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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 Z i ✐♥ ❱ ✳ ✭❲♦♦❞✇❛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✐♦♥ ✭■❱✮ I X ∈ ❱ ✐s ❛♥ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ ❢♦r X ✇✳r✳t✳ Y ✐♥ � ❱ , E , P � ✐✛ ✭❛✮ I X ✐s ❡①♦❣❡♥♦✉s ❛♥❞ t❤❡r❡ ✐s ❛ ♣❛t❤ π : I X − → X ✐♥ � ❱ , E � ✱ ✭❜✮ ❢♦r ❡✈❡r② on ✲✈❛❧✉❡ ♦❢ I X t❤❡r❡ ✐s ❛♥ X ✲✈❛❧✉❡ x s✉❝❤ t❤❛t Dep ( x , I X = on ) ❛♥❞ P ( x | I X = on ) = ✶✱ ✭❝✮ ❛❧❧ ♣❛t❤s I X − → ... − → Y ✐♥ � ❱ , E � ❤❛✈❡ t❤❡ ❢♦r♠ I X − → ... − → X − → ... − → Y ✱ ✭❞✮ I X ✐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 I Z Z ? I X X Y ❆❧❡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❡✮✱ → Z j ✐♥ E ′ ✐✛ Z i − ✭❜✮ ❢♦r ❛❧❧ Z i , Z j ∈ ❱ ✿ Z i − → Z j ✐♥ E ✱ ✭❝✮ ❢♦r ❡✈❡r② X ✲✈❛❧✉❡ x ♦❢ ❡✈❡r② X ∈ ❱ ❞✐✛❡r❡♥t ❢r♦♠ Y t❤❡r❡ ✐s ❛♥ on ✲✈❛❧✉❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥t❡r✈❡♥t✐♦♥ ✈❛r✐❛❜❧❡ I X s✉❝❤ t❤❛t Dep ( x , I X = on ) ❛♥❞ P ′ ( x | I X = 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 , I X = on | ■ ❩ = ♦♥ ) ❤♦❧❞s ✐♥ s♦♠❡ i ✲❡①♣❛♥s✐♦♥s✱ ✇❤❡r❡ I X ✐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♦♠ I X ✳ ❆❧❡①❛♥❞❡r ●❡❜❤❛rt❡r · ●❡r❤❛r❞ ❙❝❤✉r③ ❖❝❝❛♠✬s r❛③♦r ✫ ❞✐r❡❝t ❝❛✉s❛t✐♦♥ ✷✼✳✵✼✳✷✵✶✹ ✾ ✴ ✷✸