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❆ ❧♦❣✲❧✐♥❡❛r ♠♦❞❡❧ ❢♦r ✉♥s✉♣❡r✈✐s❡❞ t❡①t ♥♦r♠❛❧✐③❛t✐♦♥

❨✐ ❨❛♥❣ ❛♥❞ ❏❛❝♦❜ ❊✐s❡♥st❡✐♥

• ❡♦r❣✐❛ ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣②

▲❡①✐❝❛❧ ✈❛r✐❛t✐♦♥

◮ ❙♦❝✐❛❧ ♠❡❞✐❛ ❧❛♥❣✉❛❣❡ ✐s ❞✐✛❡r❡♥t ❢r♦♠ st❛♥❞❛r❞

❊♥❣❧✐s❤✳

◮ ❚✇❡❡t✿

ya ur website suxx brah

◮ ❙t❛♥❞❛r❞✿ ②❡❛❤✱ ②♦✉r ✇❡❜s✐t❡ s✉❝❦s ❜r♦

❲♦r❞ ✈❛r✐❛♥ts✿ ♣❤♦♥❡t✐❝ s✉❜st✐t✉t✐♦♥s✱ ❛❜❜r❡✈✐❛t✐♦♥s✱ ✉♥❝♦♥✈❡♥t✐♦♥❛❧ s♣❡❧❧✐♥❣s✱ ❡t❝✳ ◆♦r♠❛❧✐③❛t✐♦♥✿ ♠❛♣ ❢r♦♠ ♦rt❤♦❣r❛♣❤✐❝ ✈❛r✐❛♥ts t♦ st❛♥❞❛r❞ s♣❡❧❧✐♥❣s✳

▲❛r❣❡ ❧❛❜❡❧ s♣❛❝❡✳ ◆♦ ❧❛❜❡❧❡❞ ❞❛t❛✳

▲❡①✐❝❛❧ ✈❛r✐❛t✐♦♥

◮ ❙♦❝✐❛❧ ♠❡❞✐❛ ❧❛♥❣✉❛❣❡ ✐s ❞✐✛❡r❡♥t ❢r♦♠ st❛♥❞❛r❞

❊♥❣❧✐s❤✳

◮ ❚✇❡❡t✿

ya ur website suxx brah

◮ ❙t❛♥❞❛r❞✿ ②❡❛❤✱ ②♦✉r ✇❡❜s✐t❡ s✉❝❦s ❜r♦

◮ ❲♦r❞ ✈❛r✐❛♥ts✿ ♣❤♦♥❡t✐❝ s✉❜st✐t✉t✐♦♥s✱

❛❜❜r❡✈✐❛t✐♦♥s✱ ✉♥❝♦♥✈❡♥t✐♦♥❛❧ s♣❡❧❧✐♥❣s✱ ❡t❝✳ ◆♦r♠❛❧✐③❛t✐♦♥✿ ♠❛♣ ❢r♦♠ ♦rt❤♦❣r❛♣❤✐❝ ✈❛r✐❛♥ts t♦ st❛♥❞❛r❞ s♣❡❧❧✐♥❣s✳

