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  1. ♠✐①♠❝♠✿ ❛ ❙t❛t❛ ❝♦♠♠❛♥❞ ❢♦r ❡st✐♠❛t✐♥❣ ♠✐①t✉r❡ ♦❢ ▼❛r❦♦✈ ❝❤❛✐♥ ♠♦❞❡❧s ✉s✐♥❣ ▼▲ ❛♥❞ t❤❡ ❊▼ ❛❧❣♦r✐t❤♠ ▲❡❣r❛♥❞ ❉✳ ❋✳ ❙❆■◆❚✲❈❨❘ ❛♥❞ ▲❛✉r❡♥t P■❊❚ ❙▼❆❘❚✱ ❆●❘❖❈❆▼P❯❙ ❖❯❊❙❚✱ ■◆❘❆✱ ✸✺✵✵✵✱ ❘❡♥♥❡s✱ ❋r❛♥❝❡ ❋r❡♥❝❤ ❙t❛t❛ ❯s❡rs ●r♦✉♣ ♠❡❡t✐♥❣ P❛r✐s ⑤ ❏✉❧② ✻

  2. ❍❡t❡r♦❣❡♥❡♦✉s ❜❡❤❛✈✐♦✉rs ✐♥ s❡✈❡r❛❧ ❝❛s❡s t❤❛t ❛r❡ ❣❡♥❡r❛❧❧② ✉♥♦❜s❡r✈❡❞ ❛♥❞ ❝❛♥♥♦t ❜❡ ❝❛♣t✉r❡❞ ❜② ♦❜s❡r✈❛❜❧❡ ❛❣❡♥t ❝❤❛r❛❝t❡r✐st✐❝s ❙♦♠❡ ❛✈❛✐❧❛❜❧❡ ❝♦♠♠❛♥❞s ✐♥ ❙❚❆❚❆ ❛❧❧♦✇ ❡st✐♠❛t✐♥❣ ✜♥✐t❡ ♠✐①t✉r❡ ♠♦❞❡❧s t♦ ❝❛♣t✉r❡ ✉♥♦❜s❡r✈❡❞ ❤❡t❡r♦❣❡♥❡✐t② ❖✣❝✐❛❧ ❝♦♠♠❛♥❞s ✭ ❢♠♠ ✮ ❯s❡rs ✇r✐tt❡♥ ❝♦♠♠❛♥❞s ✭ ❣❧❧❛♠♠ ✱ ❧❝❧♦❣✐t ✱ ✳✳✳ ✮ ■♠♣♦ss✐❜❧❡ t♦ ❡st✐♠❛t❡ ❞✐r❡❝t❧② ❛ ♠✐①t✉r❡ ♦❢ ▼❛r❦♦✈ ❝❤❛✐♥ ♠♦❞❡❧s ✉s✐♥❣ t❤❡ ❛✈❛✐❧❛❜❧❡ ❝♦♠♠❛♥❞s ✐♥ ❙❚❆❚❆ ❇❛❝❦❣r♦✉♥❞ ◮ ▼❛r❦♦✈ ❝❤❛✐♥ ♠♦❞❡❧ ✭▼❈▼✮ ✐s ❛ ✇✐❞❡❧② ✉s❡❞ ♠♦❞❡❧❧✐♥❣ ❛♣♣r♦❛❝❤ ✐♥ s❡✈❡r❛❧ str❛♥❞s ♦❢ t❤❡ ❧✐t❡r❛t✉r❡ ▼❈▼ ❡♥❛❜❧❡s ❛♥❛❧②s✐♥❣ ❞②♥❛♠✐❝ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✇✐t❤✐♥ ❛ ❣✐✈❡♥ ♣♦♣✉❧❛t✐♦♥ ✭❢✉t✉r❡ st❛t❡s ❞❡♣❡♥❞ ♦♥ t❤❡ ♣❛st ❛❝❝♦r❞✐♥❣❧② t♦ s♦♠❡ ♣r♦❜❛❜✐❧✐t✐❡s✮ ◆✉♠❡r♦✉s ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❡❝♦♥♦♠✐❝s ✭✜r♠ ❞②♥❛♠✐❝s✱ ✉♥❡♠♣❧♦②♠❡♥t✱ ✳✳✳✮✱ ♠❡❞✐❝✐♥❡ ✭✐❧❧♥❡ss tr❡❛t♠❡♥t✮✱ s♦❝✐♦❧♦❣② ✭♣♦♣✉❧❛t✐♦♥ ♠♦❜✐❧✐t②✮✱ ✳✳✳

