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❍♦✇ ❖❝❝❛♠✬s r❛③♦r ♣r♦✈✐❞❡s ❛ ♥❡❛t ❞❡✜♥✐t✐♦♥ ♦❢ ❞✐r❡❝t ❝❛✉s❛t✐♦♥

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③

❉✉❡ss❡❧❞♦r❢ ❈❡♥t❡r ❢♦r ▲♦❣✐❝ ❛♥❞ P❤✐❧♦s♦♣❤② ♦❢ ❙❝✐❡♥❝❡ ❍❡✐♥r✐❝❤ ❍❡✐♥❡ ❯♥✐✈❡rs✐t② ❉✉❡ss❡❧❞♦r❢

✷✼✳✵✼✳✷✵✶✹

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✶ ✴ ✷✸

❖✉t❧✐♥❡

✶ ❲♦♦❞✇❛r❞✬s ✐♥t❡r✈❡♥t✐♦♥✐st t❤❡♦r② ♦❢ ❝❛✉s❛t✐♦♥ ✷ ❘❡❝♦♥str✉❝t✐♥❣ ❲♦♦❞✇❛r❞✬s t❤❡♦r② ✸ ❘❡s✉❧t ✶✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊ ⇒ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❲♦♦❞✇❛r❞ st②❧❡ ✹ ❘❡s✉❧t ✷✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊❙ ⇒ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❲♦♦❞✇❛r❞ st②❧❡ ❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✷ ✴ ✷✸

❲♦♦❞✇❛r❞✬s ✐♥t❡r✈❡♥t✐♦♥✐st t❤❡♦r② ♦❢ ❝❛✉s❛t✐♦♥

❉❡✜♥✐t✐♦♥ ✭■❱❲✮

I ✐s ❛♥ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ ❢♦r X ✇r✳t✳ Y ✐✛ ■✶✳ I ❝❛✉s❡s X✳ ■✷✳ I ❛❝ts ❛s ❛ s✇✐t❝❤ ❢♦r ❛❧❧ ♦t❤❡r ✈❛r✐❛❜❧❡s t❤❛t ❝❛✉s❡ X✳ ■✸✳ ❆♥② ❞✐r❡❝t❡❞ ♣❛t❤ ❢r♦♠ I t♦ Y ❣♦❡s t❤r♦✉❣❤ X✳ ■✹✳ I ✐s ✭st❛t✐st✐❝❛❧❧②✮ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❛♥② ✈❛r✐❛❜❧❡ Z t❤❛t ❝❛✉s❡s Y ❛♥❞ t❤❛t ✐s ♦♥ ❛ ❞✐r❡❝t❡❞ ♣❛t❤ t❤❛t ❞♦❡s ♥♦t ❣♦ t❤r♦✉❣❤ X✳ I = on ✐s ❛♥ ✐♥t❡r✈❡♥t✐♦♥ ♦♥ X ✇✳r✳t✳ Y ✐✛ I ✐s ❛♥ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ ❢♦r X ✇✳r✳t✳ Y ❛♥❞ I = on ❢♦r❝❡s X t♦ t❛❦❡ ❛ ❝❡rt❛✐♥ ✈❛❧✉❡ x✳ ✭❝❢✳ ❲♦♦❞✇❛r❞✱ ✷✵✵✸✱ ♣✳ ✾✽✮