▲❛r❣❡ ❧❛❜❡❧ s♣❛❝❡✳ ◆♦ ❧❛❜❡❧❡❞ ❞❛t❛✳

▲❡①✐❝❛❧ ✈❛r✐❛t✐♦♥

◮ ❙♦❝✐❛❧ ♠❡❞✐❛ ❧❛♥❣✉❛❣❡ ✐s ❞✐✛❡r❡♥t ❢r♦♠ st❛♥❞❛r❞

❊♥❣❧✐s❤✳

◮ ❚✇❡❡t✿

ya ur website suxx brah

◮ ❙t❛♥❞❛r❞✿ ②❡❛❤✱ ②♦✉r ✇❡❜s✐t❡ s✉❝❦s ❜r♦

◮ ❲♦r❞ ✈❛r✐❛♥ts✿ ♣❤♦♥❡t✐❝ s✉❜st✐t✉t✐♦♥s✱

❛❜❜r❡✈✐❛t✐♦♥s✱ ✉♥❝♦♥✈❡♥t✐♦♥❛❧ s♣❡❧❧✐♥❣s✱ ❡t❝✳

◮ ◆♦r♠❛❧✐③❛t✐♦♥✿ ♠❛♣ ❢r♦♠ ♦rt❤♦❣r❛♣❤✐❝

✈❛r✐❛♥ts t♦ st❛♥❞❛r❞ s♣❡❧❧✐♥❣s✳

▲❛r❣❡ ❧❛❜❡❧ s♣❛❝❡✳ ◆♦ ❧❛❜❡❧❡❞ ❞❛t❛✳

▲❡①✐❝❛❧ ✈❛r✐❛t✐♦♥

◮ ❙♦❝✐❛❧ ♠❡❞✐❛ ❧❛♥❣✉❛❣❡ ✐s ❞✐✛❡r❡♥t ❢r♦♠ st❛♥❞❛r❞

❊♥❣❧✐s❤✳

◮ ❚✇❡❡t✿

ya ur website suxx brah

◮ ❙t❛♥❞❛r❞✿ ②❡❛❤✱ ②♦✉r ✇❡❜s✐t❡ s✉❝❦s ❜r♦

◮ ❲♦r❞ ✈❛r✐❛♥ts✿ ♣❤♦♥❡t✐❝ s✉❜st✐t✉t✐♦♥s✱

❛❜❜r❡✈✐❛t✐♦♥s✱ ✉♥❝♦♥✈❡♥t✐♦♥❛❧ s♣❡❧❧✐♥❣s✱ ❡t❝✳

◮ ◆♦r♠❛❧✐③❛t✐♦♥✿ ♠❛♣ ❢r♦♠ ♦rt❤♦❣r❛♣❤✐❝

✈❛r✐❛♥ts t♦ st❛♥❞❛r❞ s♣❡❧❧✐♥❣s✳

◮ ▲❛r❣❡ ❧❛❜❡❧ s♣❛❝❡✳ ◮ ◆♦ ❧❛❜❡❧❡❞ ❞❛t❛✳

❲❤② ♥♦r♠❛❧✐③❛t✐♦♥❄

◮ ■♠♣r♦✈❡ ❞♦✇♥str❡❛♠ ◆▲P ❛♣♣❧✐❝❛t✐♦♥s✳

◮ P❛rt✲♦❢✲s♣❡❡❝❤ t❛❣❣✐♥❣ ❬✻❪ ◮ ❉❡♣❡♥❞❡♥❝② ♣❛rs✐♥❣ ❬✶✶❪ ◮ ▼❛❝❤✐♥❡ tr❛♥s❧❛t✐♦♥ ❬✼✱ ✽❪

■s t❤❡r❡ ♠♦r❡ s②st❡♠❛t✐❝ ✈❛r✐❛t✐♦♥ ✐♥ s♦❝✐❛❧ ♠❡❞✐❛❄

▼✐♥❡ ♣❛tt❡r♥s ✐♥ ❤♦✇ ✇♦r❞s ❛r❡ s♣❡❧❧❡❞✳ ❉✐s❝♦✈❡r ❝♦❤❡r❡♥t ♦rt❤♦❣r❛♣❤✐❝ st②❧❡s✳

❲❤② ♥♦r♠❛❧✐③❛t✐♦♥❄

◮ ■♠♣r♦✈❡ ❞♦✇♥str❡❛♠ ◆▲P ❛♣♣❧✐❝❛t✐♦♥s✳

◮ P❛rt✲♦❢✲s♣❡❡❝❤ t❛❣❣✐♥❣ ❬✻❪ ◮ ❉❡♣❡♥❞❡♥❝② ♣❛rs✐♥❣ ❬✶✶❪ ◮ ▼❛❝❤✐♥❡ tr❛♥s❧❛t✐♦♥ ❬✼✱ ✽❪