  3. ❙♦♠❡ ❛✈❛✐❧❛❜❧❡ ❝♦♠♠❛♥❞s ✐♥ ❙❚❆❚❆ ❛❧❧♦✇ ❡st✐♠❛t✐♥❣ ✜♥✐t❡ ♠✐①t✉r❡ ♠♦❞❡❧s t♦ ❝❛♣t✉r❡ ✉♥♦❜s❡r✈❡❞ ❤❡t❡r♦❣❡♥❡✐t② ❖✣❝✐❛❧ ❝♦♠♠❛♥❞s ✭ ❢♠♠ ✮ ❯s❡rs ✇r✐tt❡♥ ❝♦♠♠❛♥❞s ✭ ❣❧❧❛♠♠ ✱ ❧❝❧♦❣✐t ✱ ✳✳✳ ✮ ■♠♣♦ss✐❜❧❡ t♦ ❡st✐♠❛t❡ ❞✐r❡❝t❧② ❛ ♠✐①t✉r❡ ♦❢ ▼❛r❦♦✈ ❝❤❛✐♥ ♠♦❞❡❧s ✉s✐♥❣ t❤❡ ❛✈❛✐❧❛❜❧❡ ❝♦♠♠❛♥❞s ✐♥ ❙❚❆❚❆ ❇❛❝❦❣r♦✉♥❞ ◮ ▼❛r❦♦✈ ❝❤❛✐♥ ♠♦❞❡❧ ✭▼❈▼✮ ✐s ❛ ✇✐❞❡❧② ✉s❡❞ ♠♦❞❡❧❧✐♥❣ ❛♣♣r♦❛❝❤ ✐♥ s❡✈❡r❛❧ str❛♥❞s ♦❢ t❤❡ ❧✐t❡r❛t✉r❡ ▼❈▼ ❡♥❛❜❧❡s ❛♥❛❧②s✐♥❣ ❞②♥❛♠✐❝ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✇✐t❤✐♥ ❛ ❣✐✈❡♥ ♣♦♣✉❧❛t✐♦♥ ✭❢✉t✉r❡ st❛t❡s ❞❡♣❡♥❞ ♦♥ t❤❡ ♣❛st ❛❝❝♦r❞✐♥❣❧② t♦ s♦♠❡ ♣r♦❜❛❜✐❧✐t✐❡s✮ ◆✉♠❡r♦✉s ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❡❝♦♥♦♠✐❝s ✭✜r♠ ❞②♥❛♠✐❝s✱ ✉♥❡♠♣❧♦②♠❡♥t✱ ✳✳✳✮✱ ♠❡❞✐❝✐♥❡ ✭✐❧❧♥❡ss tr❡❛t♠❡♥t✮✱ s♦❝✐♦❧♦❣② ✭♣♦♣✉❧❛t✐♦♥ ♠♦❜✐❧✐t②✮✱ ✳✳✳ ◮ ❍❡t❡r♦❣❡♥❡♦✉s ❜❡❤❛✈✐♦✉rs ✐♥ s❡✈❡r❛❧ ❝❛s❡s t❤❛t ❛r❡ ❣❡♥❡r❛❧❧② ✉♥♦❜s❡r✈❡❞ ❛♥❞ ❝❛♥♥♦t ❜❡ ❝❛♣t✉r❡❞ ❜② ♦❜s❡r✈❛❜❧❡ ❛❣❡♥t ❝❤❛r❛❝t❡r✐st✐❝s