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✸ ✴ ✷✸

❲♦♦❞✇❛r❞✬s ✐♥t❡r✈❡♥t✐♦♥✐st t❤❡♦r② ♦❢ ❝❛✉s❛t✐♦♥

❉❡✜♥✐t✐♦♥ ✭❉❈❲✮

❆ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r X t♦ ❜❡ ❛ ✭t②♣❡✲❧❡✈❡❧✮ ❞✐r❡❝t ❝❛✉s❡ ♦❢ Y ✇✳r✳t✳ ❛ ✈❛r✐❛❜❧❡ s❡t ❱ ✐s t❤❛t t❤❡r❡ ❜❡ ❛ ♣♦ss✐❜❧❡ ✐♥t❡r✈❡♥t✐♦♥ ♦♥ X t❤❛t ✇✐❧❧ ❝❤❛♥❣❡ Y ♦r t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ♦❢ Y ✇❤❡♥ ♦♥❡ ❤♦❧❞s ✜①❡❞ ❛t s♦♠❡ ✈❛❧✉❡ ❛❧❧ ♦t❤❡r ✈❛r✐❛❜❧❡s Zi ✐♥ ❱✳ ✭❲♦♦❞✇❛r❞✱ ✷✵✵✸✱ ♣✳ ✺✾✮ ❖♣❡♥ q✉❡st✐♦♥s✴❝♦♥❝❡r♥s✿ ❲❤❛t ✐s ❛ ✏♣♦ss✐❜❧❡✑ ✐♥t❡r✈❡♥t✐♦♥❄ ⇒ ❧♦❣✐❝❛❧❧②✴❝♦♥❝❡♣t✉❛❧❧② ♣♦ss✐❜❧❡ ♠❛②❜❡ ♥♦t r❡str✐❝t✐✈❡ ❡♥♦✉❣❤✳✳✳ ❆♣♣❧✐❝❛❜❧❡ t♦ ❛❧❧ ❦✐♥❞s ♦❢ ✈❛r✐❛❜❧❡ s❡ts ❱❄ ⇒ ♣r♦❜❧❡♠s ✇✐t❤ s❡ts ❝♦♥t❛✐♥✐♥❣ ✈❛r✐❛❜❧❡s ❢♦r ✇❤✐❝❤ t❤❡r❡ ❛r❡ ♥♦ ✐♥t❡r✈❡♥t✐♦♥s ✐♥ t❤❡ s❡♥s❡ ♦❢ ✭■❱❲✮✳✳✳

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✹ ✴ ✷✸

❖✉t❧✐♥❡

✶ ❲♦♦❞✇❛r❞✬s ✐♥t❡r✈❡♥t✐♦♥✐st t❤❡♦r② ♦❢ ❝❛✉s❛t✐♦♥ ✷ ❘❡❝♦♥str✉❝t✐♥❣ ❲♦♦❞✇❛r❞✬s t❤❡♦r② ✸ ❘❡s✉❧t ✶✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊ ⇒ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❲♦♦❞✇❛r❞ st②❧❡ ✹ ❘❡s✉❧t ✷✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊S ⇒ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❲♦♦❞✇❛r❞ st②❧❡ ❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✺ ✴ ✷✸

❘❡❝♦♥str✉❝t✐♥❣ ❲♦♦❞✇❛r❞✬s t❤❡♦r②

❉❡✜♥✐t✐♦♥ ✭■❱✮

IX ∈ ❱ ✐s ❛♥ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ ❢♦r X ✇✳r✳t✳ Y ✐♥ ❱, E, P ✐✛ ✭❛✮ IX ✐s ❡①♦❣❡♥♦✉s ❛♥❞ t❤❡r❡ ✐s ❛ ♣❛t❤ π : IX − → X ✐♥ ❱, E✱ ✭❜✮ ❢♦r ❡✈❡r② on✲✈❛❧✉❡ ♦❢ IX t❤❡r❡ ✐s ❛♥ X✲✈❛❧✉❡ x s✉❝❤ t❤❛t Dep(x, IX = on) ❛♥❞ P(x|IX = on) = ✶✱ ✭❝✮ ❛❧❧ ♣❛t❤s IX − → ... − → Y ✐♥ ❱, E ❤❛✈❡ t❤❡ ❢♦r♠ IX − → ... − → X − → ... − → Y ✱ ✭❞✮ IX ✐s ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ ❡✈❡r② ✈❛r✐❛❜❧❡ C ✭✐♥ ❱ ♦r ♥♦t ✐♥ ❱✮ ✇❤✐❝❤ ❝❛✉s❡s Y ♦✈❡r ❛ ♣❛t❤ ♥♦t ❣♦✐♥❣ t❤r♦✉❣❤ X✳ ✭❝❢✳ ❲♦♦❞✇❛r❞✱ ✷✵✵✽✮

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✻ ✴ ✷✸

❘❡❝♦♥str✉❝t✐♥❣ ❲♦♦❞✇❛r❞✬s t❤❡♦r②

X Y IX Z IZ ?