◮ ■s t❤❡r❡ ♠♦r❡ s②st❡♠❛t✐❝ ✈❛r✐❛t✐♦♥ ✐♥ s♦❝✐❛❧

♠❡❞✐❛❄

◮ ▼✐♥❡ ♣❛tt❡r♥s ✐♥ ❤♦✇ ✇♦r❞s ❛r❡ s♣❡❧❧❡❞✳ ◮ ❉✐s❝♦✈❡r ❝♦❤❡r❡♥t ♦rt❤♦❣r❛♣❤✐❝ st②❧❡s✳

❘❡❧❛t❡❞ ✇♦r❦

▼❡t❤♦❞ ❙✉r❢❛❝❡ ❈♦♥t❡①t ❋✐♥❛❧ ❙②st❡♠ ❍❛♥ ✫ ❇❛❧❞✇✐♥ ✷✵✶✶ ❡❞✐t ❞✐st❛♥❝❡ ▲❈❙ ❧❛♥❣✉❛❣❡ ♠♦❞❡❧ s②♥t❛① ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ▲✐✉ ❡t ❛❧✳ ✷✵✶✷ ❝❤❛r❛❝t❡r✲❜❛s❡❞ tr❛♥s❧❛t✐♦♥ ❞✐str✐❜✉t✐♦♥❛❧ s✐♠✐❧❛r✐t② ❞❡❝♦❞✐♥❣ ✇✐t❤ ❧❛♥❣✉❛❣❡ ♠♦❞❡❧ ❍❛ss❛♥ ❡t ❛❧✳ ✷✵✶✸ ❡❞✐t ❞✐st❛♥❝❡ ▲❈❙ r❛♥❞♦♠ ✇❛❧❦ ❞❡❝♦❞✐♥❣ ✇✐t❤ ❧❛♥❣✉❛❣❡ ♠♦❞❡❧ ❖✉r ❛♣♣r♦❛❝❤ ❡❞✐t ❞✐st❛♥❝❡ ▲❈❙✱ ✇♦r❞ ♣❛✐rs ❧❛♥❣✉❛❣❡ ♠♦❞❡❧ ✉♥✐✜❡❞ ♠♦❞❡❧ ✇✐t❤ ❢❡❛t✉r❡s

❖✉r ❛♣♣r♦❛❝❤

◮ ❯♥s✉♣❡r✈✐s❡❞ ◮ ❋❡❛t✉r✐③❡❞ ◮ ❈♦♥t❡①t✲❞r✐✈❡♥ ◮ ❏♦✐♥t

❈❤❛r❛❝t❡r✐st✐❝s

❯♥s✉♣❡r✈✐s❡❞

◮ ▲❛❜❡❧❡❞ ❞❛t❛ ❢♦r ❚✇✐tt❡r ♥♦r♠❛❧✐③❛t✐♦♥ ✐s ❧✐♠✐t❡❞✳ ◮ ▲❛♥❣✉❛❣❡ ❝❤❛♥❣❡s✱ ❛♥♥♦t❛t✐♦♥s ♠❛② ❜❡❝♦♠❡

