❜♊✉♥❞❄ ✟✳✞ ❲❀❛t ✐s t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥ ♊❢ t❀❡ ✷✲r❡❝❡✐✈❡r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✇✐t❀ ❝♊♠♠♊♥ ♠❡ss❛❣❡❄ ✶✞

  32. ❆ ❝❧❛ss ♊❢ ♊♣❡♥ ♣r♊❜❧❡♠s ▌② r❡❢♊r♠✉❧❛t✐♊♥s ♊❢ ❛ ❢❡✇ ♊❢ t❀❡♠✳ ✺✳✶ ■s s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ♊♣t✐♠❛❧ ❢♊r ❧❡ss✲♥♊✐s② ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧s ✇✐t❀ ❢♊✉r ♩r ♠♩r❡ r❡❝❡✐✈❡rs❄ ✺✳✷ ■s s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ♊♣t✐♠❛❧ ❢♊r ♠♩r❡✲❝❛♣❛❜❧❡ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧s ✇✐t❀ t❀r❡❡ ♩r ♠♩r❡ r❡❝❡✐✈❡rs❄ ✻✳✶ ■s t❀❡ ❍❛♥✕❑♊❜❛②❛s❀✐ s❝❀❡♠❡ ✇✐t❀ ●❛✉ss✐❛♥ s✐❣♥❛❧✐♥❣ t✐❣❀t ❢♊r t❀❡ ●❛✉ss✐❛♥ ■♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧ ✇✐t❀ ✇❡❛❊ ✐♥t❡r❢❡r❡♥❝❡❄ ✻✳✹ ■s t❀❡ ❍❛♥✲❑♊❜❛②❛s❀✐ ✐♥♥❡r ❜♊✉♥❞ t✐❣❀t ✐♥ ❣❡♥❡r❛❧ ❢♊r ✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧s❄ ✜✳✷ ■s s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ♊♣t✐♠❛❧ ❢♊r t❀❡ ❣❡♥❡r❛❧ ✞✲r❡❝❡✐✈❡r ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✇✐t❀ ♊♥❡ ♠❡ss❛❣❡ t♩ ❛❧❧ t❀r❡❡ r❡❝❡✐✈❡rs ❛♥❞ ❛♥♊t❀❡r ♠❡ss❛❣❡ t♩ t✇♩ r❡❝❡✐✈❡rs❄ ✜✳✞ ❉♊❡s t❀❡ ▌❛rt♊♥ ✐♥♥❡r ❜♊✉♥❞ ❛❝❀✐❡✈❡ t❀❡ s✉♠✲❝❛♣❛❝✐t② ♊❢ t❀❡ ❜✐♥❛r② s❊❡✇✲s②♠♠❡tr✐❝ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧❄ ✜✳✹ ■s t❀❡ ▌❛rt♊♥ ✐♥♥❡r ❜♊✉♥❞ t✐❣❀t ✐♥ ❣❡♥❡r❛❧ ❢♊r ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧s❄ ✟✳✷ ❈❛♥ t❀❡ ❝♊♥✈❡rs❡ ❢♊r t❀❡ ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ❜❡ ♣r♊✈❡❞ ❞✐r❡❝t❧② ❜② ♊♣t✐♠✐③✐♥❣ t❀❡ ◆❛✐r✲❊❧ ●❛♠❛❧ ♩✉t❡r ❜♊✉♥❞❄ ✟✳✞ ❉♊❡s t❀❡ ▌❛rt♊♥ ✐♥♥❡r ❜♊✉♥❞ ❛❝❀✐❡✈❡ t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥ ♊❢ t❀❡ ✷✲r❡❝❡✐✈❡r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✇✐t❀ ❝♊♠♠♊♥ ♠❡ss❛❣❡❄ ✶✞

  33. ❚❀❡ ❝♊♠♠♊♥ t❀❡♠❡ t♩ t❀❡s❡ ✭r❡❢♊r♠✉❧❛t❡❞✮ q✉❡st✐♊♥s ❈♊♠♠♊♥ t❀❡♠❡ ■s ❛ ❝❛♥❞✐❞❛t❡ r❛t❡ r❡❣✐♊♥ ♊♣t✐♠❛❧❄ ■❞❡❛ ❢♊r t❡st✐♥❣ ♊♣t✐♠❛❧✐t②✿ • S λ ( W ⊗ W ) ? = 2 S λ ( W ) • ❉❡t❡r♠✐♥❡ s✉❜✲❛❞❞✐t✐✈✐t② ♊❢ ❛♥ ❛ss♩❝✐❛t❡❞ ♥♊♥✲❝♊♥✈❡① ❢✉♥❝t✐♊♥❛❧ ✶✹

  34. ❙t❛t✉s ♊❢ t❀❡ ♊♣❡♥ ♣r♊❜❧❡♠s ✺✳✶ ■s s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ♊♣t✐♠❛❧ ❢♊r ❧❡ss✲♥♊✐s② ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧s ✇✐t❀ ❢♊✉r ♩r ♠♩r❡ r❡❝❡✐✈❡rs❄ ✭❖P❊◆✮ ✺✳✷ ■s s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ♊♣t✐♠❛❧ ❢♊r ♠♩r❡✲❝❛♣❛❜❧❡ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧s ✇✐t❀ t❀r❡❡ ♩r ♠♩r❡ r❡❝❡✐✈❡rs❄ ✭ ◆❖ ✿ ◆❛✐r✲❳✐❛ ✬✶✷✮ ✻✳✶ ■s t❀❡ ❍❛♥✕❑♊❜❛②❛s❀✐ s❝❀❡♠❡ ✇✐t❀ ●❛✉ss✐❛♥ s✐❣♥❛❧✐♥❣ t✐❣❀t ❢♊r t❀❡ ●❛✉ss✐❛♥ ■♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧ ✇✐t❀ ✇❡❛❊ ✐♥t❡r❢❡r❡♥❝❡❄✭❖P❊◆✮ ✭ ❹❊❙ ✿ ❝♩r♥❡r ♣♊✐♥ts ✉s✐♥❣ ✐❞❡❛s ✐♥ ♠❡❛s✉r❡ tr❛♥s♣♊rt❛t✐♊♥ ❜② P♊❧②❛♥s❊✐②✲❲✉ ✬✶✺✮ ✻✳✹ ■s t❀❡ ❍❛♥✲❑♊❜❛②❛s❀✐ ✐♥♥❡r ❜♊✉♥❞ t✐❣❀t ✐♥ ❣❡♥❡r❛❧ ❢♊r ✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧s❄ ✭ ◆❖ ✿ ◆❛✐r✲❳✐❛✲❚❛③❞❛♥♣❛♥❛❀ ✬✶✺✮ ✜✳✷ ■s s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ♊♣t✐♠❛❧ ❢♊r t❀❡ ❣❡♥❡r❛❧ ✞✲r❡❝❡✐✈❡r ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✇✐t❀ ♊♥❡ ♠❡ss❛❣❡ t♩ ❛❧❧ t❀r❡❡ r❡❝❡✐✈❡rs ❛♥❞ ❛♥♊t❀❡r ♠❡ss❛❣❡ t♩ t✇♩ r❡❝❡✐✈❡rs❄ ✭ ◆❖ ✿ ◆❛✐r✲❚❛③❞❛♥♣❛♥❛❀ ✬✶✌✮ ✜✳✞ ❉♊❡s t❀❡ ▌❛rt♊♥ ✐♥♥❡r ❜♊✉♥❞ ❛❝❀✐❡✈❡ t❀❡ s✉♠✲❝❛♣❛❝✐t② ♊❢ t❀❡ ❜✐♥❛r② s❊❡✇✲s②♠♠❡tr✐❝ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧❄ ✭❖P❊◆✮ ✜✳✹ ■s t❀❡ ▌❛rt♊♥ ✐♥♥❡r ❜♊✉♥❞ t✐❣❀t ✐♥ ❣❡♥❡r❛❧ ❢♊r ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧s❄ ✭❖P❊◆✮ ✟✳✷ ❈❛♥ t❀❡ ❝♊♥✈❡rs❡ ❢♊r t❀❡ ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ❜❡ ♣r♊✈❡❞ ❞✐r❡❝t❧② ❜② ♊♣t✐♠✐③✐♥❣ t❀❡ ◆❛✐r✲❊❧ ●❛♠❛❧ ♩✉t❡r ❜♊✉♥❞❄ ✭ ❹❊❙ ✿ ●❡♥❣✲◆❛✐r ✬✶✹✮ ✟✳✞ ❉♊❡s t❀❡ ▌❛rt♊♥ ✐♥♥❡r ❜♊✉♥❞ ❛❝❀✐❡✈❡ t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥ ♊❢ t❀❡ ✷✲r❡❝❡✐✈❡r ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✇✐t❀ ❝♊♠♠♊♥ ♠❡ss❛❣❡❄ ✭ ❹❊❙ ✿ ●❡♥❣✲◆❛✐r ✬✶✹✮ ✶✺

  35. ❖✉t❧✐♥❡ • ❇r♩❛❞❝❛st ❝❀❛♥♥❡❧✿ ❊st❛❜❧✐s❀✐♥❣ ♊♣t✐♠❛❧✐t② ♊❢ ▌❛rt♊♥✬s r❡❣✐♊♥ ❢♊r ▌■▌❖ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ • ■♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧✿ ❙✉❜✲♊♣t✐♠❛❧✐t② ♊❢ t❀❡ ❍❛♥✕❑♊❜❛②❛s❀✐ r❡❣✐♊♥ • ❋❛♠✐❧② ♊❢ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠s ⋆ ❘❡❧❛t✐♊♥ t♩ ♣r♊❜❧❡♠s ♊❢ ✐♥t❡r❡st ✐♥ ♩t❀❡r ✜❡❧❞s ⋆ ❯♥✐❢②✐♥❣ ♩❜s❡r✈❛t✐♊♥s ❛♥❞ s♊♠❡ ❝♊♥❥❡❝t✉r❡s ✶✻

  36. ▌■▌❖ ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✿ ✇❀❡r❡ ❞❡♥♊t❡s t❀❡ ❛❞❞✐t✐✈❡ ♥♊✐s❡✳ ❱❡r② ✐♠♣♊rt❛♥t ❝❀❛♥♥❡❧ ❝❧❛ss ✐♥ ✇✐r❡❧❡ss ❝♊♠♠✉♥✐❝❛t✐♊♥ ▌♊❞❡❧s✿ ♠✉❧t✐✲❛♥t❡♥♥❛ tr❛♥s♠✐tt❡r✮r❡❝❡✐✈❡rs ✭❞♊✇♥❧✐♥❊✮ ▌■▌❖ ✭❱❡❝t♩r✮ ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ Y n 1 M 0 , ˆ ˆ W a ( y 1 | x ) ❉❡❝♊❞❡r ✶ M 1 X n ( M 0 , M 1 , M 2 ) ❊♥❝♊❞❡r Y n 2 M 0 , ˜ ˜ W b ( y 2 | x ) ❉❡❝♊❞❡r ✷ M 2 ✶✌

  37. ▌■▌❖ ✭❱❡❝t♩r✮ ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ Y n 1 M 0 , ˆ ˆ W a ( y 1 | x ) ❉❡❝♊❞❡r ✶ M 1 X n ( M 0 , M 1 , M 2 ) ❊♥❝♊❞❡r Y n 2 M 0 , ˜ ˜ W b ( y 2 | x ) ❉❡❝♊❞❡r ✷ M 2 ▌■▌❖ ●❛✉ss✐❛♥ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ✿ Y 1 = AX + Z Y 2 = BX + Z ✇❀❡r❡ Z ∌ N (0 , I ) ❞❡♥♊t❡s t❀❡ ❛❞❞✐t✐✈❡ ♥♊✐s❡✳ ❱❡r② ✐♠♣♊rt❛♥t ❝❀❛♥♥❡❧ ❝❧❛ss ✐♥ ✇✐r❡❧❡ss ❝♊♠♠✉♥✐❝❛t✐♊♥ ▌♊❞❡❧s✿ ♠✉❧t✐✲❛♥t❡♥♥❛ tr❛♥s♠✐tt❡r✮r❡❝❡✐✈❡rs ✭❞♊✇♥❧✐♥❊✮ ✶✌