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✼ ✴ ✷✸

❘❡❝♦♥str✉❝t✐♥❣ ❲♦♦❞✇❛r❞✬s t❤❡♦r②

❉❡✜♥✐t✐♦♥ ✭■❊✮

M′ = ❱′, E ′, P′ ✐s ❛♥ i✲❡①♣❛♥s✐♦♥ ♦❢ M = ❱, E, P ✇✳r✳t✳ Y ✐✛ ✭❛✮ ❱′ = ❱ ˙ ∪❱■✱ ✇❤❡r❡ ❱■ ❝♦♥t❛✐♥s ❢♦r ❡✈❡r② X ∈ ❱ ❞✐✛❡r❡♥t ❢r♦♠ Y ❛♥ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ I ✇✳r✳t✳ Y ✭❛♥❞ ♥♦t❤✐♥❣ ❡❧s❡✮✱ ✭❜✮ ❢♦r ❛❧❧ Zi, Zj ∈ ❱✿ Zi − → Zj ✐♥ E ′ ✐✛ Zi − → Zj ✐♥ E✱ ✭❝✮ ❢♦r ❡✈❡r② X✲✈❛❧✉❡ x ♦❢ ❡✈❡r② X ∈ ❱ ❞✐✛❡r❡♥t ❢r♦♠ Y t❤❡r❡ ✐s ❛♥

• n✲✈❛❧✉❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ IX s✉❝❤ t❤❛t

Dep(x, IX = on) ❛♥❞ P′(x|IX = on) = ✶✱ ✭❞✮ P′

■=♦✛ ↑ ❱ = P✱

✭❡✮ P′(■ = ♦♥), P′(■ = ♦✛) > ✵✳

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✽ ✴ ✷✸

❘❡❝♦♥str✉❝t✐♥❣ ❲♦♦❞✇❛r❞✬s t❤❡♦r②

❲❡ r❡❝♦♥str✉❝t ❲♦♦❞✇❛r❞✬s ✭✷✵✵✸✮ ❞❡✜♥✐t✐♦♥ ♦❢ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❛s ❛ ♣❛rt✐❛❧ ❞❡✜♥✐t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✭❉❈✮

■❢ t❤❡r❡ ❡①✐st i✲❡①♣❛♥s✐♦♥s ❱′, E ′, P′ ♦❢ ❱, E, P ✇✳r✳t✳ Y ✱ t❤❡♥✿ X ∈ ❱ ✐s ❛ ❞✐r❡❝t ❝❛✉s❡ ♦❢ Y ∈ ❱ ✇✳r✳t✳ ❱ ✐✛ Dep(Y , IX = on|■❩ = ♦♥) ❤♦❧❞s ✐♥ s♦♠❡ i✲❡①♣❛♥s✐♦♥s✱ ✇❤❡r❡ IX ✐s ❛♥ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ ❢♦r X ✇✳r✳t✳ Y ✐♥ ❱′, E ′, P′ ❛♥❞ ■❩ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡s ✐♥ ❱′, E ′, P′ ❞✐✛❡r❡♥t ❢r♦♠ IX✳

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✾ ✴ ✷✸

❖✉t❧✐♥❡

✶ ❲♦♦❞✇❛r❞✬s ✐♥t❡r✈❡♥t✐♦♥✐st t❤❡♦r② ♦❢ ❝❛✉s❛t✐♦♥ ✷ ❘❡❝♦♥str✉❝t✐♥❣ ❲♦♦❞✇❛r❞✬s t❤❡♦r② ✸ ❘❡s✉❧t ✶✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊ ⇒ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❲♦♦❞✇❛r❞ st②❧❡ ✹ ❘❡s✉❧t ✷✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊❙ ⇒ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❲♦♦❞✇❛r❞ st②❧❡ ❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✶✵ ✴ ✷✸

