❘❛s❝❤ ▼♦❞❡❧s ❛♥❞ t❤❡ ❘ ♣❛❝❦❛❣❡ ❡❘▼ ❘❡✐♥❤♦❧❞ ❍❛t③✐♥❣❡r ■♥st✐t✉t❡ ❢♦r ❙t❛t✐st✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝s ❲❯ ❱✐❡♥♥❛ ▼✉♥✐❝❤ ✷✵✶✵ ✶
■♥tr♦ ❲❤❛t ✐s ❡❘♠❄ ● ❡❘♠ s❤♦rt ❢♦r ❡①t❡♥❞❡❞ ❘❛s❝❤ ♠♦❞❡❧❧✐♥❣ ● ✐s ❛♥ ❘ ♣❛❝❦❛❣❡ ● ✐s ♦♣❡♥ s♦✉r❝❡✿ ♥♦ ❧✐❝❡♥s❡ ❢❡❡s✱ s♦✉r❝❡ ❝♦❞❡ ❛✈❛✐❧❛❜❧❡✱ ●P▲✿ s❤❛r❡✱ ❝❤❛♥❣❡✱ ❛♥❞ r❡❞✐str✐❜✉t❡ ✉♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s ● ❢♦r ❘❛s❝❤ ❢❛♠✐❧② ♠♦❞❡❧s✿ ✉t✐❧✐t✐❡s ❢♦r ✜tt✐♥❣✱ t❡st✐♥❣✱ ❛♥❞ ❞✐s♣❧❛②✐♥❣ r❡s✉❧ts ● ❝✉rr❡♥t❧② ✐♠♣❧❡♠❡♥t❡❞ ♠♦❞❡❧s✿ ▲P❈▼✱ P❈▼✱ ▲❘❙▼✱ ❘❙▼✱ ▲▲❚▼✱ ❘▼✱ ✭▲▲❘❆✮ ● ✉s❡s ❈▼▲ ❡st✐♠❛t✐♦♥ ▼✉♥✐❝❤ ✷✵✶✵ ✷
■♥tr♦ ❲❤❛t ✐s ■t❡♠ ❘❡s♣♦♥s❡ ❚❤❡♦r② ✭■❘❚✮❄ ■❘❚ ✐s ❜✉✐❧t ❛r♦✉♥❞ t❤❡ ❝❡♥tr❛❧ ✐❞❡❛✿ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ s✉❜❥❡❝t✬s ❝❡rt❛✐♥ r❡❛❝t✐♦♥ t♦ ❛ st✐♠✉❧✉s ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ❛s ❛ ❢✉♥❝t✐♦♥ ❝❤❛r❛❝t❡r✐s✐♥❣ t❤❡ s✉❜❥❡❝t✬s ❧♦❝❛t✐♦♥ ♦♥ ❛ ❧❛t❡♥t tr❛✐t ♣❧✉s ♦♥❡ ♦r ♠♦r❡ ♣❛r❛♠❡t❡rs ❝❤❛r❛❝t❡r✐s✐♥❣ t❤❡ st✐♠✉❧✉s 1.0 0.8 Probability 0.6 0.4 0.2 0.0 −4 −2 0 2 4 Latent Dimension ▼✉♥✐❝❤ ✷✵✶✵ ✸
❚❤❡ ❘❛s❝❤ ▼♦❞❡❧ ❚❤❡ ❘❛s❝❤ ▼♦❞❡❧ ✭❘▼✮ ✭❘❛s❝❤✱ ✶✾✻✵✮ exp ( θ v − β i ) P ( X vi = 1 ∣ θ v ,β i ) = 1 + exp ( θ v − β i ) X vi ✳ ✳ ✳ ♣❡rs♦♥ v ❣✐✈❡s ❝♦rr❡❝t ❛♥s✇❡r t♦ ✐t❡♠ i θ v ✳ ✳ ✳ ❵❛❜✐❧✐t②✬ ♦❢ ♣❡rs♦♥ v β i ✳ ✳ ✳ ❵❞✐✣❝✉❧t②✬ ♦❢ ✐t❡♠ i I 1 I 2 I 3 I 4 r v ❘❛✇ ❙❝♦r❡s✿ ✶ ✵ ✵ ✵ ✶ P 1 ✶ ✵ ✶ ✵ ✷ P 2 ✶ ✶ ✵ ✵ ✷ P 3 ∑ i x vi = r v ✵ ✶ ✶ ✶ ✸ P 4 ∑ v x vi = s i ✸ ✷ ✷ ✶ ✕ s i ▼✉♥✐❝❤ ✷✵✶✵ ✹
❚❤❡ ❘❛s❝❤ ▼♦❞❡❧ ❙❡✈❡r❛❧ ■❈❈s 1.0 0.8 Probability to Solve 0.6 Item 1 Item 2 Item 3 0.4 0.2 0.0 −4 −2 0 2 4 Latent Dimension ▼✉♥✐❝❤ ✷✵✶✵ ✺