  4. ❇❛❝❦❣r♦✉♥❞ ◮ ▼❛r❦♦✈ ❝❤❛✐♥ ♠♦❞❡❧ ✭▼❈▼✮ ✐s ❛ ✇✐❞❡❧② ✉s❡❞ ♠♦❞❡❧❧✐♥❣ ❛♣♣r♦❛❝❤ ✐♥ s❡✈❡r❛❧ str❛♥❞s ♦❢ t❤❡ ❧✐t❡r❛t✉r❡ ▼❈▼ ❡♥❛❜❧❡s ❛♥❛❧②s✐♥❣ ❞②♥❛♠✐❝ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✇✐t❤✐♥ ❛ ❣✐✈❡♥ ♣♦♣✉❧❛t✐♦♥ ✭❢✉t✉r❡ st❛t❡s ❞❡♣❡♥❞ ♦♥ t❤❡ ♣❛st ❛❝❝♦r❞✐♥❣❧② t♦ s♦♠❡ ♣r♦❜❛❜✐❧✐t✐❡s✮ ◆✉♠❡r♦✉s ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❡❝♦♥♦♠✐❝s ✭✜r♠ ❞②♥❛♠✐❝s✱ ✉♥❡♠♣❧♦②♠❡♥t✱ ✳✳✳✮✱ ♠❡❞✐❝✐♥❡ ✭✐❧❧♥❡ss tr❡❛t♠❡♥t✮✱ s♦❝✐♦❧♦❣② ✭♣♦♣✉❧❛t✐♦♥ ♠♦❜✐❧✐t②✮✱ ✳✳✳ ◮ ❍❡t❡r♦❣❡♥❡♦✉s ❜❡❤❛✈✐♦✉rs ✐♥ s❡✈❡r❛❧ ❝❛s❡s t❤❛t ❛r❡ ❣❡♥❡r❛❧❧② ✉♥♦❜s❡r✈❡❞ ❛♥❞ ❝❛♥♥♦t ❜❡ ❝❛♣t✉r❡❞ ❜② ♦❜s❡r✈❛❜❧❡ ❛❣❡♥t ❝❤❛r❛❝t❡r✐st✐❝s ◮ ❙♦♠❡ ❛✈❛✐❧❛❜❧❡ ❝♦♠♠❛♥❞s ✐♥ ❙❚❆❚❆ ❛❧❧♦✇ ❡st✐♠❛t✐♥❣ ✜♥✐t❡ ♠✐①t✉r❡ ♠♦❞❡❧s t♦ ❝❛♣t✉r❡ ✉♥♦❜s❡r✈❡❞ ❤❡t❡r♦❣❡♥❡✐t② ❖✣❝✐❛❧ ❝♦♠♠❛♥❞s ✭ ❢♠♠ ✮ ❯s❡rs ✇r✐tt❡♥ ❝♦♠♠❛♥❞s ✭ ❣❧❧❛♠♠ ✱ ❧❝❧♦❣✐t ✱ ✳✳✳ ✮ ◮ ■♠♣♦ss✐❜❧❡ t♦ ❡st✐♠❛t❡ ❞✐r❡❝t❧② ❛ ♠✐①t✉r❡ ♦❢ ▼❛r❦♦✈ ❝❤❛✐♥ ♠♦❞❡❧s ✉s✐♥❣ t❤❡ ❛✈❛✐❧❛❜❧❡ ❝♦♠♠❛♥❞s ✐♥ ❙❚❆❚❆

  5. Pr♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ♦❢ ▼▼❈▼ ✿ ♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡❧♦♥❣✐♥❣ t♦ t②♣❡ ❚❤❡ ♠✐①❡❞ ▼❛r❦♦✈ ❝❤❛✐♥ ♠♦❞❡❧ ✭▼▼❈▼✮ ◮ ▼▼❈▼ ❞❡s❝r✐❜❡s t❤❡ ❞②♥❛♠✐❝s ♦❢ N ❛❣❡♥ts ♦♥ ❛ ✜♥✐t❡ st❛t❡ s♣❛❝❡ K ♦✈❡r ❛ t✐♠❡ ♣❡r✐♦❞ T ✇✐t❤ ❤❡t❡r♦❣❡♥❡♦✉s tr❛♥s✐t✐♦♥ ♣r♦❝❡ss❡s ◮ Pr♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ♦❢ ▼❈▼ T i � f ( y i ) = P ( y it = k | y it − 1 = j ) , ∀ i ∈ N ; ∀ j, k ∈ K t =1 y i = ( y i 0 , y i 1 , · · · , y iTi ); T i ≤ T