st❛❧❡ ❛♥❞ ✐❧❧✲s✉✐t❡❞ t♦ ♥❡✇ s♣❡❧❧✐♥❣s ❛♥❞ ✇♦r❞s✳

❚❡r♠ ❢r❡q✉❡♥❝② ♦❢ ✏❛❢✑ ♦✈❡r ♠♦♥t❤s ❚❡r♠ ❢r❡q✉❡♥❝② ♦❢ ✏❧♠❧✑ ♦✈❡r ♠♦♥t❤s

❈❤❛r❛❝t❡r✐st✐❝s

❋❡❛t✉r✐③❡❞✿ ♣❡r♠✐tt✐♥❣ ♦✈❡r❧❛♣♣✐♥❣ ❢❡❛t✉r❡s

◮ ❙tr✐♥❣ s✐♠✐❧❛r✐t② ❢❡❛t✉r❡s✳ ◮ ▲❡①✐❝❛❧ ❢❡❛t✉r❡s t❤❛t ♠❡♠♦r✐③❡ ❝♦♥✈❡♥t✐♦♥❛❧✐③❡❞

✇♦r❞ ♣❛✐rs✱ ❡✳❣✳ ②♦✉✴u✱ t♦✴2✳

❈❤❛r❛❝t❡r✐st✐❝s

❈♦♥t❡①t✲❞r✐✈❡♥ give me suttin to believe in

◮ ❯♥s✉♣❡r✈✐s❡❞ ♥♦r♠❛❧✐③❛t✐♦♥ ♥❡❡❞s str♦♥❣ ❝✉❡ ♦❢

❧♦❝❛❧ ❝♦♥t❡①t✳

◮ ❙tr✐♥❣ s✐♠✐❧❛r✐t② ♥❡❡❞s t♦ ❜❡ ♦✈❡r❝♦♠❡ ❜②

❝♦♥t❡①t✉❛❧ ♣r❡❢❡r❡♥❝❡✳

◮ ❊❞✐t✲❉✐st✭❜✉tt♦♥✴s✉✐t✐♥❣✱ suttin✮ = ✷ ◮ ❊❞✐t✲❉✐st✭s♦♠❡t❤✐♥❣✱ suttin✮ = ✺

❈❤❛r❛❝t❡r✐st✐❝s

❏♦✐♥t gimme suttin 2 beleive innnn

◮ ◆♦ ✇♦r❞s ❛r❡ ✐♥ t❤❡ st❛♥❞❛r❞ ✈♦❝❛❜✉❧❛r②✳ ◮ ❏♦✐♥t ✐♥❢❡r❡♥❝❡ ✐s ♥❡❡❞❡❞ t♦ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢

❝♦♥t❡①t ✐♥❢♦r♠❛t✐♦♥✳

◆♦r♠❛❧✐③❛t✐♦♥ ✐♥ ❛ ♣r♦❜❛❜✐❧✐st✐❝ ♠♦❞❡❧

▼❛①✐♠✐③❡ ❧✐❦❡❧✐❤♦♦❞ ♦❢ ♦❜s❡r✈❡❞ ✭❚✇✐tt❡r✮ t❡①t✱ ♠❛r❣✐♥❛❧✐③✐♥❣ ♦✈❡r ♥♦r♠❛❧✐③❛t✐♦♥s✳

P(s) P(ya ur website suxx brah)

t

s t ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ ②❛❦ ②♦✉r ✇❡❜s✐t❡ ✳ ✳ ✳ ②❡❛ ❝✉r❡ ✇❡❜s✐t❡ ✳ ✳ ✳

◆♦r♠❛❧✐③❛t✐♦♥ ✐♥ ❛ ♣r♦❜❛❜✐❧✐st✐❝ ♠♦❞❡❧

▼❛①✐♠✐③❡ ❧✐❦❡❧✐❤♦♦❞ ♦❢ ♦❜s❡r✈❡❞ ✭❚✇✐tt❡r✮ t❡①t✱ ♠❛r❣✐♥❛❧✐③✐♥❣ ♦✈❡r ♥♦r♠❛❧✐③❛t✐♦♥s✳

P(s) P(ya ur website suxx brah) =

• t

P(s, t) P(ya ur website . . . , ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ ) + P(ya ur website . . . , ②❛❦ ②♦✉r ✇❡❜s✐t❡ ✳ ✳ ✳ ) + P(ya ur website . . . , ②❡❛ ❝✉r❡ ✇❡❜s✐t❡ ✳ ✳ ✳ ) + . . .

❆ ♥♦✐s② ❝❤❛♥♥❡❧ ♠♦❞❡❧

P(s, t) = Pt(t)Pe(s|t) Pe(ya ur . . . |②❛♠ ✉r♥ ✳ ✳ ✳ ) × Pt(②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ ) ❚❤❡ ❧❛♥❣✉❛❣❡ ♠♦❞❡❧ ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ♦✤✐♥❡✳ ❚❤❡ ❡♠✐ss✐♦♥ ♠♦❞❡❧ ✐s ❧♦❝❛❧❧② ♥♦r♠❛❧✐③❡❞ ❛♥❞ ❧♦❣✲❧✐♥❡❛r✳ s t ②❛♠ ✉r♥ ❡①♣ ❢ ❡①♣ ❢ ②♦✉r ②♦✉r