❘❡s✉❧t ✶✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊ ⇒ ❲♦♦❞✇❛r❞

❉❡✜♥✐t✐♦♥ ✭❈▼❈✮

❆ ❝❛✉s❛❧ ♠♦❞❡❧ ❱, E, P s❛t✐s✜❡s t❤❡ ❝❛✉s❛❧ ▼❛r❦♦✈ ❝♦♥❞✐t✐♦♥ ✐✛ ❡✈❡r② X ∈ ❱ ✐s ♣r♦❜❛❜✐❧✐st✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❛❧❧ ✐ts ♥♦♥✲❡✛❡❝ts ❝♦♥❞✐t✐♦♥❛❧ ♦♥ ✐ts ❝❛✉s❛❧ ♣❛r❡♥ts ✭❝❢✳ ❙♣✐rt❡s ❡t ❛❧✳✱ ✷✵✵✵✱ ♣✳ ✷✾✮✳ ❈▼❈ ✐s ❛ss✉♠❡❞ t♦ ❤♦❧❞ ❢♦r ❝❛✉s❛❧ ♠♦❞❡❧s ✇❤♦s❡ ✈❛r✐❛❜❧❡ s❡ts ❛r❡ ❝❛✉s❛❧❧② s✉✣❝✐❡♥t✿

❉❡✜♥✐t✐♦♥ ✭❝❛✉s❛❧ s✉✣❝✐❡♥❝②✮

❱ ✐s ❝❛✉s❛❧❧② s✉✣❝✐❡♥t ✐✛ ❡✈❡r② ❝♦♠♠♦♥ ❝❛✉s❡ C ♦❢ ✈❛r✐❛❜❧❡s ✐♥ ❱ ✐s ✐♥ ❱ ♦r t❛❦❡s t❤❡ s❛♠❡ ✈❛❧✉❡ c ❢♦r ❛❧❧ ✐♥❞✐✈✐❞✉❛❧s ✐♥ t❤❡ ❞♦♠❛✐♥✳ ✭❝❢✳ ❙♣✐rt❡s ❡t ❛❧✱ ✷✵✵✵✱ ♣✳ ✷✷✮

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✶✶ ✴ ✷✸

❘❡s✉❧t ✶✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊ ⇒ ❲♦♦❞✇❛r❞

❋♦r ❛❝②❝❧✐❝ ❣r❛♣❤s✱ ❈▼❈ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ d✲s❡♣❛r❛t✐♦♥ ❝r✐t❡r✐♦♥ ✭❝❢✳ ❱❡r♠❛✱ ✶✾✽✻❀ P❡❛r❧✱ ✶✾✽✽✱ ♣✳ ✶✶✾❢✮✿

❉❡✜♥✐t✐♦♥ ✭❞✲s❡♣❛r❛t✐♦♥ ❝r✐t❡r✐♦♥✮

❱, E, P s❛t✐s✜❡s t❤❡ ❞✲s❡♣❛r❛t✐♦♥ ❝r✐t❡r✐♦♥ ✐✛ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s ❢♦r ❛❧❧ X, Y ∈ ❱ ❛♥❞ ❩ ⊆ ❱\{X, Y }✿ ■❢ X ❛♥❞ Y ❛r❡ d✲s❡♣❛r❛t❡❞ ❜② ❩ ✐♥ ❱, E✱ t❤❡♥ Indep(X, Y |❩)✳

❉❡✜♥✐t✐♦♥ ✭d✲s❡♣❛r❛t✐♦♥✱ d✲❝♦♥♥❡❝t✐♦♥✮

X ❛♥❞ Y ❛r❡ d✲s❡♣❛r❛t❡❞ ❜② ❩ ✐♥ ❱, E ✐✛ X ❛♥❞ Y ❛r❡ ♥♦t d✲❝♦♥♥❡❝t❡❞ ❣✐✈❡♥ ❩ ✐♥ ❱, E✳ X ❛♥❞ Y ❛r❡ d✲❝♦♥♥❡❝t❡❞ ❣✐✈❡♥ ❩ ✐♥ ❱, E ✐✛ X ❛♥❞ Y ❛r❡ ❝♦♥♥❡❝t❡❞ ❜② ❛ ❝❛✉s❛❧ ♣❛t❤ π ✐♥ ❱, E s✉❝❤ t❤❛t ♥♦ ♥♦♥✲❝♦❧❧✐❞❡r ♦♥ π ✐s ✐♥ ❩✱ ✇❤✐❧❡ ❛❧❧ ❝♦❧❧✐❞❡rs ♦♥ π ❛r❡ ✐♥ ❩ ♦r ❤❛✈❡ ❛♥ ❡✛❡❝t ✐♥ ❩✳