  38. ●✐st ✿ ❙❀♊✇✐♥❣ ♊♣t✐♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ r❛♥❞♊♠ ✈❛r✐❛❜❧❡s ❢♊r ❛ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠ ❘❡♠❛r❊ ✿ ■❞❡❛s ❞♩ ♥♊t ❡①t❡♥❞ t♩ s❀♊✇ ♊♣t✐♠❛❧✐t② ✇❀❡♥ t❀❡r❡ ✐s ❝♊♠♠♊♥ ♠❡ss❛❣❡✱ ✐✳❡✳ ❖♣t✐♠❛❧✐t② ✐♥ ❣❡♥❡r❛❧ ✭●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹✮ ●✐st ✿ ❉❡✈❡❧♊♣ ❛ t❡❝❀♥✐q✉❡ ❢♊r ♣r♊✈✐♥❣ ♊♣t✐♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ r❛♥❞♊♠ ✈❛r✐❛❜❧❡s ✭❢r♩♠ s✉❜✲❛❞❞✐t✐✈✐t②✮ ❊①♣❧❛✐♥ ♩✉r t❡❝❀♥✐q✉❡ ♊♥ ✭❢♊r s✐♠♣❧✐❝✐t②✮ ❍✐st♩r② ❖♣t✐♠❛❧✐t② ♊❢ ▌❛rt♊♥✬s ❜♊✉♥❞✱ R ( W a , W b ) ✱ ✇❛s ❡st❛❜❧✐s❀❡❞✿ • ❙❝❛❧❛r ❝❛s❡ ✭❇❡r❣♠❛♥s ✬✌✞✮ ✭❊♥tr♊♣② P♊✇❡r ■♥❡q✉❛❧✐t②✮ • ❘❡✈❡rs❡❧② ❞❡❣r❛❞❡❞ s❡tt✐♥❣ ✭P♩❧t②r❡✈ ✬✌✜✱ ❊❧ ●❛♠❛❧ ✬✜✶✮ • ❖♣t✐♠❛❧✐t② ♊♥ R 0 = 0 ✭❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❀❛♠❛✐ ✬✵✻✮ ⋆ ❇✉✐❧❞s ♊♥ ✐❞❡❛s ✐♥ P♩❧t②r❡✈ ⋆ ❚♩✉r ❞❡ ❢♊r❝❡ ✐♥ ♊♣t✐♠✐③❛t✐♊♥ ✶✜

  39. ❖♣t✐♠❛❧✐t② ✐♥ ❣❡♥❡r❛❧ ✭●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹✮ ●✐st ✿ ❉❡✈❡❧♊♣ ❛ t❡❝❀♥✐q✉❡ ❢♊r ♣r♊✈✐♥❣ ♊♣t✐♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ r❛♥❞♊♠ ✈❛r✐❛❜❧❡s ✭❢r♩♠ s✉❜✲❛❞❞✐t✐✈✐t②✮ ❊①♣❧❛✐♥ ♩✉r t❡❝❀♥✐q✉❡ ♊♥ ✭❢♊r s✐♠♣❧✐❝✐t②✮ ❍✐st♩r② ❖♣t✐♠❛❧✐t② ♊❢ ▌❛rt♊♥✬s ❜♊✉♥❞✱ R ( W a , W b ) ✱ ✇❛s ❡st❛❜❧✐s❀❡❞✿ • ❙❝❛❧❛r ❝❛s❡ ✭❇❡r❣♠❛♥s ✬✌✞✮ ✭❊♥tr♊♣② P♊✇❡r ■♥❡q✉❛❧✐t②✮ • ❘❡✈❡rs❡❧② ❞❡❣r❛❞❡❞ s❡tt✐♥❣ ✭P♩❧t②r❡✈ ✬✌✜✱ ❊❧ ●❛♠❛❧ ✬✜✶✮ • ❖♣t✐♠❛❧✐t② ♊♥ R 0 = 0 ✭❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❀❛♠❛✐ ✬✵✻✮ ⋆ ❇✉✐❧❞s ♊♥ ✐❞❡❛s ✐♥ P♩❧t②r❡✈ ⋆ ❚♩✉r ❞❡ ❢♊r❝❡ ✐♥ ♊♣t✐♠✐③❛t✐♊♥ ⋆ ●✐st ✿ ❙❀♊✇✐♥❣ ♊♣t✐♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ r❛♥❞♊♠ ✈❛r✐❛❜❧❡s ❢♊r ❛ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠ ⋆ ❘❡♠❛r❊ ✿ ■❞❡❛s ❞♩ ♥♊t ❡①t❡♥❞ t♩ s❀♊✇ ♊♣t✐♠❛❧✐t② ✇❀❡♥ t❀❡r❡ ✐s ❝♊♠♠♊♥ ♠❡ss❛❣❡✱ ✐✳❡✳ R 0 ï¿œ = 0 ✶✜

  40. ❊①♣❧❛✐♥ ♩✉r t❡❝❀♥✐q✉❡ ♊♥ ✭❢♊r s✐♠♣❧✐❝✐t②✮ ❍✐st♩r② ❖♣t✐♠❛❧✐t② ♊❢ ▌❛rt♊♥✬s ❜♊✉♥❞✱ R ( W a , W b ) ✱ ✇❛s ❡st❛❜❧✐s❀❡❞✿ • ❙❝❛❧❛r ❝❛s❡ ✭❇❡r❣♠❛♥s ✬✌✞✮ ✭❊♥tr♊♣② P♊✇❡r ■♥❡q✉❛❧✐t②✮ • ❘❡✈❡rs❡❧② ❞❡❣r❛❞❡❞ s❡tt✐♥❣ ✭P♩❧t②r❡✈ ✬✌✜✱ ❊❧ ●❛♠❛❧ ✬✜✶✮ • ❖♣t✐♠❛❧✐t② ♊♥ R 0 = 0 ✭❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❀❛♠❛✐ ✬✵✻✮ ⋆ ❇✉✐❧❞s ♊♥ ✐❞❡❛s ✐♥ P♩❧t②r❡✈ ⋆ ❚♩✉r ❞❡ ❢♊r❝❡ ✐♥ ♊♣t✐♠✐③❛t✐♊♥ ⋆ ●✐st ✿ ❙❀♊✇✐♥❣ ♊♣t✐♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ r❛♥❞♊♠ ✈❛r✐❛❜❧❡s ❢♊r ❛ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠ ⋆ ❘❡♠❛r❊ ✿ ■❞❡❛s ❞♩ ♥♊t ❡①t❡♥❞ t♩ s❀♊✇ ♊♣t✐♠❛❧✐t② ✇❀❡♥ t❀❡r❡ ✐s ❝♊♠♠♊♥ ♠❡ss❛❣❡✱ ✐✳❡✳ R 0 ï¿œ = 0 • ❖♣t✐♠❛❧✐t② ✐♥ ❣❡♥❡r❛❧ ✭●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹✮ ⋆ ●✐st ✿ ❉❡✈❡❧♊♣ ❛ t❡❝❀♥✐q✉❡ ❢♊r ♣r♊✈✐♥❣ ♊♣t✐♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ r❛♥❞♊♠ ✈❛r✐❛❜❧❡s ✭❢r♩♠ s✉❜✲❛❞❞✐t✐✈✐t②✮ ✶✜

  41. ❍✐st♩r② ❖♣t✐♠❛❧✐t② ♊❢ ▌❛rt♊♥✬s ❜♊✉♥❞✱ R ( W a , W b ) ✱ ✇❛s ❡st❛❜❧✐s❀❡❞✿ • ❙❝❛❧❛r ❝❛s❡ ✭❇❡r❣♠❛♥s ✬✌✞✮ ✭❊♥tr♊♣② P♊✇❡r ■♥❡q✉❛❧✐t②✮ • ❘❡✈❡rs❡❧② ❞❡❣r❛❞❡❞ s❡tt✐♥❣ ✭P♩❧t②r❡✈ ✬✌✜✱ ❊❧ ●❛♠❛❧ ✬✜✶✮ • ❖♣t✐♠❛❧✐t② ♊♥ R 0 = 0 ✭❲❡✐♥❣❛rt❡♥ ❛♥❞ ❙t❡✐♥❜❡r❣ ❛♥❞ ❙❀❛♠❛✐ ✬✵✻✮ ⋆ ❇✉✐❧❞s ♊♥ ✐❞❡❛s ✐♥ P♩❧t②r❡✈ ⋆ ❚♩✉r ❞❡ ❢♊r❝❡ ✐♥ ♊♣t✐♠✐③❛t✐♊♥ ⋆ ●✐st ✿ ❙❀♊✇✐♥❣ ♊♣t✐♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ r❛♥❞♊♠ ✈❛r✐❛❜❧❡s ❢♊r ❛ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠ ⋆ ❘❡♠❛r❊ ✿ ■❞❡❛s ❞♩ ♥♊t ❡①t❡♥❞ t♩ s❀♊✇ ♊♣t✐♠❛❧✐t② ✇❀❡♥ t❀❡r❡ ✐s ❝♊♠♠♊♥ ♠❡ss❛❣❡✱ ✐✳❡✳ R 0 ï¿œ = 0 • ❖♣t✐♠❛❧✐t② ✐♥ ❣❡♥❡r❛❧ ✭●❡♥❣ ❛♥❞ ◆❛✐r ✬✶✹✮ ⋆ ●✐st ✿ ❉❡✈❡❧♊♣ ❛ t❡❝❀♥✐q✉❡ ❢♊r ♣r♊✈✐♥❣ ♊♣t✐♠❛❧✐t② ♊❢ ●❛✉ss✐❛♥ r❛♥❞♊♠ ✈❛r✐❛❜❧❡s ✭❢r♩♠ s✉❜✲❛❞❞✐t✐✈✐t②✮ ❊①♣❧❛✐♥ ♩✉r t❡❝❀♥✐q✉❡ ♊♥ R 0 = 0 ✭❢♊r s✐♠♣❧✐❝✐t②✮ ✶✜

  42. ✭◆❛✐r ✬✶✞✮ ✉♣♣❡r ❝♊♥❝❛✈❡ ❡♥✈❡❧♊♣❡✿ ◆♩t ❡❛s② t♩ ❝♊♠♣✉t❡ ✭✐♥ ❣❡♥❡r❛❧✮ ❋♩r ✱ ❧❡t ❖✉t❡r ❜♊✉♥❞ ✭❑♩r♥❡r✲▌❛rt♊♥ ✬✌✟✮ ❚❀❡ s❡t ♊❢ r❛t❡ ♣❛✐rs ( R 1 , R 2 ) s❛t✐s❢②✐♥❣ R 2 ≀ I ( U ; Y 2 ) R 1 + R 2 ≀ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) ❢♊r s♊♠❡ p ( u, x ) ✱ ✇❀❡r❡ E( ï¿œ X ï¿œ 2 ) ≀ P ❢♊r♠s ❛♥ ♩✉t❡r ❜♊✉♥❞ t♩ t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥✳ ❉❡♥♊t❡ t❀✐s r❡❣✐♊♥ ❛s O ( W a , W b ) ✳ ✶✟

  43. ✭◆❛✐r ✬✶✞✮ ✉♣♣❡r ❝♊♥❝❛✈❡ ❡♥✈❡❧♊♣❡✿ ◆♩t ❡❛s② t♩ ❝♊♠♣✉t❡ ✭✐♥ ❣❡♥❡r❛❧✮ ❖✉t❡r ❜♊✉♥❞ ✭❑♩r♥❡r✲▌❛rt♊♥ ✬✌✟✮ ❚❀❡ s❡t ♊❢ r❛t❡ ♣❛✐rs ( R 1 , R 2 ) s❛t✐s❢②✐♥❣ R 2 ≀ I ( U ; Y 2 ) R 1 + R 2 ≀ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) ❢♊r s♊♠❡ p ( u, x ) ✱ ✇❀❡r❡ E( ï¿œ X ï¿œ 2 ) ≀ P ❢♊r♠s ❛♥ ♩✉t❡r ❜♊✉♥❞ t♩ t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥✳ ❉❡♥♊t❡ t❀✐s r❡❣✐♊♥ ❛s O ( W a , W b ) ✳ ❋♩r λ > 1 ✱ ❧❡t S λ ( W a , W b ) := ( R 1 ,R 2 ) ∈O R 1 + λR 2 max = max p ( u,x ) λI ( U ; Y 2 ) + I ( X ; Y 1 | U ) ✶✟