❆ ♥♦✐s② ❝❤❛♥♥❡❧ ♠♦❞❡❧

P(s, t) = Pt(t)Pe(s|t) Pe(ya ur . . . |②❛♠ ✉r♥ ✳ ✳ ✳ ) × Pt(②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ )

◮ ❚❤❡ ❧❛♥❣✉❛❣❡ ♠♦❞❡❧ ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ♦✤✐♥❡✳ ◮ ❚❤❡ ❡♠✐ss✐♦♥ ♠♦❞❡❧ ✐s ❧♦❝❛❧❧② ♥♦r♠❛❧✐③❡❞ ❛♥❞ ❧♦❣✲❧✐♥❡❛r✳

Pe(s|t) =

• n

Pe(sn|tn) Pe(ya|②❛♠)Pe(ur|✉r♥) . . . Pe(sn|tn) = ❡①♣ (θ′❢ (sn, tn)) Z(tn) ❡①♣ (θ′❢ (ur, ②♦✉r)) Z(②♦✉r)

❋❡❛t✉r❡s

❙tr✐♥❣ s✐♠✐❧❛r✐t②

◮ ❈♦♠❜✐♥❡ ❡❞✐t ❞✐st❛♥❝❡ ❛♥❞ ❧♦♥❣❡st✲❝♦♠♠♦♥

s✉❜s❡q✉❡♥❝❡ ✭▲❈❙✮ ❬✸❪✿

◮ ❇✐♥ t♦ ❝r❡❛t❡ ❜✐♥❛r② ❢❡❛t✉r❡s✱ ❡✳❣✳✱

top✺(s, t) =

• ✶,

t ∈ Top✺(s) ✵, ♦t❤❡r✇✐s❡ ❲♦r❞ ♣❛✐rs ❆ss✐❣♥ ✇❡✐❣❤ts t♦ ❝♦♠♠♦♥❧② ♦❝❝✉rr✐♥❣ ✇♦r❞ ♣❛✐rs✱ ❡✳❣✳ ②♦✉✴ ✱ s✉❝❦s✴ ✳ ❚❤✐s ❛❧❧♦✇s t❤❡ ♠♦❞❡❧ t♦ ✏♠❡♠♦r✐③❡✑ ❢r❡q✉❡♥t s✉❜st✐t✉t✐♦♥s✳

❋❡❛t✉r❡s

❙tr✐♥❣ s✐♠✐❧❛r✐t②

◮ ❈♦♠❜✐♥❡ ❡❞✐t ❞✐st❛♥❝❡ ❛♥❞ ❧♦♥❣❡st✲❝♦♠♠♦♥

s✉❜s❡q✉❡♥❝❡ ✭▲❈❙✮ ❬✸❪✿

◮ ❇✐♥ t♦ ❝r❡❛t❡ ❜✐♥❛r② ❢❡❛t✉r❡s✱ ❡✳❣✳✱

top✺(s, t) =

• ✶,

t ∈ Top✺(s) ✵, ♦t❤❡r✇✐s❡ ❲♦r❞ ♣❛✐rs

◮ ❆ss✐❣♥ ✇❡✐❣❤ts t♦ ❝♦♠♠♦♥❧② ♦❝❝✉rr✐♥❣ ✇♦r❞

♣❛✐rs✱ ❡✳❣✳ ②♦✉✴u✱ s✉❝❦s✴suxx✳

◮ ❚❤✐s ❛❧❧♦✇s t❤❡ ♠♦❞❡❧ t♦ ✏♠❡♠♦r✐③❡✑ ❢r❡q✉❡♥t

s✉❜st✐t✉t✐♦♥s✳

▲❡❛r♥✐♥❣

❖✉r ❣♦❛❧ ✐s t♦ ❧❡❛r♥ t♦ ✇❡✐❣❤t t❤❡s❡ ❢❡❛t✉r❡s✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❧✐❦❡❧✐❤♦♦❞✳✳✳