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✶✷ ✴ ✷✸

❘❡s✉❧t ✶✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊ ⇒ ❲♦♦❞✇❛r❞

❖❝❝❛♠✬s r❛③♦r ✭♦r t❤❡ ❝❛✉s❛❧ ♠✐♥✐♠❛❧✐t②✴♣r♦❞✉❝t✐✈✐t② ❝♦♥❞✐t✐♦♥✮✿

❉❡✜♥✐t✐♦♥ ✭▼✐♥✮

■❢ ❱, E, P s❛t✐s✜❡s ❈▼❈✱ t❤❡♥ ❱, E, P s❛t✐s✜❡s ▼✐♥ ✐✛ ♥♦ s✉❜♠♦❞❡❧ ❱, E ′, P ✇✐t❤ E ′ ⊂ E ❛❧s♦ s❛t✐s✜❡s ❈▼❈ ✭❝❢✳ ❙♣✐rt❡s ❡t ❛❧✳✱ ✷✵✵✵✱ ♣✳ ✸✶✮✳

❉❡✜♥✐t✐♦♥ ✭Pr♦❞✮

❱, E, P s❛t✐s✜❡s Pr♦❞ ✐✛ Dep(X, Y |Par(Y )\{X}) ❤♦❧❞s ❢♦r ❛❧❧ X, Y ∈ ❱ ✇✐t❤ X − → Y ✐♥ ❱, E✳ ✭❙❝❤✉r③ ❛♥❞ ●❡❜❤❛rt❡r✱ ✷✵✶✹✮

❚❤❡♦r❡♠ ✭✶✮

❋♦r ❛❝②❝❧✐❝ ❝❛✉s❛❧ ♠♦❞❡❧s s❛t✐s❢②✐♥❣ ❈▼❈✱ ▼✐♥ ✐s ❡q✉✐✈❛❧❡♥t ✇✐t❤ Pr♦❞✳

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✶✸ ✴ ✷✸

❘❡s✉❧t ✶✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊ ⇒ ❲♦♦❞✇❛r❞

❚❤❡♦r❡♠ ✭✷✮

■❢ ❱, E, P ✐s ❛♥ ❛❝②❝❧✐❝ ❝❛✉s❛❧ ♠♦❞❡❧ ❛♥❞ ❢♦r ❡✈❡r② Y ∈ ❱ t❤❡r❡ ✐s ❛♥ i✲❡①♣❛♥s✐♦♥ ❱′, E ′, P′ ♦❢ ❱, E, P ✇✳r✳t✳ Y s❛t✐s❢②✐♥❣ ❈▼❈ ❛♥❞ ▼✐♥✱ t❤❡♥ ❢♦r ❛❧❧ X, Y ∈ ❱ ✭✇✐t❤ X = Y ✮ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✭✐✮ X − → Y ✐♥ ❱, E✳ ✭✐✐✮ Dep(Y , IX = on|■❩ = ♦♥) ❤♦❧❞s ✐♥ s♦♠❡ i✲❡①♣❛♥s✐♦♥s ♦❢ ❱, E, P ✇✳r✳t✳ Y ✱ ✇❤❡r❡ IX ✐s ❛♥ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ ❢♦r X ✇✳r✳t✳ Y ✐♥ ❱′, E ′, P′ ❛♥❞ ■❩ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡s ✐♥ ❱′, E ′, P′ ❞✐✛❡r❡♥t ❢r♦♠ IX✳ ⇒ ❉✐r❡❝t ❝❛✉s❛t✐♦♥ ❛ ❧❛ ❲♦♦❞✇❛r❞ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❣r❛♣❤ t❤❡♦r❡t✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✐♥ s②st❡♠s ❱, E, P ✇✐t❤ i✲❡①♣❛♥s✐♦♥s ✇✳r✳t✳ ❡✈❡r② Y ∈ ❱ s❛t✐s❢②✐♥❣ ❈▼❈ ❛♥❞ ▼✐♥✳