  6. ❚❤❡ ♠✐①❡❞ ▼❛r❦♦✈ ❝❤❛✐♥ ♠♦❞❡❧ ✭▼▼❈▼✮ ◮ ▼▼❈▼ ❞❡s❝r✐❜❡s t❤❡ ❞②♥❛♠✐❝s ♦❢ N ❛❣❡♥ts ♦♥ ❛ ✜♥✐t❡ st❛t❡ s♣❛❝❡ K ♦✈❡r ❛ t✐♠❡ ♣❡r✐♦❞ T ✇✐t❤ ❤❡t❡r♦❣❡♥❡♦✉s tr❛♥s✐t✐♦♥ ♣r♦❝❡ss❡s ◮ Pr♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ♦❢ ▼❈▼ T i � f ( y i ) = P ( y it = k | y it − 1 = j ) , ∀ i ∈ N ; ∀ j, k ∈ K t =1 y i = ( y i 0 , y i 1 , · · · , y iTi ); T i ≤ T ◮ Pr♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ♦❢ ▼▼❈▼ G � f ( y i ) = π g f g ( y i ) g =1 0 ≤ π g ≤ 1 ✿ ♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡❧♦♥❣✐♥❣ t♦ t②♣❡ g

  7. Pr♦❜❛❜✐❧✐t② ♦❢ t②♣❡ ♠❡♠❜❡rs❤✐♣ ❋r❛❝t✐♦♥❛❧ ♠✉❧t✐♥♦♠✐❛❧ ❧♦❣✐t ❢♦r ♣❛r❛♠❡tr✐❝ s♣❡❝✐✜❝❛t✐♦♥ ◆♦♥✲♣❛r❛♠❡tr✐❝ ❡st✐♠❛t✐♦♥ ✐♠♣❧✐❡s t❤❛t ❛r❡ t❤❡ s❛♠❡ ❢♦r ❛❧❧ ❛❣❡♥ts ❚❤❡ ♠♦❞❡❧ s♣❡❝✐✜❝❛t✐♦♥ ◮ ▼✉❧t✐♥♦♠✐❛❧ ❧♦❣✐t s♣❡❝✐✜❝❛t✐♦♥ ♦❢ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ′ exp( β jk | g x it − 1 ) P ( y it = k | y it − 1 = j, g ) = l =1 exp( β ′ � K jl | g x it − 1 ) β jj | g = 0 ❢♦r ✐❞❡♥t✐✜❝❛t✐♦♥

  8. ❚❤❡ ♠♦❞❡❧ s♣❡❝✐✜❝❛t✐♦♥ ◮ ▼✉❧t✐♥♦♠✐❛❧ ❧♦❣✐t s♣❡❝✐✜❝❛t✐♦♥ ♦❢ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ′ exp( β jk | g x it − 1 ) P ( y it = k | y it − 1 = j, g ) = l =1 exp( β ′ � K jl | g x it − 1 ) β jj | g = 0 ❢♦r ✐❞❡♥t✐✜❝❛t✐♦♥ ◮ Pr♦❜❛❜✐❧✐t② ♦❢ t②♣❡ ♠❡♠❜❡rs❤✐♣ ❋r❛❝t✐♦♥❛❧ ♠✉❧t✐♥♦♠✐❛❧ ❧♦❣✐t ❢♦r ♣❛r❛♠❡tr✐❝ s♣❡❝✐✜❝❛t✐♦♥ ′ exp( λ g z i ) P ( g i = g | z i ) = ( ∀ g ∈ G − 1) h =1 exp( λ ′ � G h z i ) ◆♦♥✲♣❛r❛♠❡tr✐❝ ❡st✐♠❛t✐♦♥ ✐♠♣❧✐❡s t❤❛t P ( g i = g ) ❛r❡ t❤❡ s❛♠❡ ❢♦r ❛❧❧ ❛❣❡♥ts

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