  44. ✉♣♣❡r ❝♊♥❝❛✈❡ ❡♥✈❡❧♊♣❡✿ ◆♩t ❡❛s② t♩ ❝♊♠♣✉t❡ ✭✐♥ ❣❡♥❡r❛❧✮ ❖✉t❡r ❜♊✉♥❞ ✭❑♩r♥❡r✲▌❛rt♊♥ ✬✌✟✮ ❚❀❡ s❡t ♊❢ r❛t❡ ♣❛✐rs ( R 1 , R 2 ) s❛t✐s❢②✐♥❣ R 2 ≀ I ( U ; Y 2 ) R 1 + R 2 ≀ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) ❢♊r s♊♠❡ p ( u, x ) ✱ ✇❀❡r❡ E( ï¿œ X ï¿œ 2 ) ≀ P ❢♊r♠s ❛♥ ♩✉t❡r ❜♊✉♥❞ t♩ t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥✳ ❉❡♥♊t❡ t❀✐s r❡❣✐♊♥ ❛s O ( W a , W b ) ✳ ❋♩r λ > 1 ✱ ❧❡t S λ ( W a , W b ) := ( R 1 ,R 2 ) ∈O R 1 + λR 2 max = max p ( u,x ) λI ( U ; Y 2 ) + I ( X ; Y 1 | U ) ï¿œ ï¿œ λI ( X ; Z ) + C µ X [ I ( X ; Y ) − λI ( X ; Z )] = max ✭◆❛✐r ✬✶✞✮ p ( x ) ✶✟

  45. ❖✉t❡r ❜♊✉♥❞ ✭❑♩r♥❡r✲▌❛rt♊♥ ✬✌✟✮ ❚❀❡ s❡t ♊❢ r❛t❡ ♣❛✐rs ( R 1 , R 2 ) s❛t✐s❢②✐♥❣ R 2 ≀ I ( U ; Y 2 ) R 1 + R 2 ≀ I ( U ; Y 2 ) + I ( X ; Y 1 | U ) ❢♊r s♊♠❡ p ( u, x ) ✱ ✇❀❡r❡ E( ï¿œ X ï¿œ 2 ) ≀ P ❢♊r♠s ❛♥ ♩✉t❡r ❜♊✉♥❞ t♩ t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥✳ ❉❡♥♊t❡ t❀✐s r❡❣✐♊♥ ❛s O ( W a , W b ) ✳ ❋♩r λ > 1 ✱ ❧❡t S λ ( W a , W b ) := ( R 1 ,R 2 ) ∈O R 1 + λR 2 max = max p ( u,x ) λI ( U ; Y 2 ) + I ( X ; Y 1 | U ) ï¿œ ï¿œ λI ( X ; Z ) + C µ X [ I ( X ; Y ) − λI ( X ; Z )] = max ✭◆❛✐r ✬✶✞✮ p ( x ) x x f ( x ) ✉♣♣❡r ❝♊♥❝❛✈❡ ❡♥✈❡❧♊♣❡✿ C x [ f ] ✶✟ ◆♩t ❡❛s② t♩ ❝♊♠♣✉t❡ ✭✐♥ ❣❡♥❡r❛❧✮

  46. ❖♥❡ ❝❛♥ s❀♊✇ t❀❛t ✐❢ ●❛✉ss✐❛♥s ♠❛①✐♠✐③❡ C µ X [ h ( Y 1 ) − λh ( Y 2 )] t❀❡♥ ▌❛rt♊♥✬s ✐♥♥❡r ❜♊✉♥❞ ✐s ♊♣t✐♠❛❧ ✭♊♥ R 0 = 0 ✮ ❍❡r❡ h ( X ) ✐s t❀❡ ❞✐✛❡r❡♥t✐❛❧ ❡♥tr♊♣②✿ ï¿œ − h ( X ) := f ( x ) log f ( x ) dx, ✇❀❡r❡ f ( x ) ✐s t❀❡ ❞❡♥s✐t② ❢✉♥❝t✐♊♥ ♊❢ X ✳ ❆ s✐♠✐❧❛r ✭♠♩r❡✲✐♥✈♊❧✈❡❞✮ ♣r♊❜❧❡♠ s❀♊✇s ✉♣ ✇❀❡♥ R 0 ï¿œ = 0 ❆♥ ✐❞❡♥t✐❝❛❧ t❡❝❀♥✐q✉❡ ✭t♩ t❀❡ ♊♥❡ ■ ❛♠ ❣♊✐♥❣ t♩ ❞❡♠♊♥str❛t❡✮ ❡st❛❜❧✐s❀❡s t❀❛t ❛❧s♩ ✷✵

  47. ▲❡♠♠❛ ✿ ✐s s✉❜✲❛❞❞✐t✐✈❡✳ Pr♊♊❢ ✿ ❋♩r ❛♥② ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t② ✈✐❛ s✉❜✲❛❞❞✐t✐✈✐t② ✭●❡♥❣✲◆❛✐r ✬✶✹✮ ▌❛①✐♠✐③❡✱ ❢♊r λ > 1 ✱ t❀❡ ✈❛❧✉❡ ♊❢ t❀❡ ❢✉♥❝t✐♊♥❛❧ C µ X [ h ( AX + Z ) − λh ( BX + Z )] ♊✈❡r X : ❊ ( XX T ) ï¿œ K ✱ ✇❀❡r❡ A, B ❛r❡ ✐♥✈❡rt✐❜❧❡ ♠❛tr✐❝❡s ❛♥❞ Z ∌ N (0 , I ) ✳ ❲❡ ✇✐❧❧ s❡❡ t❀❛t t❀❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ✐s h ( AX ∗ + Z ) − λh ( BX ∗ + Z ) , ✇❀❡r❡ X ∗ ∌ N (0 , K ′ ) ❢♊r s♊♠❡ K ′ ï¿œ K ✳ ✷✶

  48. ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t② ✈✐❛ s✉❜✲❛❞❞✐t✐✈✐t② ✭●❡♥❣✲◆❛✐r ✬✶✹✮ ▌❛①✐♠✐③❡✱ ❢♊r λ > 1 ✱ t❀❡ ✈❛❧✉❡ ♊❢ t❀❡ ❢✉♥❝t✐♊♥❛❧ C µ X [ h ( AX + Z ) − λh ( BX + Z )] ♊✈❡r X : ❊ ( XX T ) ï¿œ K ✱ ✇❀❡r❡ A, B ❛r❡ ✐♥✈❡rt✐❜❧❡ ♠❛tr✐❝❡s ❛♥❞ Z ∌ N (0 , I ) ✳ ❲❡ ✇✐❧❧ s❡❡ t❀❛t t❀❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ✐s h ( AX ∗ + Z ) − λh ( BX ∗ + Z ) , ✇❀❡r❡ X ∗ ∌ N (0 , K ′ ) ❢♊r s♊♠❡ K ′ ï¿œ K ✳ ▲❡♠♠❛ ✿ C µ X [ h ( AX + Z ) − λh ( BX + Z )] ✐s s✉❜✲❛❞❞✐t✐✈❡✳ Pr♊♊❢ ✿ ❋♩r ❛♥② µ X 1 ,X 2 h ( AX 1 + Z 1 , AX 2 + Z 2 | U ) − λh ( BX 1 + Z 1 , BX 2 + Z 2 | U ) = h ( AX 1 + Z 1 | U, AX 2 + Z 2 ) − λh ( BX 1 + Z 1 | U, AX 2 + Z 2 ) + h ( AX 2 + Z 2 | U, BX 1 + Z 1 ) − λh ( BX 2 + Z 2 | U, BX 1 + Z 1 ) − ( λ − 1) I ( AX 2 + Z 2 ; BX 1 + Z 1 | U ) ✷✶

  49. ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t② ✈✐❛ s✉❜✲❛❞❞✐t✐✈✐t② ✭●❡♥❣✲◆❛✐r ✬✶✹✮ ▌❛①✐♠✐③❡✱ ❢♊r λ > 1 ✱ t❀❡ ✈❛❧✉❡ ♊❢ t❀❡ ❢✉♥❝t✐♊♥❛❧ C µ X [ h ( AX + Z ) − λh ( BX + Z )] ♊✈❡r X : ❊ ( XX T ) ï¿œ K ✱ ✇❀❡r❡ A, B ❛r❡ ✐♥✈❡rt✐❜❧❡ ♠❛tr✐❝❡s ❛♥❞ Z ∌ N (0 , I ) ✳ ❲❡ ✇✐❧❧ s❡❡ t❀❛t t❀❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ✐s h ( AX ∗ + Z ) − λh ( BX ∗ + Z ) , ✇❀❡r❡ X ∗ ∌ N (0 , K ′ ) ❢♊r s♊♠❡ K ′ ï¿œ K ✳ ▲❡♠♠❛ ✿ C µ X [ h ( AX + Z ) − λh ( BX + Z )] ✐s s✉❜✲❛❞❞✐t✐✈❡✳ Pr♊♊❢ ✿ ❋♩r ❛♥② µ X 1 ,X 2 C µ X 1 ,X 2 [ h ( AX 1 + Z 1 , AX 2 + Z 2 ) − λh ( BX 1 + Z 1 , BX 2 + Z 2 )] ≀ C µ X 1 [ h ( AX 1 + Z 1 ) − λh ( BX 1 + Z 1 )] + C µ X 2 [ h ( AX 2 + Z 2 ) − λh ( BX 2 + Z 2 )] ✷✶

  50. ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t② ✈✐❛ s✉❜✲❛❞❞✐t✐✈✐t② ✭●❡♥❣✲◆❛✐r ✬✶✹✮ ▌❛①✐♠✐③❡✱ ❢♊r λ > 1 ✱ t❀❡ ✈❛❧✉❡ ♊❢ t❀❡ ❢✉♥❝t✐♊♥❛❧ C µ X [ h ( AX + Z ) − λh ( BX + Z )] ♊✈❡r X : ❊ ( XX T ) ï¿œ K ✱ ✇❀❡r❡ A, B ❛r❡ ✐♥✈❡rt✐❜❧❡ ♠❛tr✐❝❡s ❛♥❞ Z ∌ N (0 , I ) ✳ ❲❡ ✇✐❧❧ s❡❡ t❀❛t t❀❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ ✐s h ( AX ∗ + Z ) − λh ( BX ∗ + Z ) , ✇❀❡r❡ X ∗ ∌ N (0 , K ′ ) ❢♊r s♊♠❡ K ′ ï¿œ K ✳ ▲❡♠♠❛ ✿ C µ X [ h ( AX + Z ) − λh ( BX + Z )] ✐s s✉❜✲❛❞❞✐t✐✈❡✳ Pr♊♊❢ ✿ ❋♩r ❛♥② µ X 1 ,X 2 h ( AX 1 + Z 1 , AX 2 + Z 2 | U ) − λh ( BX 1 + Z 1 , BX 2 + Z 2 | U ) = h ( AX 1 + Z 1 | U, AX 2 + Z 2 ) − λh ( BX 1 + Z 1 | U, AX 2 + Z 2 ) + h ( AX 2 + Z 2 | U, BX 1 + Z 1 ) − λh ( BX 2 + Z 2 | U, BX 1 + Z 1 ) − ( λ − 1) I ( AX 2 + Z 2 ; BX 1 + Z 1 | U ) ✷✶

  51. ■♠♣❧✐❡s t❀❛t ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ✿ ❛r❡ ●❛✉ss✐❛♥ ❈❀❛r❛❝t❡r✐③❛t✐♊♥ ♊❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ Pr♊♊❢✿ ❯s✐♥❣ ❝❀❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♊♥s ✭❋♩✉r✐❡r tr❛♥s❢♊r♠s✮ ❚❀✐s t❡❝❀♥✐q✉❡ ❀❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♩t❀❡rs ✐♥ ✈❛r✐♩✉s ♩t❀❡r ✐♥st❛♥❝❡s✳ ◆♩t❡ ✿ ❚❀❡r❡ ❛r❡ s♊♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❀ ✇♩r❊ ♊❢ ▲✐❡❜ ❛♥❞ ❇❛rt❀❡ ✭✟✵s✮ ❚❀❡② ❛❧s♩ ✉s❡ r♩t❛t✐♊♥s ✭❜✉t ♥♊t ✐♥❢♊r♠❛t✐♊♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮ ◆♩t❡ ✿ ❚❀✉s✱ ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ✿ ✭❢r♩♠ ❝♊♥str✉❝t✐♊♥✮ ✭❢r♩♠ ♣r♊♊❢ ♊❢ s✉❜✲❛❞❞✐t✐✈✐t②✮ ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t②✿ ❝t❞✳✳ ▲❡t ( U † , X † ) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max µ X C µ X [ h ( AX + Z ) − λh ( BX + Z )] = h ( AX † + Z | U † ) − λh ( BX † + Z | U † ) . ▲❡t ( X a , U a ) ❛♥❞ ( X b , U b ) ❜❡ ✐✳✐✳❞✳ ❛❝❝♩r❞✐♥❣ t♩ ( U † , X † ) ✳ ✷✷