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✶✹ ✴ ✷✸

❖✉t❧✐♥❡

✶ ❲♦♦❞✇❛r❞✬s ✐♥t❡r✈❡♥t✐♦♥✐st t❤❡♦r② ♦❢ ❝❛✉s❛t✐♦♥ ✷ ❘❡❝♦♥str✉❝t✐♥❣ ❲♦♦❞✇❛r❞✬s t❤❡♦r② ✸ ❘❡s✉❧t ✶✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊ ⇒ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❲♦♦❞✇❛r❞ st②❧❡ ✹ ❘❡s✉❧t ✷✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊❙ ⇒ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❲♦♦❞✇❛r❞ st②❧❡ ❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✶✺ ✴ ✷✸

❘❡s✉❧t ✷✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊❙ ⇒ ❲♦♦❞✇❛r❞

❉❡✜♥✐t✐♦♥ ✭■❱❙✮

IX ∈ ❱ ✐s ❛ st♦❝❤❛st✐❝ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ ❢♦r X ✇✳r✳t✳ Y ✐♥ ❱, E, P ✐✛ ✭❛✮ IX ✐s ❡①♦❣❡♥♦✉s ❛♥❞ t❤❡r❡ ✐s ❛ ♣❛t❤ π : IX − → X ✐♥ ❱, E✱ ✭❜✮ ❢♦r ❡✈❡r② on✲✈❛❧✉❡ ♦❢ IX t❤❡r❡ ✐s ❛♥ X✲✈❛❧✉❡ x s✉❝❤ t❤❛t Dep(x, IX = on)✱ ✭❝✮ ❛❧❧ ♣❛t❤s IX − → ... − → Y ✐♥ ❱, E ❤❛✈❡ t❤❡ ❢♦r♠ IX − → ... − → X − → ... − → Y ✱ ✭❞✮ IX ✐s ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ ❡✈❡r② ✈❛r✐❛❜❧❡ C ✭✐♥ ❱ ♦r ♥♦t ✐♥ ❱✮ ✇❤✐❝❤ ❝❛✉s❡s Y ♦✈❡r ❛ ♣❛t❤ ♥♦t ❣♦✐♥❣ t❤r♦✉❣❤ X✳

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✶✻ ✴ ✷✸

❘❡s✉❧t ✷✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊❙ ⇒ ❲♦♦❞✇❛r❞

❉❡✜♥✐t✐♦♥ ✭■❊❙✮

M′ = ❱′, E ′, P′ ✐s ❛ st♦❝❤❛st✐❝ i✲❡①♣❛♥s✐♦♥ ♦❢ M = ❱, E, P ❢♦r X ✇✳r✳t✳ Y ✐✛ ✭❛✮ ❱′ = ❱ ˙ ∪❱IX ✱ ✇❤❡r❡ ❱IX ❝♦♥t❛✐♥s ❛ st♦❝❤❛st✐❝ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ IX ❢♦r X ✇✳r✳t✳ Y ✭❛♥❞ ♥♦t❤✐♥❣ ❡❧s❡✮✱ ✭❜✮ ❢♦r ❛❧❧ Zi, Zj ∈ ❱✿ Zi − → Zj ✐♥ E ′ ✐✛ Zi − → Zj ✐♥ E✱ ✭❝✮ ❢♦r ❡✈❡r② X✲✈❛❧✉❡ x ♦❢ ❡✈❡r② X ∈ ❱ ❞✐✛❡r❡♥t ❢r♦♠ Y t❤❡r❡ ✐s ❛♥