❈❘❋ ❧♦❣ ② ① ① ②

② ①

① ② ❖✉r ♠♦❞❡❧ ✭▼❘❋✮ ❧♦❣ s

t s

s t

s t

s t

▲❡❛r♥✐♥❣

❖✉r ❣♦❛❧ ✐s t♦ ❧❡❛r♥ t♦ ✇❡✐❣❤t t❤❡s❡ ❢❡❛t✉r❡s✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❧✐❦❡❧✐❤♦♦❞✳✳✳

❈❘❋ ℓ(θ) = ❧♦❣ P(②|①) ∂ℓ ∂θ = f (①, ②) − E②|①;θ[f (①, ②)] ❖✉r ♠♦❞❡❧ ✭▼❘❋✮ ❧♦❣ s

t s

s t

s t

s t

▲❡❛r♥✐♥❣

❖✉r ❣♦❛❧ ✐s t♦ ❧❡❛r♥ t♦ ✇❡✐❣❤t t❤❡s❡ ❢❡❛t✉r❡s✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❧✐❦❡❧✐❤♦♦❞✳✳✳

❈❘❋ ℓ(θ) = ❧♦❣ P(②|①) ∂ℓ ∂θ = f (①, ②) − E②|①;θ[f (①, ②)] ❖✉r ♠♦❞❡❧ ✭▼❘❋✮ ℓ(θ) = ❧♦❣ P(s) ∂ℓ ∂θ = Et|s[f (s, t) − Es′|t[f (s, t)]]

❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣

■♥ ❛ ❧♦❝❛❧❧②✲♥♦r♠❛❧✐③❡❞ ♠♦❞❡❧✱ t❤❡s❡ ❡①♣❡❝t❛t✐♦♥s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ♠❛r❣✐♥❛❧s P(tn|s✶:N) ❬✶❪✳ ②❛ ✉r ✇❡❜s✐t❡ s✉①① ❜r❛❤ ②❛♠ ✉r♥ ✇❡❜s✐t❡ s✉❝❦s ❜r❛

• ②♦✉

②♦✉r s✉❝❦ ❜r❛t

• ②❡❛❤

②♦✉✬r❡ s✐① ❜r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❖✉t❡r ❡①♣❡❝t❛t✐♦♥

t s✿ ❢♦r✇❛r❞✲❜❛❝❦✇❛r❞

■♥♥❡r ❡①♣❡❝t❛t✐♦♥

s t✿ s✉♠ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡

❛t ❡❛❝❤ ✳ ❚✐♠❡ ❝♦♠♣❧❡①✐t②✿

✷ ✷

✳✳✳ ❇✉t ✶✵✹

❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣

■♥ ❛ ❧♦❝❛❧❧②✲♥♦r♠❛❧✐③❡❞ ♠♦❞❡❧✱ t❤❡s❡ ❡①♣❡❝t❛t✐♦♥s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ♠❛r❣✐♥❛❧s P(tn|s✶:N) ❬✶❪✳ ②❛ ✉r ✇❡❜s✐t❡ s✉①① ❜r❛❤ ②❛♠ ✉r♥ ✇❡❜s✐t❡ s✉❝❦s ❜r❛

• ②♦✉

②♦✉r s✉❝❦ ❜r❛t

• ②❡❛❤

②♦✉✬r❡ s✐① ❜r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

◮ ❖✉t❡r ❡①♣❡❝t❛t✐♦♥ Et|s✿ ❢♦r✇❛r❞✲❜❛❝❦✇❛r❞ ◮ ■♥♥❡r ❡①♣❡❝t❛t✐♦♥ Es′|t✿ s✉♠ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡

sn ❛t ❡❛❝❤ n✳ ❚✐♠❡ ❝♦♠♣❧❡①✐t②✿

✷ ✷

✳✳✳ ❇✉t ✶✵✹

❉②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣

■♥ ❛ ❧♦❝❛❧❧②✲♥♦r♠❛❧✐③❡❞ ♠♦❞❡❧✱ t❤❡s❡ ❡①♣❡❝t❛t✐♦♥s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❡ ♠❛r❣✐♥❛❧s P(tn|s✶:N) ❬✶❪✳ ②❛ ✉r ✇❡❜s✐t❡ s✉①① ❜r❛❤ ②❛♠ ✉r♥ ✇❡❜s✐t❡ s✉❝❦s ❜r❛