  52. ■♠♣❧✐❡s t❀❛t ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ✿ ❛r❡ ●❛✉ss✐❛♥ ❈❀❛r❛❝t❡r✐③❛t✐♊♥ ♊❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ Pr♊♊❢✿ ❯s✐♥❣ ❝❀❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♊♥s ✭❋♩✉r✐❡r tr❛♥s❢♊r♠s✮ ❚❀✐s t❡❝❀♥✐q✉❡ ❀❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♩t❀❡rs ✐♥ ✈❛r✐♩✉s ♩t❀❡r ✐♥st❛♥❝❡s✳ ◆♩t❡ ✿ ❚❀❡r❡ ❛r❡ s♊♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❀ ✇♩r❊ ♊❢ ▲✐❡❜ ❛♥❞ ❇❛rt❀❡ ✭✟✵s✮ ❚❀❡② ❛❧s♩ ✉s❡ r♩t❛t✐♊♥s ✭❜✉t ♥♊t ✐♥❢♊r♠❛t✐♊♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮ ❚❀❡r❡❢♊r❡ ✿ ✇❡ ❣❡t t❀❛t ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ✿ ✳ ◆♩t❡ ✿ ❚❀✉s✱ ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ✿ ✭❢r♩♠ ❝♊♥str✉❝t✐♊♥✮ ✭❢r♩♠ ♣r♊♊❢ ♊❢ s✉❜✲❛❞❞✐t✐✈✐t②✮ ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t②✿ ❝t❞✳✳ ▲❡t ( U † , X † ) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max µ X C µ X [ h ( AX + Z ) − λh ( BX + Z )] = h ( AX † + Z | U † ) − λh ( BX † + Z | U † ) . ▲❡t ( X a , U a ) ❛♥❞ ( X b , U b ) ❜❡ ✐✳✐✳❞✳ ❛❝❝♩r❞✐♥❣ t♩ ( U † , X † ) ✳ ❙❡tt✐♥❣ U = ( U a , U b ) ✱ X + = X a + X b ❛♥❞ X − = X a − X b t❀❡ ♣r♊♊❢ ♊❢ s✉❜✲❛❞❞✐t✐✈✐t② √ √ 2 2 ②✐❡❧❞s ï¿œ ï¿œ 2 V = C µ X 1 ,X 2 [ h ( AX 1 + Z 1 , AX 2 + Z 2 ) − λh ( BX 1 + Z 1 , BX 2 + Z 2 )] ï¿œ ( µ X + ,X − ) ï¿œ ï¿œ ≀ C µ X 1 [ h ( AX 1 + Z 1 ) − λh ( BX 1 + Z 1 )] ï¿œ µ X + ï¿œ ï¿œ + C µ X 2 [ h ( AX 2 + Z 2 ) − λh ( BX 2 + Z 2 )] ï¿œ µ X − − ( λ − 1) I ( AX − + Z 2 ; BX + + Z 1 | U a , U b ) ≀ V + V ✷✷

  53. ■♠♣❧✐❡s t❀❛t ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ✿ ❛r❡ ●❛✉ss✐❛♥ ❈❀❛r❛❝t❡r✐③❛t✐♊♥ ♊❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ Pr♊♊❢✿ ❯s✐♥❣ ❝❀❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♊♥s ✭❋♩✉r✐❡r tr❛♥s❢♊r♠s✮ ❚❀✐s t❡❝❀♥✐q✉❡ ❀❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♩t❀❡rs ✐♥ ✈❛r✐♩✉s ♩t❀❡r ✐♥st❛♥❝❡s✳ ◆♩t❡ ✿ ❚❀❡r❡ ❛r❡ s♊♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❀ ✇♩r❊ ♊❢ ▲✐❡❜ ❛♥❞ ❇❛rt❀❡ ✭✟✵s✮ ❚❀❡② ❛❧s♩ ✉s❡ r♩t❛t✐♊♥s ✭❜✉t ♥♊t ✐♥❢♊r♠❛t✐♊♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮ ◆♩t❡ ✿ ❚❀✉s✱ ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ✿ ✭❢r♩♠ ❝♊♥str✉❝t✐♊♥✮ ✭❢r♩♠ ♣r♊♊❢ ♊❢ s✉❜✲❛❞❞✐t✐✈✐t②✮ ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t②✿ ❝t❞✳✳ ▲❡t ( U † , X † ) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max µ X C µ X [ h ( AX + Z ) − λh ( BX + Z )] = h ( AX † + Z | U † ) − λh ( BX † + Z | U † ) . ▲❡t ( X a , U a ) ❛♥❞ ( X b , U b ) ❜❡ ✐✳✐✳❞✳ ❛❝❝♩r❞✐♥❣ t♩ ( U † , X † ) ✳ ❙❡tt✐♥❣ U = ( U a , U b ) ✱ X + = X a + X b ❛♥❞ X − = X a − X b t❀❡ ♣r♊♊❢ ♊❢ s✉❜✲❛❞❞✐t✐✈✐t② √ √ 2 2 ②✐❡❧❞s ï¿œ ï¿œ 2 V = C µ X 1 ,X 2 [ h ( AX 1 + Z 1 , AX 2 + Z 2 ) − λh ( BX 1 + Z 1 , BX 2 + Z 2 )] ï¿œ ( µ X + ,X − ) ï¿œ ï¿œ ≀ C µ X 1 [ h ( AX 1 + Z 1 ) − λh ( BX 1 + Z 1 )] ï¿œ µ X + ï¿œ ï¿œ + C µ X 2 [ h ( AX 2 + Z 2 ) − λh ( BX 2 + Z 2 )] ï¿œ µ X − − ( λ − 1) I ( AX − + Z 2 ; BX + + Z 1 | U a , U b ) ≀ V + V ❚❀❡r❡❢♊r❡ ✿ ✇❡ ❣❡t t❀❛t ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ( U a , U b ) ✿ X + ⊥ X − ✳ ✷✷

  54. ■♠♣❧✐❡s t❀❛t ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ✿ ❛r❡ ●❛✉ss✐❛♥ ❈❀❛r❛❝t❡r✐③❛t✐♊♥ ♊❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ Pr♊♊❢✿ ❯s✐♥❣ ❝❀❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♊♥s ✭❋♩✉r✐❡r tr❛♥s❢♊r♠s✮ ❚❀✐s t❡❝❀♥✐q✉❡ ❀❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♩t❀❡rs ✐♥ ✈❛r✐♩✉s ♩t❀❡r ✐♥st❛♥❝❡s✳ ◆♩t❡ ✿ ❚❀❡r❡ ❛r❡ s♊♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❀ ✇♩r❊ ♊❢ ▲✐❡❜ ❛♥❞ ❇❛rt❀❡ ✭✟✵s✮ ❚❀❡② ❛❧s♩ ✉s❡ r♩t❛t✐♊♥s ✭❜✉t ♥♊t ✐♥❢♊r♠❛t✐♊♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮ ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t②✿ ❝t❞✳✳ ▲❡t ( U † , X † ) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max µ X C µ X [ h ( AX + Z ) − λh ( BX + Z )] = h ( AX † + Z | U † ) − λh ( BX † + Z | U † ) . ▲❡t ( X a , U a ) ❛♥❞ ( X b , U b ) ❜❡ ✐✳✐✳❞✳ ❛❝❝♩r❞✐♥❣ t♩ ( U † , X † ) ✳ ◆♩t❡ ✿ ❚❀✉s✱ ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ( U a , U b ) ✿ • X a ⊥ X b ✭❢r♩♠ ❝♊♥str✉❝t✐♊♥✮ • ( X a + X b ) ⊥ ( X a − X b ) ✭❢r♩♠ ♣r♊♊❢ ♊❢ s✉❜✲❛❞❞✐t✐✈✐t②✮ ✷✷

  55. ❚❀✐s t❡❝❀♥✐q✉❡ ❀❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♩t❀❡rs ✐♥ ✈❛r✐♩✉s ♩t❀❡r ✐♥st❛♥❝❡s✳ ◆♩t❡ ✿ ❚❀❡r❡ ❛r❡ s♊♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❀ ✇♩r❊ ♊❢ ▲✐❡❜ ❛♥❞ ❇❛rt❀❡ ✭✟✵s✮ ❚❀❡② ❛❧s♩ ✉s❡ r♩t❛t✐♊♥s ✭❜✉t ♥♊t ✐♥❢♊r♠❛t✐♊♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮ ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t②✿ ❝t❞✳✳ ▲❡t ( U † , X † ) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max µ X C µ X [ h ( AX + Z ) − λh ( BX + Z )] = h ( AX † + Z | U † ) − λh ( BX † + Z | U † ) . ▲❡t ( X a , U a ) ❛♥❞ ( X b , U b ) ❜❡ ✐✳✐✳❞✳ ❛❝❝♩r❞✐♥❣ t♩ ( U † , X † ) ✳ ◆♩t❡ ✿ ❚❀✉s✱ ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ( U a , U b ) ✿ • X a ⊥ X b ✭❢r♩♠ ❝♊♥str✉❝t✐♊♥✮ • ( X a + X b ) ⊥ ( X a − X b ) ✭❢r♩♠ ♣r♊♊❢ ♊❢ s✉❜✲❛❞❞✐t✐✈✐t②✮ • ■♠♣❧✐❡s t❀❛t ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ( U a , U b ) ✿ X a , X b ❛r❡ ●❛✉ss✐❛♥ ⋆ ❈❀❛r❛❝t❡r✐③❛t✐♊♥ ♊❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ ⋆ Pr♊♊❢✿ ❯s✐♥❣ ❝❀❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♊♥s ✭❋♩✉r✐❡r tr❛♥s❢♊r♠s✮ ✷✷

  56. ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t②✿ ❝t❞✳✳ ▲❡t ( U † , X † ) ❜❡ ❛ ♠❛①✐♠✐③❡r✱ ✐✳❡✳ V = max µ X C µ X [ h ( AX + Z ) − λh ( BX + Z )] = h ( AX † + Z | U † ) − λh ( BX † + Z | U † ) . ▲❡t ( X a , U a ) ❛♥❞ ( X b , U b ) ❜❡ ✐✳✐✳❞✳ ❛❝❝♩r❞✐♥❣ t♩ ( U † , X † ) ✳ ◆♩t❡ ✿ ❚❀✉s✱ ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ( U a , U b ) ✿ • X a ⊥ X b ✭❢r♩♠ ❝♊♥str✉❝t✐♊♥✮ • ( X a + X b ) ⊥ ( X a − X b ) ✭❢r♩♠ ♣r♊♊❢ ♊❢ s✉❜✲❛❞❞✐t✐✈✐t②✮ • ■♠♣❧✐❡s t❀❛t ❝♊♥❞✐t✐♊♥❡❞ ♊♥ ( U a , U b ) ✿ X a , X b ❛r❡ ●❛✉ss✐❛♥ ⋆ ❈❀❛r❛❝t❡r✐③❛t✐♊♥ ♊❢ ●❛✉ss✐❛♥s ✭❇❡r♥st❡✐♥ ✬✹✵s✮ ⋆ Pr♊♊❢✿ ❯s✐♥❣ ❝❀❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♊♥s ✭❋♩✉r✐❡r tr❛♥s❢♊r♠s✮ ❚❀✐s t❡❝❀♥✐q✉❡ ❀❛s ❜❡❡♥ s✉❜s❡q✉❡♥t❧② ✉s❡❞ ❜② ♩t❀❡rs ✐♥ ✈❛r✐♩✉s ♩t❀❡r ✐♥st❛♥❝❡s✳ ◆♩t❡ ✿ ❚❀❡r❡ ❛r❡ s♊♠❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❀ ✇♩r❊ ♊❢ ▲✐❡❜ ❛♥❞ ❇❛rt❀❡ ✭✟✵s✮ ❚❀❡② ❛❧s♩ ✉s❡ r♩t❛t✐♊♥s ✭❜✉t ♥♊t ✐♥❢♊r♠❛t✐♊♥ ♠❡❛s✉r❡s ❛♥❞ ✐ts ❛❧❣❡❜r❛✮ ✷✷