• n✲✈❛❧✉❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ IX s✉❝❤ t❤❛t

Dep(x, IX = on)✱ ✭❞✮ P′

IX =off ↑ ❱ = P✱

✭❡✮ P′(IX = on), P′(IX = off ) > ✵✳

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✶✼ ✴ ✷✸

❘❡s✉❧t ✷✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊❙ ⇒ ❲♦♦❞✇❛r❞

❚♦ ❡st❛❜❧✐s❤ ❛ ❞✐r❡❝t ❝❛✉s❛❧ r❡❧❛t✐♦♥s❤✐♣ X − → Y ✱ ❲♦♦❞✇❛r❞ ✭✷✵✵✸✮ ♥❡❡❞s t♦ ❜❧♦❝❦ ♣r♦❜❛❜✐❧✐t② ♣r♦♣❛❣❛t✐♦♥ ❜❡t✇❡❡♥ X ❛♥❞ Y ♦✈❡r ✐♥❞✐r❡❝t ♣❛t❤s✳ ❆❧t❡r♥❛t✐✈❡❧② ♦♥❡ ❝❛♥ ❜❧♦❝❦ ❛❧❧ ✐♥❞✐r❡❝t ♣❛t❤s ❜❡t✇❡❡♥ X ❛♥❞ Y ❜② ❝♦♥❞✐t✐♦♥❛❧✐③✐♥❣ ♦♥ Par(Y )\{X}✳

X IX Z ? Y

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✶✽ ✴ ✷✸

❘❡s✉❧t ✷✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊❙ ⇒ ❲♦♦❞✇❛r❞

❲❡ r❡❝♦♥str✉❝t t❤❡ st♦❝❤❛st✐❝ ✈❛r✐❛♥t ♦❢ ❲♦♦❞✇❛r❞✬s ✭✷✵✵✸✮ ❞❡✜♥✐t✐♦♥ ♦❢ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❛s ❛ ♣❛rt✐❛❧ ❞❡✜♥✐t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✭❉❈❙✮

■❢ t❤❡r❡ ❡①✐st st♦❝❤❛st✐❝ i✲❡①♣❛♥s✐♦♥s ❱′, E ′, P′ ♦❢ ❱, E, P ❢♦r X ✇✳r✳t✳ Y ✱ t❤❡♥✿ X ∈ ❱ ✐s ❛ ❞✐r❡❝t ❝❛✉s❡ ♦❢ Y ∈ ❱ ✇✳r✳t✳ ❱ ✐✛ Dep(Y , IX = on|Par(Y )\{X}) ❤♦❧❞s ✐♥ s♦♠❡ st♦❝❤❛st✐❝ i✲❡①♣❛♥s✐♦♥s✱ ✇❤❡r❡ IX ✐s ❛♥ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ ❢♦r X ✇✳r✳t✳ Y ✐♥ ❱′, E ′, P′✳

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✶✾ ✴ ✷✸

❘❡s✉❧t ✷✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊❙ ⇒ ❲♦♦❞✇❛r❞

❚❤❡♦r❡♠ ✭✸✮

■❢ ❱, E, P ✐s ❛♥ ❛❝②❝❧✐❝ ❝❛✉s❛❧ ♠♦❞❡❧ ❛♥❞ ❢♦r ❛❧❧ X, Y ∈ ❱ ✭✇✐t❤ X = Y ✮ t❤❡r❡ ✐s ❛ st♦❝❤❛st✐❝ i✲❡①♣❛♥s✐♦♥ ❱′, E ′, P′ ♦❢ ❱, E, P ❢♦r X ✇✳r✳t✳ Y s❛t✐s❢②✐♥❣ ❈▼❈ ❛♥❞ ▼✐♥✱ t❤❡♥ ❢♦r ❛❧❧ X, Y ∈ ❱ ✭✇✐t❤ X = Y ✮ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✭✐✮ X − → Y ✐♥ ❱, E✳ ✭✐✐✮ Dep(Y , IX = on|Par(Y )\{X}) ❤♦❧❞s ✐♥ s♦♠❡ st♦❝❤❛st✐❝ i✲❡①♣❛♥s✐♦♥s ♦❢ ❱, E, P ❢♦r X ✇✳r✳t✳ Y ✳ ⇒ ❚❤❡ st♦❝❤❛st✐❝ ✈❡rs✐♦♥ ♦❢ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❛ ❧❛ ❲♦♦❞✇❛r❞ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❣r❛♣❤ t❤❡♦r❡t✐❝❛❧ ♥♦t✐♦♥ ♦❢ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✐♥ s②st❡♠s ❱, E, P ✇✐t❤ st♦❝❤❛st✐❝ i✲❡①♣❛♥s✐♦♥s ❢♦r ❡✈❡r② X ∈ ❱ ✇✳r✳t✳ ❡✈❡r② Y ∈ ❱ ✭✇✐t❤ X = Y ✮ s❛t✐s❢②✐♥❣ ❈▼❈ ❛♥❞ ▼✐♥✳