❚❤❡ ❘❛s❝❤ ▼♦❞❡❧ ❘❛s❝❤ ▼♦❞❡❧ ❆ss✉♠♣t✐♦♥s ✴ Pr♦♣❡rt✐❡s ✉♥✐❞✐♠❡♥s✐♦♥❛❧✐t② P ( X vi = 1 ∣ θ v ,β i ,ϕ ϕ ϕ ) = P ( X vi = 1 ∣ θ v ,β i ) r❡s♣♦♥s❡ ♣r♦❜❛❜✐❧✐t② ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ♦t❤❡r ✈❛r✐❛❜❧❡s ϕ ϕ ϕ s✉✣❝✐❡♥❝② f ( x vi ,...,x vk ∣ θ v ) = g ( r v ∣ θ v ) h ( x vi ,...,x vk ) r❛✇ s❝♦r❡ r v = ∑ i x vi ✭s✉♠ ♦❢ r❡s♣♦♥s❡s✮ ❝♦♥t❛✐♥s ❛❧❧ ✐♥❢♦r♠❛✲ t✐♦♥ ♦♥ ❛❜✐❧✐t②✱ r❡❣❛r❞❧❡ss ✇❤✐❝❤ ✐t❡♠s ❤❛✈❡ ❜❡❡♥ s♦❧✈❡❞ ❝♦♥❞✐t✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡ X vi � X vj ∣ θ v , ∀ i,j ❢♦r ✜①❡❞ θ t❤❡r❡ ✐s ♥♦ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❛♥② t✇♦ ✐t❡♠s ♠♦♥♦t♦♥✐❝✐t② ❢♦r θ v > θ w ∶ f ( x vi ∣ θ v ,β i ) > f ( x wi ∣ θ w ,β i ) , ∀ θ v ,θ w r❡s♣♦♥s❡ ♣r♦❜❛❜✐❧✐t② ✐♥❝r❡❛s❡s ✇✐t❤ ❤✐❣❤❡r ✈❛❧✉❡s ♦❢ θ ▼✉♥✐❝❤ ✷✵✶✵ ✻
❚❤❡ ❘❛s❝❤ ▼♦❞❡❧ P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥ ■t❡♠ P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥ ▸ ❧✐❦❡❧✐❤♦♦❞ ❜❛s❡❞ ♠❡t❤♦❞s✿ ❞✐✛❡r ✐♥ t❤❡✐r tr❡❛t♠❡♥t ♦❢ ♣❡rs♦♥ ♣❛r❛♠❡t❡rs ● ❥♦✐♥t ▼▲ ❡st✐♠❛t✐♦♥ ✭❏▼▲✮ ● ❝♦♥❞✐t✐♦♥❛❧ ▼▲ ❡st✐♠❛t✐♦♥ ✭❈▼▲✮ ● ♠❛r❣✐♥❛❧ ▼▲ ❡st✐♠❛t✐♦♥✭▼▼▲✮ ▸ ♦t❤❡r ♠❡t❤♦❞s ❛✈❛✐❧❛❜❧❡✿ ❧❡ss ♦❢t❡♥ ✉s❡❞ ♥♦t ❝♦✈❡r❡❞ ❤❡r❡ P❡rs♦♥ P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥ ● ▼▲ ❛♥❞ ✇❡✐❣❤t❡❞ ▼▲ ❡st✐♠❛t✐♦♥ ● ❇❛②❡s ❛♣♣r♦❛❝❤❡s ▼✉♥✐❝❤ ✷✵✶✵ ✼
❚❤❡ ❘❛s❝❤ ▼♦❞❡❧ ❏♦✐♥t ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✭❏▼▲✮ s✉✣❝✐❡♥t st❛t✐st✐❝s ❛r❡✿ r v = ∑ i x vi ❢♦r θ v L u = exp (∑ v θ v r v ) exp (−∑ i β i s i ) ∏ v ∏ i ( 1 + exp ( θ v − β i )) s i = ∑ v x vi ❢♦r β i ♣r♦❜❧❡♠✿ ✐t❡♠ ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s ✐♥❝♦♥s✐st❡♥t ❛s n → ∞ ❜✐❛s❡❞ ✐♥ ✜♥✐t❡ s❛♠♣❧❡s ✇✐t❤ k ( k − 1 ) ▼❛r❣✐♥❛❧ ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✭▼▼▲✮ ✐♥t❡❣r❛t❡ ♦✉t t❤❡ ♣❡rs♦♥ ♣❛r❛♠❡t❡r ⎥ n r ⎡ ⎤ ⎥ ⎢ L m = ∏ exp ( θr ) exp (− ∑ β i s i ) ∫ ⎥ ⎢ dG ( θ ) ⎥ ⎢ ∏ k ⎦ ⎢ i = 1 ( 1 + exp ( θ − β i )) r i ⎣ ❞✐str✐❜✉t✐♦♥ ❢♦r θ ♠✉st ❜❡ s♣❡❝✐✜❡❞✱ ✉s✉❛❧❧② θ ∼ N ( 0 , 1 ) ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ✐♥ ❘ ✉s✐♥❣ t❤❡ ❧t♠ ♣❛❝❦❛❣❡ ✭❘✐③♦♣♦✉❧♦s✱✷✵✵✾✮ ▼✉♥✐❝❤ ✷✵✶✵ ✽