  57. ❆✣r♠❛t✐✈❡ ✐❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❢✉♥❝t✐♊♥❛❧ s✉❜✲❛❞❞✐t✐✈❡❄ ❲❀② s❀♊✉❧❞ s♊♠❡♊♥❡ ❝❛r❡❄ ■❢ tr✉❡✱ s♊❧✈❡s t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥ ❢♊r t❀❡ ●❛✉ss✐❛♥ ❩✲✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧ ❘❡❧❛t❡❞ t♩ r❡✈❡rs❡ ❊P■s✱ ❀②♣❡r♣❧❛♥❡ ❝♊♥❥❡❝t✉r❡✱ ❡t❝✳ ❆♥ ♊♣❡♥ q✉❡st✐♊♥ ❲❡ ❀❛✈❡ s❡❡♥ ✭②❡st❡r❞❛② ❛♥❞ t♊❞❛②✮ ❀♊✇ s✉❜✲❛❞❞✐t✐✈✐t② ✐♠♣❧✐❡s ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t② ❖♣❡♥ q✉❡st✐♊♥ ❋♩r α, a ∈ (0 , 1) ✱ ❞♩ ●❛✉ss✐❛♥s ♠❛①✐♠✐③❡ t❀❡ ❢✉♥❝t✐♊♥❛❧ αh ( X 2 + aX 1 + Z ) + (1 − α ) h ( X 1 + Z ) − h ( aX 1 + Z ) ♊✈❡r X 1 ⊥ X 2 ✱ s✉❜❥❡❝t t♩ E( X 2 1 ) ≀ P 1 ✱ E( X 2 2 ) ≀ P 2 ✳ ❍❡r❡ Z ∌ N (0 , 1) ✐s ✐♥❞❡♣❡♥❞❡♥t ♊❢ X 1 , X 2 ✳ ✷✞

  58. ❲❀② s❀♊✉❧❞ s♊♠❡♊♥❡ ❝❛r❡❄ ■❢ tr✉❡✱ s♊❧✈❡s t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥ ❢♊r t❀❡ ●❛✉ss✐❛♥ ❩✲✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧ ❘❡❧❛t❡❞ t♩ r❡✈❡rs❡ ❊P■s✱ ❀②♣❡r♣❧❛♥❡ ❝♊♥❥❡❝t✉r❡✱ ❡t❝✳ ❆♥ ♊♣❡♥ q✉❡st✐♊♥ ❲❡ ❀❛✈❡ s❡❡♥ ✭②❡st❡r❞❛② ❛♥❞ t♊❞❛②✮ ❀♊✇ s✉❜✲❛❞❞✐t✐✈✐t② ✐♠♣❧✐❡s ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t② ❖♣❡♥ q✉❡st✐♊♥ ❋♩r α, a ∈ (0 , 1) ✱ ❞♩ ●❛✉ss✐❛♥s ♠❛①✐♠✐③❡ t❀❡ ❢✉♥❝t✐♊♥❛❧ αh ( X 2 + aX 1 + Z ) + (1 − α ) h ( X 1 + Z ) − h ( aX 1 + Z ) ♊✈❡r X 1 ⊥ X 2 ✱ s✉❜❥❡❝t t♩ E( X 2 1 ) ≀ P 1 ✱ E( X 2 2 ) ≀ P 2 ✳ ❍❡r❡ Z ∌ N (0 , 1) ✐s ✐♥❞❡♣❡♥❞❡♥t ♊❢ X 1 , X 2 ✳ ❆✣r♠❛t✐✈❡ ✐❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❢✉♥❝t✐♊♥❛❧ s✉❜✲❛❞❞✐t✐✈❡❄ ï¿œ ï¿œ C µ X 1 αh ( X 2 + aX 1 + Z ) + (1 − α ) h ( X 1 + Z ) − h ( aX 1 + Z ) ✷✞

  59. ❆♥ ♊♣❡♥ q✉❡st✐♊♥ ❲❡ ❀❛✈❡ s❡❡♥ ✭②❡st❡r❞❛② ❛♥❞ t♊❞❛②✮ ❀♊✇ s✉❜✲❛❞❞✐t✐✈✐t② ✐♠♣❧✐❡s ●❛✉ss✐❛♥ ♊♣t✐♠❛❧✐t② ❖♣❡♥ q✉❡st✐♊♥ ❋♩r α, a ∈ (0 , 1) ✱ ❞♩ ●❛✉ss✐❛♥s ♠❛①✐♠✐③❡ t❀❡ ❢✉♥❝t✐♊♥❛❧ αh ( X 2 + aX 1 + Z ) + (1 − α ) h ( X 1 + Z ) − h ( aX 1 + Z ) ♊✈❡r X 1 ⊥ X 2 ✱ s✉❜❥❡❝t t♩ E( X 2 1 ) ≀ P 1 ✱ E( X 2 2 ) ≀ P 2 ✳ ❍❡r❡ Z ∌ N (0 , 1) ✐s ✐♥❞❡♣❡♥❞❡♥t ♊❢ X 1 , X 2 ✳ ❆✣r♠❛t✐✈❡ ✐❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❢✉♥❝t✐♊♥❛❧ s✉❜✲❛❞❞✐t✐✈❡❄ ï¿œ ï¿œ C µ X 1 αh ( X 2 + aX 1 + Z ) + (1 − α ) h ( X 1 + Z ) − h ( aX 1 + Z ) ❲❀② s❀♊✉❧❞ s♊♠❡♊♥❡ ❝❛r❡❄ • ■❢ tr✉❡✱ s♊❧✈❡s t❀❡ ❝❛♣❛❝✐t② r❡❣✐♊♥ ❢♊r t❀❡ ●❛✉ss✐❛♥ ❩✲✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧ • ❘❡❧❛t❡❞ t♩ r❡✈❡rs❡ ❊P■s✱ ❀②♣❡r♣❧❛♥❡ ❝♊♥❥❡❝t✉r❡✱ ❡t❝✳ ✷✞

  60. ❖✉t❧✐♥❡ • ❇r♩❛❞❝❛st ❝❀❛♥♥❡❧✿ ❊st❛❜❧✐s❀✐♥❣ ♊♣t✐♠❛❧✐t② ♊❢ ▌❛rt♊♥✬s ❢♊r ▌■▌❖ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ • ■♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧✿ ❙✉❜✲♊♣t✐♠❛❧✐t② ♊❢ t❀❡ ❍❛♥✕❑♊❜❛②❛s❀✐ r❡❣✐♊♥ • ❋❛♠✐❧② ♊❢ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠s ⋆ ❘❡❧❛t✐♊♥ t♩ ♣r♊❜❧❡♠s ♊❢ ✐♥t❡r❡st ✐♥ ♩t❀❡r ✜❡❧❞s ⋆ ❯♥✐❢②✐♥❣ ♩❜s❡r✈❛t✐♊♥s ❛♥❞ s♊♠❡ ❝♊♥❥❡❝t✉r❡s ✷✹

  61. ■♥t❡r❢❡r❡♥❝❡ ❈❀❛♥♥❡❧ ✭❆❀❧s✇❡❞❡ ✬✌✹✮ X n Y n 1 1 W a ( y 1 | x 1 , x 2 ) ˆ M 1 ❊♥❝♊❞❡r ✶ ❉❡❝♊❞❡r ✶ M 1 Y n 2 ˆ W b ( y 2 | x 1 , x 2 ) M 2 ❊♥❝♊❞❡r ✷ ❉❡❝♊❞❡r ✷ M 2 X n 2 ✷✺

  62. ❋✐rst st❡♣✿ ❋✐♥❞ ❛ ❝❀❛♥♥❡❧ ❝❧❛ss ✇❀❡r❡ ❍❑ r❡❣✐♊♥ s✐♠♣❧✐✜❡s ❆◆❉ ②❡t ♥♊t t♩♩ tr✐✈✐❛❧ ❍❛♥✲❑♊❜❛②❛s❀✐ ❛❝❀✐❡✈❛❜❧❡ r❡❣✐♊♥ ✭✶✟✜✶✮ á ❧❛ ❈❀♊♥❣ ❡t✳ ❛❧✳ ❆ r❛t❡✲♣❛✐r ( R 1 , R 2 ) ✐s ❛❝❀✐❡✈❛❜❧❡ ❢♊r t❀❡ ✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧ ✐❢ R 1 < I ( X 1 ; Y 1 | U 2 , Q ) , R 2 < I ( X 2 ; Y 2 | U 1 , Q ) , R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | Q ) + I ( X 2 ; Y 2 | U 1 , U 2 , Q ) , R 1 + R 2 < I ( X 2 , U 1 ; Y 2 | Q ) + I ( X 1 ; Y 1 | U 1 , U 2 , Q ) , R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | U 1 , Q ) + I ( X 2 , U 1 ; Y 2 | U 2 , Q ) , 2 R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | Q ) + I ( X 1 ; Y 1 | U 1 , U 2 , Q ) + I ( X 2 , U 1 ; Y 2 | U 2 , Q ) , R 1 + 2 R 2 < I ( X 2 , U 1 ; Y 2 | Q ) + I ( X 2 ; Y 2 | U 1 , U 2 , Q ) + I ( X 1 , U 2 ; Y 1 | U 1 , Q ) ❢♊r s♊♠❡ ♣♠❢ p ( q ) p ( u 1 , x 1 | q ) p ( u 2 , x 2 | q ) ✱ ✇❀❡r❡ | U 1 | ≀ | X 1 | + 4 ✱ | U 2 | ≀ | X 2 | + 4 ✱ ❛♥❞ | Q | ≀ 7 ✳ ❉❡♥♊t❡ t❀❡ ✭❝❧♩s✉r❡ ♊❢✮ r❡❣✐♊♥ ❛s R ( W a , W b ) ✳ ◆✉♠❡r✐❝❛❧❧② ✐♥❢❡❛s✐❜❧❡ t♩ ❝♊♠♣✉t❡ R ( W a W b ) ❡✈❡♥ ❢♊r ❣❡♥❡r✐❝ ❜✐♥❛r②✲✐♥♣✉t ❜✐♥❛r②✲♊✉t♣✉t ✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧s ✷✻

  63. ❍❛♥✲❑♊❜❛②❛s❀✐ ❛❝❀✐❡✈❛❜❧❡ r❡❣✐♊♥ ✭✶✟✜✶✮ á ❧❛ ❈❀♊♥❣ ❡t✳ ❛❧✳ ❆ r❛t❡✲♣❛✐r ( R 1 , R 2 ) ✐s ❛❝❀✐❡✈❛❜❧❡ ❢♊r t❀❡ ✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧ ✐❢ R 1 < I ( X 1 ; Y 1 | U 2 , Q ) , R 2 < I ( X 2 ; Y 2 | U 1 , Q ) , R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | Q ) + I ( X 2 ; Y 2 | U 1 , U 2 , Q ) , R 1 + R 2 < I ( X 2 , U 1 ; Y 2 | Q ) + I ( X 1 ; Y 1 | U 1 , U 2 , Q ) , R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | U 1 , Q ) + I ( X 2 , U 1 ; Y 2 | U 2 , Q ) , 2 R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | Q ) + I ( X 1 ; Y 1 | U 1 , U 2 , Q ) + I ( X 2 , U 1 ; Y 2 | U 2 , Q ) , R 1 + 2 R 2 < I ( X 2 , U 1 ; Y 2 | Q ) + I ( X 2 ; Y 2 | U 1 , U 2 , Q ) + I ( X 1 , U 2 ; Y 1 | U 1 , Q ) ❢♊r s♊♠❡ ♣♠❢ p ( q ) p ( u 1 , x 1 | q ) p ( u 2 , x 2 | q ) ✱ ✇❀❡r❡ | U 1 | ≀ | X 1 | + 4 ✱ | U 2 | ≀ | X 2 | + 4 ✱ ❛♥❞ | Q | ≀ 7 ✳ ❉❡♥♊t❡ t❀❡ ✭❝❧♩s✉r❡ ♊❢✮ r❡❣✐♊♥ ❛s R ( W a , W b ) ✳ ◆✉♠❡r✐❝❛❧❧② ✐♥❢❡❛s✐❜❧❡ t♩ ❝♊♠♣✉t❡ R ( W a W b ) ❡✈❡♥ ❢♊r ❣❡♥❡r✐❝ ❜✐♥❛r②✲✐♥♣✉t ❜✐♥❛r②✲♊✉t♣✉t ✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧s ❋✐rst st❡♣✿ • ❋✐♥❞ ❛ ❝❀❛♥♥❡❧ ❝❧❛ss ✇❀❡r❡ ❍❑ r❡❣✐♊♥ s✐♠♣❧✐✜❡s ❆◆❉ ②❡t ♥♊t t♩♩ tr✐✈✐❛❧ ✷✻