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✷✵ ✴ ✷✸

❈♦♥❝❧✉s✐♦♥

✶ ❲♦♦❞✇❛r❞✬s ✐♥t❡r✈❡♥t✐♦♥✐st t❤❡♦r② ♦❢ ❝❛✉s❛t✐♦♥ ✷ ❘❡❝♦♥str✉❝t✐♥❣ ❲♦♦❞✇❛r❞✬s t❤❡♦r② ✸ ❘❡s✉❧t ✶✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊ ⇒ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❲♦♦❞✇❛r❞ st②❧❡ ✹ ❘❡s✉❧t ✷✿ ❈▼❈ ✰ ▼✐♥ ✰ ■❊❙ ⇒ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ❲♦♦❞✇❛r❞ st②❧❡ ❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✷✶ ✴ ✷✸

▼❛♥② t❤❛♥❦s✦

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✷✷ ✴ ✷✸

❘❡❢❡r❡♥❝❡s

❏✳ P❡❛r❧ ✭✶✾✽✽✮✳ Pr♦❜❛❜✐❧✐st✐❝ ❘❡❛s♦♥✐♥❣ ✐♥ ❊①♣❡rt ❙②st❡♠s✳ ❙❛♥ ▼❛t❡♦✱ ▼❆✿ ▼♦r❣❛♥ ❑❛✉❢♠❛♥♥✳

• ✳ ❙❝❤✉r③✱ ❛♥❞ ❆✳ ●❡❜❤❛rt❡r ✭✷✵✶✹✮✳ ❈❛✉s❛❧✐t② ❛s ❛ t❤❡♦r❡t✐❝❛❧ ❝♦♥❝❡♣t✿

❊①♣❧❛♥❛t♦r② ✇❛rr❛♥t ❛♥❞ ❡♠♣✐r✐❝❛❧ ❝♦♥t❡♥t ♦❢ t❤❡ t❤❡♦r② ♦❢ ❝❛✉s❛❧ ♥❡ts✳ ❚♦ ❛♣♣❡❛r ✐♥ ❙②♥t❤❡s❡✳ P✳ ❙♣✐rt❡s✱ ❈✳ ●❧②♠♦✉r✱ ❛♥❞ ❘✳ ❙❝❤❡✐♥❡s ✭✷✵✵✵✮✳ ❈❛✉s❛t✐♦♥✱ Pr❡❞✐❝t✐♦♥✱ ❛♥❞ ❙❡❛r❝❤✳ ❈❛♠❜r✐❞❣❡✱ ▼❆✿ ▼■❚ Pr❡ss✳ ❚✳ ❙✳ ❱❡r♠❛ ✭✶✾✽✻✮✳ ❈❛✉s❛❧ ♥❡t✇♦r❦s✿ ❙❡♠❛♥t✐❝s ❛♥❞ ❡①♣r❡ss✐✈❡♥❡ss✳ ❚❡❝❤♥✐❝❛❧ r❡♣♦rt ❘✲✻✺✱ ❈♦❣♥✐t✐✈❡ ❙②st❡♠s ▲❛❜♦r❛t♦r②✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❛❧✐❢♦r♥✐❛✱ ▲♦s ❆♥❣❡❧❡s✳ ❏✳ ❲♦♦❞✇❛r❞ ✭✷✵✵✸✮✳ ▼❛❦✐♥❣ ❚❤✐♥❣s ❍❛♣♣❡♥✳ ❖①❢♦r❞✿ ❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❏✳ ❲♦♦❞✇❛r❞ ✭✷✵✵✽✮✳ ❘❡s♣♦♥s❡ t♦ ❙tr❡✈❡♥s✳ P❤✐❧♦s♦♣❤② ❛♥❞ P❤❡♥♦♠❡♥♦❧♦❣✐❝❛❧ ❘❡s❡❛r❝❤✱ ▲❳❳❱■■✭✶✮✱ ✶✾✸✲✷✶✷✳

❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✷✸ ✴ ✷✸