❚❤❡ ❘❛s❝❤ ▼♦❞❡❧ ❈♦♥❞✐t✐♦♥❛❧ ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ✭❈▼▲✮ ❝♦♥❞✐t✐♦♥ ♦♥ r v x i β i ) n r L c = exp (−∑ β i s i )/∏ exp (−∑ ∑ r x ∣ r i i ✕ ♣❡rs♦♥ ♣❛r❛♠❡t❡rs ❞♦ ♥♦t ♦❝❝✉r ✐♥ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❧✐❦❡❧✐❤♦♦❞ ✕ ✐t❡♠s ❝❛♥ ❜❡ ❝♦♠♣❛r❡❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♣❡rs♦♥s ✭s❡♣❛r❛t✐♦♥✮ ✕ ❧❡❛❞s t♦ s♣❡❝✐✜❝ ♦❜❥❡❝t✐✈✐t② ✕ ♣❡rs♦♥ ❢r❡❡ ✐t❡♠ ❝❛❧✐❜r❛t✐♦♥ ✕ ❵s❛♠♣❧❡✲✐♥❞❡♣❡♥❞❡♥❝❡✬✿ ❛❝t✉❛❧ s❛♠♣❧❡ ♥♦t ♦❢ r❡❧❡✈❛♥❝❡ ❢♦r ✐♥❢❡r❡♥❝❡ ♦♥ ✐t❡♠ ♣❛r❛♠❡t❡rs ❈▼▲ ❡st✐♠❛t❡s ❛r❡ ✉♥❜✐❛s❡❞ ❛♥❞ ❝♦♥s✐st❡♥t ❛s n → ∞ ❢♦r ❡st✐♠❛❜✐❧✐t② s❡t β 1 = 0 ♦r ∑ β i = 0 ✐t❡♠s ✇✐t❤ s❝♦r❡ s i = 0 ♦r n ❛♥❞ ♣❡rs♦♥ ✇✐t❤ r v = 0 ♦r k ❛r❡ r❡♠♦✈❡❞ ♣r✐♦r t♦ ❡st✐♠❛t✐♦♥ ▼✉♥✐❝❤ ✷✵✶✵ ✾
❚❤❡ ❘❛s❝❤ ▼♦❞❡❧ ▼▼▲ ✈s ❈▼▲ ▼▼▲ ❆❞✈❛♥t❛❣❡s✿ ● ❣✐✈❡s ❛❧s♦ ❡st✐♠❛t❡s ❢♦r ♣❡rs♦♥s ✇✐t❤ r v = 0 ♦r r v = k ● ✇❤❡♥ r❡s❡❛r❝❤ ❛✐♠s ❛t ♣❡rs♦♥ ❞✐str✐❜✉t✐♦♥ ● ❛❧❧♦✇s ❡st✐♠❛t✐♦♥ ♦❢ ❛❞❞✐t✐♦♥❛❧ ♣❛r❛♠❡t❡rs ✭✷P▲✱ ✸P▲ ♠♦❞❡❧s✮ ● ❢❛st❡r ✇✐t❤ ❧❛r❣❡ k ❈▼▲ ❆❞✈❛♥t❛❣❡s✿ ● ✇❤❡♥ ❘▼ ✐s ✉s❡❞ ❛s ♠❡❛s✉r❡♠❡♥t ♠♦❞❡❧ ✭s❝❛❧❡ ❝♦♥str✉❝t✐♦♥✮ ● ▼▼▲ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❜✐❛s❡❞ ✐❢ G ( θ ) ✐♥❝♦rr❡❝t❧② s♣❡❝✐✜❡❞ ● ❈▼▲ ❝❧♦s❡r t♦ ❝♦♥❝❡♣t ♦❢ ♣❡rs♦♥✲❢r❡❡ ❛ss❡ss♠❡♥t ● ❛❧❧♦✇s ❢♦r s♣❡❝✐✜❝ ♦❜❥❡❝t✐✈✐t② ● s❡✈❡r❛❧ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ♥♦t ❛✈❛✐❧❛❜❧❡ ✇✐t❤ ▼▼▲ ❞✐str✐❜✉t✐♦♥❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❈▼▲ ❛♥❞ ▼▼▲ ❡st✐♠❛t❡s ❛r❡ t❤❡ s❛♠❡ ❛s②♠t♦t✐❝❛❧❧② ▼✉♥✐❝❤ ✷✵✶✵ ✶✵
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