  64. ❈❧❡❛♥ ❩ ■♥t❡r❢❡r❡♥❝❡ ❈❀❛♥♥❡❧ ✭❈❩■❈✮ ▌♊❞❡❧ X n Y n 1 1 ˆ M 1 W a ( y 1 | x 1 , x 2 ) ❊♥❝♊❞❡r ✶ ❉❡❝♊❞❡r ✶ M 1 Y n 2 = X n 2 ˆ M 2 ❊♥❝♊❞❡r ✷ ❉❡❝♊❞❡r ✷ M 2 X n 2 ❈❧❡❛♥ ❩✲✐♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧ ▲❡♠♠❛ ✿ ❆ r❛t❡✲♣❛✐r ( R 1 , R 2 ) ❜❡❧♊♥❣s t♩ ❍❛♥✲❑♊❜❛②❛s❀✐ r❡❣✐♊♥ ✐❢ ❛♥❞ ♊♥❧② ✐❢ R 1 < I ( X 1 ; Y 1 | U 2 , Q ) , R 2 < H ( X 2 | Q ) , R 1 + R 2 < I ( X 1 , U 2 ; Y 1 | Q ) + H ( X 2 | U 2 , Q ) , ❢♊r s♊♠❡ ♣♠❢ p ( q ) p ( x 1 | q ) p ( u 2 , x 2 | q ) ✱ ✇❀❡r❡ | U 2 | ≀ | X 2 | ❛♥❞ | Q | ≀ 2 ✳ ❉❡♥♊t❡ r❡❣✐♊♥✿ R ( W a ) ✷✌

  65. ▲❡♠♠❛ ✭s✉❜✲❛❞❞✐t✐✈✐t②✮ ✭◆❛✐r✲❳✐❛✲❚❛③❞❛♥♣❛♥❛❀ ✬✶✺✮✿ ■♠♣❧✐❡s ♊♣t✐♠❛❧✐t② ♊❢ ✱ ✳ ❋♩r ✱ ✐s ❣✐✈❡♥ ❜② ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ❊q✉✐✈❛❧❡♥t t♩ t❡st ✐❢ S λ ( W a ⊗ W a ) = 2 S λ ( W a ) , ∀ W a , λ ≥ 0 , ✇❀❡r❡ S λ ( W a ) := ( R 1 ,R 2 ) ∈R ( W a ) λR 1 + R 2 . max ✷✜

  66. ▲❡♠♠❛ ✭s✉❜✲❛❞❞✐t✐✈✐t②✮ ✭◆❛✐r✲❳✐❛✲❚❛③❞❛♥♣❛♥❛❀ ✬✶✺✮✿ ■♠♣❧✐❡s ♊♣t✐♠❛❧✐t② ♊❢ ✱ ✳ ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ❊q✉✐✈❛❧❡♥t t♩ t❡st ✐❢ S λ ( W a ⊗ W a ) = 2 S λ ( W a ) , ∀ W a , λ ≥ 0 , ✇❀❡r❡ S λ ( W a ) := ( R 1 ,R 2 ) ∈R ( W a ) λR 1 + R 2 . max ❋♩r λ ∈ [0 , 1] ✱ S λ ( W a ) ✐s ❣✐✈❡♥ ❜② ï¿œ ï¿œ (1 − λ ) H ( X 2 ) + λI ( X 1 , U 2 ; Y 1 ) + λH ( X 2 | U 2 ) max p 1 ( x 1 ) p 2 ( u 2 ,x 2 ) ï¿œ ï¿œ = max H ( X 2 ) + λI ( X 1 ; Y 1 ) p 1 ( x 1 ) p 2 ( x 2 ) ✷✜

  67. ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ❊q✉✐✈❛❧❡♥t t♩ t❡st ✐❢ S λ ( W a ⊗ W a ) = 2 S λ ( W a ) , ∀ W a , λ ≥ 0 , ✇❀❡r❡ S λ ( W a ) := ( R 1 ,R 2 ) ∈R ( W a ) λR 1 + R 2 . max ❋♩r λ ∈ [0 , 1] ✱ S λ ( W a ) ✐s ❣✐✈❡♥ ❜② ï¿œ ï¿œ (1 − λ ) H ( X 2 ) + λI ( X 1 , U 2 ; Y 1 ) + λH ( X 2 | U 2 ) max p 1 ( x 1 ) p 2 ( u 2 ,x 2 ) ï¿œ ï¿œ = max H ( X 2 ) + λI ( X 1 ; Y 1 ) p 1 ( x 1 ) p 2 ( x 2 ) ▲❡♠♠❛ ✭s✉❜✲❛❞❞✐t✐✈✐t②✮ ✭◆❛✐r✲❳✐❛✲❚❛③❞❛♥♣❛♥❛❀ ✬✶✺✮✿ H ( X 21 , X 22 ) + λI ( X 11 , X 12 ; Y 11 , Y 12 ) ï¿œ ï¿œ ï¿œ ï¿œ ≀ H ( X 21 ) + λI ( X 11 ; Y 11 ) + H ( X 22 ) + λI ( X 12 ; Y 12 ) − (1 − λ ) I ( X 21 ; X 22 ) . ■♠♣❧✐❡s ♊♣t✐♠❛❧✐t② ♊❢ S λ ( W a ) ✱ λ ∈ [0 , 1] ✳ ✷✜

  68. ◗✉❡st✐♊♥ ✿ ❈❛♥ ✇❡ ♥✉♠❡r✐❝❛❧❧② t❡st ✐❢ ❄ ✐s ❛ ❜✐♥❛r② r❛♥❞♊♠ ✈❛r✐❛❜❧❡ ✭✐✳❡✳ ❝♊♥❝❛✈❡ ❡♥✈❡❧♊♣❡ ♊✈❡r s✐♥❣❧❡ ✈❛r✐❛❜❧❡✮ ✿ ❀❛s ❛t ♠♩st ✷ ✐♥✢❡①✐♊♥ ♣♊✐♥ts ❲❀❛t ❛❜♩✉t λ > 1 ❄ ❋♩r λ ≥ 1 ✱ S λ ( W a ) ✐s ❣✐✈❡♥ ❜② ï¿œ ï¿œ I ( X 1 , U 2 ; Y 1 ) + H ( X 2 | U 2 ) + ( λ − 1) I ( X 1 ; Y 1 | U 2 ) max p 1 ( x 1 ) p 2 ( u 2 ,x 2 ) ï¿œ ï¿œ I ( X 1 , X 2 ; Y 1 ) + C X 2 [( λ − 1) I ( X 1 ; Y 1 ) + H ( X 2 ) − I ( X 2 ; Y 1 | X 1 )] = max p 1 ( x 1 ) p 2 ( x 2 ) ✷✟

  69. ❲❀❛t ❛❜♩✉t λ > 1 ❄ ❋♩r λ ≥ 1 ✱ S λ ( W a ) ✐s ❣✐✈❡♥ ❜② ï¿œ ï¿œ I ( X 1 , U 2 ; Y 1 ) + H ( X 2 | U 2 ) + ( λ − 1) I ( X 1 ; Y 1 | U 2 ) max p 1 ( x 1 ) p 2 ( u 2 ,x 2 ) ï¿œ ï¿œ I ( X 1 , X 2 ; Y 1 ) + C X 2 [( λ − 1) I ( X 1 ; Y 1 ) + H ( X 2 ) − I ( X 2 ; Y 1 | X 1 )] = max p 1 ( x 1 ) p 2 ( x 2 ) ◗✉❡st✐♊♥ ✿ ❈❛♥ ✇❡ ♥✉♠❡r✐❝❛❧❧② t❡st ✐❢ S λ ( W a ⊗ W a ) = 2 S λ ( W a ) ❄ X 2 ✐s ❛ ❜✐♥❛r② r❛♥❞♊♠ ✈❛r✐❛❜❧❡ ✭✐✳❡✳ ❝♊♥❝❛✈❡ ❡♥✈❡❧♊♣❡ ♊✈❡r s✐♥❣❧❡ ✈❛r✐❛❜❧❡✮ ( λ − 1) I ( X 1 ; Y 1 ) + H ( X 2 ) − I ( X 2 ; Y 1 | X 1 ) ✿ ❀❛s ❛t ♠♩st ✷ ✐♥✢❡①✐♊♥ ♣♊✐♥ts ✷✟

  70. ❲❀❛t ❛❜♩✉t λ > 1 ❄ H ( X 2 ) + ( λ − 1) H ( Y 1 ) − λH ( Y 1 | X 1 ) P 2 ( X 2 ) [ H ( X 2 ) + ( λ − 1) H ( Y 1 ) − λH ( Y 1 | X 1 )] C P ( X 2 ) ❚❀❡ s❀❛♣❡ ♊❢ ❝♊♥❝❛✈❡ ❡♥✈❡❧♊♣❡ ❢♊r ❛ ❣❡♥❡r✐❝ ❜✐♥❛r② ❈❩■❈ ✷✟

  71. ❙✉❜✲♊♣t✐♠❛❧✐t② ♊❢ t❀❡ ❍❛♥✲❑♊❜❛②❛s❀✐ r❡❣✐♊♥ A ❍❑ 1 2 A ❚■◆ ( W ⊗ 2 ) W ( Y 1 = 0 | X 1 , X 2 ) λ ( W ) λ λ ï¿œ 1 ï¿œ 0 . 5 ✷ ✶✳✶✵✌✺✶✻ ✶✳✶✵✜✶✹✶ 1 0 ï¿œ 0 . 12 ï¿œ 0 . 89 ✟ ✶✳✵✌✹✹✜✹ ✶✳✵✌✺✺✹✹ 0 . 21 0 . 62 ï¿œ 0 . 01 ï¿œ 0 . 58 ✶✷ ✶✳✷✜✟✜✞✵ ✶✳✷✟✞✌✻✵ 0 . 20 0 . 74 ï¿œ 0 . 78 ï¿œ 0 . 07 ✶✹ ✶✳✹✷✻✺✷✻ ✶✳✹✞✷✹✶✟ 0 . 46 0 . 05 ï¿œ 0 . 91 ï¿œ 0 . 22 ✶✺ ✶✳✞✷✞✌✻✻ ✶✳✞✞✟✵✻✺ 0 . 66 0 . 15 ï¿œ 0 . 91 ï¿œ 0 . 13 ✶✻ ✶✳✺✶✺✹✷✶ ✶✳✺✞✹✌✷✹ 0 . 62 0 . 06 ï¿œ 0 . 38 ï¿œ 0 . 87 ✶✜ ✶✳✹✹✟✟✺✟ ✶✳✹✻✜✺✌✌ 0 . 12 0 . 79 ❈♊✉♥t❡r❡①❛♠♣❧❡s t♩ t❀❡ ♊♣t✐♠❛❧✐t② ♊❢ ❍❛♥✲❑♊❜❛②❛s❀✐ r❡❣✐♊♥✳ ◆♩t❡ ✿ ❋♩r t❀❡ ✜rst ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ t❀❡ ❝♊♥❝❛✈❡ ❡♥✈❡❧♊♣❡ ❛♥❛❧②t✐❝❛❧❧②✳ ✞✵

  72. ▲❡t ❛♥❞ r❡s♣❡❝t✐✈❡❧② ❞❡♥♊t❡ ❛♥❞ ✇❀❡r❡ ✐s t❀❡ ❜✐♥❛r② ❡♥tr♊♣② ❢✉♥❝t✐♊♥ P❛rt✐❝✉❧❛r ❈❀❛♥♥❡❧ 1 2 0 0 0 0 X 1 Y 1 X 1 Y 1 1 2 1 1 1 1 X 2 = 0 X 2 = 1 • ❲❡ ❝♊♠♣✉t❡ max HK λR 1 + R 2 ❢♊r λ = 2 ï¿œ ￜￜ ï¿œ max H ( Y 1 ) + H ( X 2 ) + 2 H ( Y 1 ) − H ( Y 1 | X 1 ) C p 1 ( x 1 ) p 2 ( x 2 ) p 2 ( x 2 ) ✞✶

  73. P❛rt✐❝✉❧❛r ❈❀❛♥♥❡❧ 1 2 0 0 0 0 X 1 Y 1 X 1 Y 1 1 2 1 1 1 1 X 2 = 0 X 2 = 1 • ❲❡ ❝♊♠♣✉t❡ max HK λR 1 + R 2 ❢♊r λ = 2 f ( p, q ) ï¿œ ￜￜ ï¿œ max H ( Y 1 ) + H ( X 2 ) + 2 H ( Y 1 ) − H ( Y 1 | X 1 ) C p 1 ( x 1 ) p 2 ( x 2 ) p 2 ( x 2 ) • ▲❡t p ❛♥❞ q r❡s♣❡❝t✐✈❡❧② ❞❡♥♊t❡ Pr ( X 1 = 0) ❛♥❞ Pr ( X 2 = 0) q ) − 2 ph b ( q + 1 p ) h b ( q ) + h b ( q + p f ( p, q ) = (1 − 2¯ 2 ¯ ) 2 ✇❀❡r❡ h b ( . ) ✐s t❀❡ ❜✐♥❛r② ❡♥tr♊♣② ❢✉♥❝t✐♊♥ ✞✶

  74. P❛rt✐❝✉❧❛r ❝❀❛♥♥❡❧ ❝♊♥t✐♥✉❡❞ f ( p, q ) ✐s ❝♊♥❝❛✈❡ ✐♥ q ❢♊r p ≥ 1 2 ❛♥❞ ❢♊r 0 ≀ p < 1 2  f ( p, q ) q > 1 − 2 p  q [ f ( p, q )] = C f ( p, 1 − 2 p ) − f ( p, 0) q + f ( p, 0) q ∈ [0 , 1 − 2 p ]  1 − 2 p ✞✷

  75. P❛rt✐❝✉❧❛r ❝❀❛♥♥❡❧ ❝♊♥t✐♥✉❡❞ f ( p, q ) ✐s ❝♊♥❝❛✈❡ ✐♥ q ❢♊r p ≥ 1 2 ❛♥❞ ❢♊r 0 ≀ p < 1 2  f ( p, q ) q > 1 − 2 p  q [ f ( p, q )] = C f ( p, 1 − 2 p ) − f ( p, 0) q + f ( p, 0) q ∈ [0 , 1 − 2 p ]  1 − 2 p ✵✳✞ ✵✳✷✺ ✵✳✷ ✵✳✶✺ ✵✳✶ f (0 . 2 , q ) ✵✳✵✺ C q [ f (0 . 2 , q )] q ✵ 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 ✞✷

  76. P❛rt✐❝✉❧❛r ❝❀❛♥♥❡❧ ❝♊♥t✐♥✉❡❞ f ( p, q ) ✐s ❝♊♥❝❛✈❡ ✐♥ q ❢♊r p ≥ 1 2 ❛♥❞ ❢♊r 0 ≀ p < 1 2  f ( p, q ) q > 1 − 2 p  q [ f ( p, q )] = C f ( p, 1 − 2 p ) − f ( p, 0) q + f ( p, 0) q ∈ [0 , 1 − 2 p ]  1 − 2 p ❈♩r♩❧❧❛r② ▌❛①✐♠✉♠ ♊❢ 2 R 1 + R 2 ❢♊r t❀❡ ❍❛♥✕❑♊❜❛②❛s❀✐ r❡❣✐♊♥ ✐s ❡q✉❛❧ t♩ t❀❡ ♠❛①✐♠✉♠ ♊❢ T ( p, q ) ❢♊r ( p, q ) ∈ [0 , 1] × [0 , 1] ✱ ✇❀❡r❡  h b ( q + p 2 ¯ q ) + f ( p, q ) q ≥ min { 0 , 1 − 2 p }  T ( p, q ) = q ) + f ( p, 1 − 2 p ) − f ( p, 0) h b ( q + p 2 ¯ q + f ( p, 0) o.w.,  1 − 2 p q ) − 2 ph b ( q +1 p ) h b ( q ) + h b ( q + p ✇❀❡r❡ f ( p, q ) = (1 − 2¯ 2 ¯ 2 ) ✞✷

  77. P❧♩t ♊❢ T ( p, q ) ◆✉♠❡r✐❝❛❧ s❡❛r❝❀ ✐♥❞✐❝❛t❡s✿ max p,q T ( p, q ) = 1 . 107516 .. ❛t p = 0 . 5078 .. ❛♥❞ q = 0 . 4365 .. ✾✾

  78. P❛rt✐❝✉❧❛r ❝❀❛♥♥❡❧ ❝♊♥t✐♥✉❡❞ • ■♥t❡r✈❛❧ ❛r✐t❀♠❡t✐❝ ✐s ❛ ♠❡t❀♊❞ t♩ ♩❜t❛✐♥ ❢♊r♠❛❧ ❜♊✉♥❞s ❢♊r ❢✉♥❝t✐♊♥s ❝♊♥s✐st✐♥❣ ♊❢ ❜❛s✐❝ ❛r✐t❀♠❡t✐❝ ❢✉♥❝t✐♊♥s ❛♥❞ ❝♊♠♠♊♥❧② ✉s❡❞ ❢✉♥❝t✐♊♥s s✉❝❀ ❛s ❧♊❣❛r✐t❀♠s ❛♥❞ tr✐❣♊♥♊♠❡tr✐❝ ❢✉♥❝t✐♊♥s✳ • T ( p, q ) ♊♥❧② ✐♥❝❧✉❞❡s ❜❛s✐❝ ❛r✐t❀♠❡t✐❝ ❢✉♥❝t✐♊♥s ❛♥❞ ❧♊❣❛r✐t❀♠✳ • ❲❡ ✉s❡❞ ❏✉❧✐❛ ❜❛s❡❞ ✐♠♣❧❡♠❡♥t❛t✐♊♥ ♊❢ t❀✐s ❢♊r♠❛❧ ♠❡t❀♊❞ t♩ ♩❜t❛✐♥ max T ( p, q ) ∈ [1 . 10751 , 1 . 10769] • ❚❀❡ ✷✲❧❡tt❡r ❚■◆ ❛❝❀✐❡✈❡s 2 R 1 + R 2 = 1 . 108141 ❛t t❀❡ ❞✐str✐❜✉t✐♊♥ P (( X 11 , X 12 ) = (0 , 0)) = p P (( X 11 , X 12 ) = (1 , 1)) = 1 − p P (( X 21 , X 22 ) = (0 , 0)) = 0 . 36 q P (( X 21 , X 22 ) = (1 , 1)) = 1 − 1 . 64 q P (( X 21 , X 22 ) = (0 , 1)) = 0 . 64 q P (( X 21 , X 22 ) = (1 , 0)) = 0 . 64 q ✇❀❡r❡ p = 0 . 507829413 ✱ q = 0 . 436538150 • ❘❡♣❡t✐t✐♊♥ ❝♊❞✐♥❣ ❛❝r♩ss t✐♠❡ s❡❡♠s t♩ ♩✉t♣❡r❢♊r♠ ♠❡♠♊r②❧❡ss ❝♊❞✐♥❣ ✞✹

  79. ■❢ s✉❜✲❛❞❞✐t✐✈❡✱ t❀❡♥ ▌❛rt♊♥✬s r❡❣✐♊♥ ✐s ♊♣t✐♠❛❧ ❢♊r ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ ■❢ ❡①❛♠♣❧❡ ✇❀❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡✱ t❀❡♥ ♊♥❡ s❀♊✉❧❞ ❜❡ ❛❜❧❡ t♩ ❞❡❞✉❝❡ ❛ ❝❀❛♥♥❡❧ ✇❀❡r❡ ▌❛rt♊♥✬s r❡❣✐♊♥ ✐s ♥♊t ♊♣t✐♠❛❧ ❘❡♠❛r❊s✿ ❈♊♥❥❡❝t✉r❡❞ t♩ ❜❡ s✉❜✲❛❞❞✐t✐✈❡ ✭❆♥❛♥t❀❛r❛♠✲●♊❀❛r✐✲◆❛✐r ✬✶✞✮ ❚♩ ❡✈❛❧✉❛t❡ t❀❡ ❝♊♥❝❛✈❡ ❡♥✈❡❧♊♣❡ ❙✉✣❝❡s t♩ ❝♊♥s✐❞❡r ✿ ✳ ❲❡ ❞✐❞ ♥♊t ❣❡t ❛♥② ❝♊♥tr❛❞✐❝t✐♊♥ t♩ s✉❜✲❛❞❞✐✈✐t② ❢♊r ❜✐♥❛r② ✐♥♣✉t ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧s ❈❛♥ ♣r♊✈❡ s✉❜✲❛❞❞✐t✐✈✐t② ✇❀❡♥ ♩r ✳ ❲❀❛t ❛❜♩✉t ▌❛rt♊♥✬s r❡❣✐♊♥ ❢♊r t❀❡ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧❄ ■s t❀❡ ❢♊❧❧♊✇✐♥❣ ❢✉♥❝t✐♊♥❛❧ s✉❜✲❛❞❞✐t✐✈❡ ♩r ✐s t❀❡r❡ ❛♥ ❡①❛♠♣❧❡ ✇❀❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡❄ ▲❡t W a ( y | x ) ❛♥❞ W b ( z | x ) ❜❡ ❣✐✈❡♥ ❝❀❛♥♥❡❧s✱ α ∈ [0 , 1] ✱ ❛♥❞ λ ≥ 1 ✳ ï¿œ ï¿œ C µ X ( λ − α ) H ( Y ) − αH ( Z ) + max p ( u,v | x ) { λI ( U ; Y ) + I ( V ; Z ) − I ( U ; V ) } ✞✺

  80. ❘❡♠❛r❊s✿ ❈♊♥❥❡❝t✉r❡❞ t♩ ❜❡ s✉❜✲❛❞❞✐t✐✈❡ ✭❆♥❛♥t❀❛r❛♠✲●♊❀❛r✐✲◆❛✐r ✬✶✞✮ ❚♩ ❡✈❛❧✉❛t❡ t❀❡ ❝♊♥❝❛✈❡ ❡♥✈❡❧♊♣❡ ❙✉✣❝❡s t♩ ❝♊♥s✐❞❡r ✿ ✳ ❲❡ ❞✐❞ ♥♊t ❣❡t ❛♥② ❝♊♥tr❛❞✐❝t✐♊♥ t♩ s✉❜✲❛❞❞✐✈✐t② ❢♊r ❜✐♥❛r② ✐♥♣✉t ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧s ❈❛♥ ♣r♊✈❡ s✉❜✲❛❞❞✐t✐✈✐t② ✇❀❡♥ ♩r ✳ ❲❀❛t ❛❜♩✉t ▌❛rt♊♥✬s r❡❣✐♊♥ ❢♊r t❀❡ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧❄ ■s t❀❡ ❢♊❧❧♊✇✐♥❣ ❢✉♥❝t✐♊♥❛❧ s✉❜✲❛❞❞✐t✐✈❡ ♩r ✐s t❀❡r❡ ❛♥ ❡①❛♠♣❧❡ ✇❀❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡❄ ▲❡t W a ( y | x ) ❛♥❞ W b ( z | x ) ❜❡ ❣✐✈❡♥ ❝❀❛♥♥❡❧s✱ α ∈ [0 , 1] ✱ ❛♥❞ λ ≥ 1 ✳ ï¿œ ï¿œ C µ X ( λ − α ) H ( Y ) − αH ( Z ) + max p ( u,v | x ) { λI ( U ; Y ) + I ( V ; Z ) − I ( U ; V ) } • ■❢ s✉❜✲❛❞❞✐t✐✈❡✱ t❀❡♥ ▌❛rt♊♥✬s r❡❣✐♊♥ ✐s ♊♣t✐♠❛❧ ❢♊r ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ • ■❢ ∃ ❡①❛♠♣❧❡ ✇❀❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡✱ t❀❡♥ ♊♥❡ s❀♊✉❧❞ ❜❡ ❛❜❧❡ t♩ ❞❡❞✉❝❡ ❛ ❝❀❛♥♥❡❧ ✇❀❡r❡ ▌❛rt♊♥✬s r❡❣✐♊♥ ✐s ♥♊t ♊♣t✐♠❛❧ ✞✺

  81. ❲❀❛t ❛❜♩✉t ▌❛rt♊♥✬s r❡❣✐♊♥ ❢♊r t❀❡ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧❄ ■s t❀❡ ❢♊❧❧♊✇✐♥❣ ❢✉♥❝t✐♊♥❛❧ s✉❜✲❛❞❞✐t✐✈❡ ♩r ✐s t❀❡r❡ ❛♥ ❡①❛♠♣❧❡ ✇❀❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡❄ ▲❡t W a ( y | x ) ❛♥❞ W b ( z | x ) ❜❡ ❣✐✈❡♥ ❝❀❛♥♥❡❧s✱ α ∈ [0 , 1] ✱ ❛♥❞ λ ≥ 1 ✳ ï¿œ ï¿œ C µ X ( λ − α ) H ( Y ) − αH ( Z ) + max p ( u,v | x ) { λI ( U ; Y ) + I ( V ; Z ) − I ( U ; V ) } • ■❢ s✉❜✲❛❞❞✐t✐✈❡✱ t❀❡♥ ▌❛rt♊♥✬s r❡❣✐♊♥ ✐s ♊♣t✐♠❛❧ ❢♊r ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ • ■❢ ∃ ❡①❛♠♣❧❡ ✇❀❡r❡ ✐t ✐s s✉♣❡r✲❛❞❞✐t✐✈❡✱ t❀❡♥ ♊♥❡ s❀♊✉❧❞ ❜❡ ❛❜❧❡ t♩ ❞❡❞✉❝❡ ❛ ❝❀❛♥♥❡❧ ✇❀❡r❡ ▌❛rt♊♥✬s r❡❣✐♊♥ ✐s ♥♊t ♊♣t✐♠❛❧ ❘❡♠❛r❊s✿ • ❈♊♥❥❡❝t✉r❡❞ t♩ ❜❡ s✉❜✲❛❞❞✐t✐✈❡ ✭❆♥❛♥t❀❛r❛♠✲●♊❀❛r✐✲◆❛✐r ✬✶✞✮ • ❚♩ ❡✈❛❧✉❛t❡ t❀❡ ❝♊♥❝❛✈❡ ❡♥✈❡❧♊♣❡ ⋆ ❙✉✣❝❡s t♩ ❝♊♥s✐❞❡r ( U, V ) ✿ | U | + | V | ≀ | X | + 1 ✳ ⋆ ❲❡ ❞✐❞ ♥♊t ❣❡t ❛♥② ❝♊♥tr❛❞✐❝t✐♊♥ t♩ s✉❜✲❛❞❞✐✈✐t② ❢♊r ❜✐♥❛r② ✐♥♣✉t ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧s • ❈❛♥ ♣r♊✈❡ s✉❜✲❛❞❞✐t✐✈✐t② ✇❀❡♥ α = 0 ♩r α = 1 ✳ ✞✺

  82. ❚❀✐s ✐❞❡❛ ✇❛s ❛❧s♩ ✉s❡❞ t♩ r❡s♊❧✈❡ ✜✳✷ ■s s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ♊♣t✐♠❛❧ ❢♊r t❀❡ ❣❡♥❡r❛❧ ✞✲r❡❝❡✐✈❡r ❉▌✲❇❈ ✇✐t❀ ♊♥❡ ♠❡ss❛❣❡ t♩ ❛❧❧ t❀r❡❡ r❡❝❡✐✈❡rs ❛♥❞ ❛♥♊t❀❡r ♠❡ss❛❣❡ t♩ t✇♩ r❡❝❡✐✈❡rs❄ ◆❖ ✭◆❛✐r✱❚❛③❞❛♥♣❛♥❛❀ ✬✶✌✮ ❘❡♠❛r❊s • ■❞❡❛✿ ❚♩ ❞❡♠♊♥str❛t❡ s✉♣❡r✲❛❞❞✐t✐✈✐t② • ❉✐✣❝✉❧t② ✿ ■❞❡♥t✐❢② ❛ s✉✣❝✐❡♥t❧② s✐♠♣❧❡ ❝❧❛ss ✇❀❡r❡ ⋆ ❊✈❛❧✉❛t✐♊♥ ♊❢ t❀❡ r❡❣✐♊♥ ✐s ♣♊ss✐❜❧❡ ✿ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ⋆ ❙✉♣❡r✲❛❞❞✐t✐✈✐t② ❀♊❧❞s ✞✻

  83. ❘❡♠❛r❊s • ■❞❡❛✿ ❚♩ ❞❡♠♊♥str❛t❡ s✉♣❡r✲❛❞❞✐t✐✈✐t② • ❉✐✣❝✉❧t② ✿ ■❞❡♥t✐❢② ❛ s✉✣❝✐❡♥t❧② s✐♠♣❧❡ ❝❧❛ss ✇❀❡r❡ ⋆ ❊✈❛❧✉❛t✐♊♥ ♊❢ t❀❡ r❡❣✐♊♥ ✐s ♣♊ss✐❜❧❡ ✿ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ⋆ ❙✉♣❡r✲❛❞❞✐t✐✈✐t② ❀♊❧❞s ❚❀✐s ✐❞❡❛ ✇❛s ❛❧s♩ ✉s❡❞ t♩ r❡s♊❧✈❡ ✜✳✷ ■s s✉♣❡r♣♊s✐t✐♊♥ ❝♊❞✐♥❣ ♊♣t✐♠❛❧ ❢♊r t❀❡ ❣❡♥❡r❛❧ ✞✲r❡❝❡✐✈❡r ❉▌✲❇❈ ✇✐t❀ ♊♥❡ ♠❡ss❛❣❡ t♩ ❛❧❧ t❀r❡❡ r❡❝❡✐✈❡rs ❛♥❞ ❛♥♊t❀❡r ♠❡ss❛❣❡ t♩ t✇♩ r❡❝❡✐✈❡rs❄ ◆❖ ✭◆❛✐r✱❚❛③❞❛♥♣❛♥❛❀ ✬✶✌✮ ✞✻

  84. ❖✉t❧✐♥❡ • ❇r♩❛❞❝❛st ❝❀❛♥♥❡❧✿ ❊st❛❜❧✐s❀✐♥❣ ♊♣t✐♠❛❧✐t② ♊❢ ▌❛rt♊♥✬s ❢♊r ▌■▌❖ ❜r♩❛❞❝❛st ❝❀❛♥♥❡❧ • ■♥t❡r❢❡r❡♥❝❡ ❝❀❛♥♥❡❧✿ ❙✉❜✲♊♣t✐♠❛❧✐t② ♊❢ t❀❡ ❍❛♥✕❑♊❜❛②❛s❀✐ r❡❣✐♊♥ • ❋❛♠✐❧② ♊❢ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠s ⋆ ❘❡❧❛t✐♊♥ t♩ ♣r♊❜❧❡♠s ♊❢ ✐♥t❡r❡st ✐♥ ♩t❀❡r ✜❡❧❞s ⋆ ❯♥✐❢②✐♥❣ ♩❜s❡r✈❛t✐♊♥s ❛♥❞ s♊♠❡ ❝♊♥❥❡❝t✉r❡s ✞✌

  85. ❯s✐♥❣ ❋❡♥❝❀❡❧ ❞✉❛❧✐t② t❀✐s ✐s s❛♠❡ ❛s ❆ s♣❡❝✐✜❝ ❢❛♠✐❧② ♊❢ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠s ❙❀♊✇s ✉♣ ✿ ❚❡st✐♥❣ t❀❡ ♊♣t✐♠❛❧✐t② ♊❢ ❝♊❞✐♥❣ s❝❀❡♠❡s ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ✭✉s✉❛❧❧②✮ r❡❞✉❝❡s t♩ t❡st✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♊❢ ï¿œ ï¿œ ï¿œ C Îœ X , α S ∈ R . α S H ( X S ) S ⊆ [ n ] ✞✜

  86. ❆ s♣❡❝✐✜❝ ❢❛♠✐❧② ♊❢ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠s ❙❀♊✇s ✉♣ ✿ ❚❡st✐♥❣ t❀❡ ♊♣t✐♠❛❧✐t② ♊❢ ❝♊❞✐♥❣ s❝❀❡♠❡s ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ✭✉s✉❛❧❧②✮ r❡❞✉❝❡s t♩ t❡st✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♊❢ ï¿œ ï¿œ ï¿œ C Îœ X , α S ∈ R . α S H ( X S ) S ⊆ [ n ] ❯s✐♥❣ ❋❡♥❝❀❡❧ ❞✉❛❧✐t② t❀✐s ✐s s❛♠❡ ❛s ï¿œ G 1 ( γ 1 ) := max α S H ( X S ) − E( γ 1 ( X )) µ X S ⊆ [ n ] ï¿œ α S H ( X S ) − E( γ 2 ( X )) G 2 ( γ 2 ) := max µ X S ⊆ [ n ] ï¿œ G 12 ( γ 1 , γ 2 ) := max α S H ( X 1 S , X 2 S ) − E( γ 1 ( X 1 )) − E( γ 2 ( X 2 )) µ X 1 , X 2 S ⊆ [ n ] ■s G 12 ( γ 1 , γ 2 ) = G 1 ( γ 1 ) + G 2 ( γ 2 ) ∀ γ 1 , γ 2 ❄ ✐✳❡✳ ■s t❀❡ ♠❛①✐♠✐③❡r ♊❢ G 12 ❛ ♣r♩❞✉❝t ❞✐str✐❜✉t✐♊♥❄ ✞✜

  87. ❆ s♣❡❝✐✜❝ ❢❛♠✐❧② ♊❢ ♥♊♥✲❝♊♥✈❡① ♊♣t✐♠✐③❛t✐♊♥ ♣r♊❜❧❡♠s ❙❀♊✇s ✉♣ ✿ ❚❡st✐♥❣ t❀❡ ♊♣t✐♠❛❧✐t② ♊❢ ❝♊❞✐♥❣ s❝❀❡♠❡s ❚❡st✐♥❣ ♊♣t✐♠❛❧✐t② ✭✉s✉❛❧❧②✮ r❡❞✉❝❡s t♩ t❡st✐♥❣ s✉❜✲❛❞❞✐t✐✈✐t② ♊❢ ï¿œ ï¿œ ï¿œ C Îœ X , α S ∈ R . α S H ( X S ) S ⊆ [ n ] ❯s✐♥❣ ❋❡♥❝❀❡❧ ❞✉❛❧✐t② t❀✐s ✐s s❛♠❡ ❛s ï¿œ G 1 ( γ 1 ) := max α S H ( X S ) − E( γ 1 ( X )) µ X S ⊆ [ n ] ï¿œ α S H ( X S ) − E( γ 2 ( X )) G 2 ( γ 2 ) := max µ X S ⊆ [ n ] ï¿œ G 12 ( γ 1 , γ 2 ) := max α S H ( X 1 S , X 2 S ) − E( γ 1 ( X 1 )) − E( γ 2 ( X 2 )) µ X 1 , X 2 S ⊆ [ n ] ❆r❡ t❀❡r❡ ♩t❀❡r ✜❡❧❞s ✇❀❡r❡ t❀❡ s❛♠❡ ❢❛♠✐❧② s❀♊✇s ✉♣❄ ✞✜

  88. ❚❀✐s ✭s❡r❡♥❞✐♣✐t♩✉s✮ r❡❞✐s❝♊✈❡r② ♊❢ t❀❡ ❧✐♥❊ ❜❡t✇❡❡♥ ❀②♣❡r❝♊♥tr❛❝t✐✈✐t② ❛♥❞ ✐♥❢♊r♠❛t✐♊♥ ♠❡❛s✉r❡s ❛♥❞ t❀❡s❡ ❡q✉✐✈❛❧❡♥t ❝❀❛r❛❝t❡r✐③❛t✐♊♥s ✐s s♣✉rr✐♥❣ ❛ ❧♩t ♊❢ ✇♩r❊ ❍②♣❡r❝♊♥tr❛❝t✐✈✐t② ❙t✉❞✐❡❞ ✐♥ ❢✉♥❝t✐♊♥❛❧ ❛♥❛❧②s✐s✱ ❝s t❀❡♊r②✱ ❡t❝✳ ❉❡✜♥✐t✐♊♥ ( X, Y ) ∌ µ XY ✐s ( p, q ) ✲❀②♣❡r❝♊♥tr❛❝t✐✈❡ ❢♊r 1 ≀ q ≀ p ✐❢ ï¿œ Tg ï¿œ p ≀ ï¿œ g ï¿œ q ∀ g ( Y ) ✇❀❡r❡ T ✐s t❀❡ ▌❛r❩♩✈ ♊♣❡r❛t♩r ❝❀❛r❛❝t❡r✐③❡❞ ❜② µ Y | X 1 p ✳ ❍❡r❡ ï¿œ Z ï¿œ p = E ( | Z | p ) ✞✟

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