• ②♦✉

②♦✉r s✉❝❦ ❜r❛t

• ②❡❛❤

②♦✉✬r❡ s✐① ❜r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

◮ ❖✉t❡r ❡①♣❡❝t❛t✐♦♥ Et|s✿ ❢♦r✇❛r❞✲❜❛❝❦✇❛r❞ ◮ ■♥♥❡r ❡①♣❡❝t❛t✐♦♥ Es′|t✿ s✉♠ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡

sn ❛t ❡❛❝❤ n✳

◮ ❚✐♠❡ ❝♦♠♣❧❡①✐t②✿ O(NV ✷

T + NV ✷ TVS) ✳✳✳

❇✉t VT, VS > ✶✵✹

❙❡q✉❡♥t✐❛❧ ▼♦♥t❡ ❈❛r❧♦

❆ r❛♥❞♦♠✐③❡❞ ❛❧❣♦r✐t❤♠ t♦ ❛♣♣r♦①✐♠❛t❡ P(t|s) ❛s ❛ ✇❡✐❣❤t❡❞ s✉♠✱ P(t|s) ≈

• k

ω(k)δt(k)(t)

P(t|ya ur website . . . ) ≈        ✵.✷ × δ(②♦✉ ②♦✉r ✇❡❜s✐t❡ ✳ ✳ ✳ ) ✵.✻ × δ(②❡❛❤ ②♦✉r ✇❡❜s✐t❡ ✳ ✳ ✳ ) . . . ✵.✶ × δ(②♦✉ ②♦✉✬r❡ ✇❡❜s✐t❡ ✳ ✳ ✳ )

◮ ❡✣❝✐❡♥t ❛♥❞ s✐♠♣❧❡ ◮ ❡❛s② t♦ ♣❛r❛❧❧❡❧✐③❡ ◮ ♥✉♠❜❡r ♦❢ s❛♠♣❧❡s K ♣r♦✈✐❞❡s ✐♥t✉✐t✐✈❡ t✉♥✐♥❣

❜❡t✇❡❡♥ ❛❝❝✉r❛❝② ❛♥❞ s♣❡❡❞

❙❡q✉❡♥t✐❛❧ ■♠♣♦rt❛♥❝❡ ❙❛♠♣❧✐♥❣ ✭❙■❙✮

❆t ❡❛❝❤ n✱ ❢♦r ❡❛❝❤ s❛♠♣❧❡ k

◮ ❉r❛✇ t(k)

n

❢r♦♠ t❤❡ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ Q(t)✳

◮ ❯♣❞❛t❡ ω(k)

n

s♦ t❤❛t ω(k)

n

∝P(t(k)

✶:n|s✶:n)

Q(t(k)

✶:n)

= . . . =

❊♠✐ss✐♦♥ ♠♦❞❡❧

• Pe(sn|t(k)

n ) ▲❛♥❣✉❛❣❡ ♠♦❞❡❧

• Pt(t(k)

n |t(k) n−✶)

Q(t(k)

n |t(k) n−✶, sn)

• Pr♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥

ω(k)

n−✶

❈♦♠♣✉t✐♥❣ t❤❡ ❣r❛❞✐❡♥t ❢r♦♠ ❙■❙

◮ ❖✉t❡r ❡①♣❡❝t❛t✐♦♥✿

Et|s[f (s, t)] =

K

• k

ω(k)

N N

• n

❢ (sn, t(k)

n )

■♥♥❡r ❡①♣❡❝t❛t✐♦♥✿ ❞r❛✇ s❛♠♣❧❡s ♦❢ ✿

t s s t

s t ✶ ❢ ✭■♥ ♣r❛❝t✐❝❡✱ ✇❡ s❡t ✶✵ ❛♥❞ ✶✮

❈♦♠♣✉t✐♥❣ t❤❡ ❣r❛❞✐❡♥t ❢r♦♠ ❙■❙

◮ ❖✉t❡r ❡①♣❡❝t❛t✐♦♥✿

Et|s[f (s, t)] =

K

• k

ω(k)

N N

• n

❢ (sn, t(k)

n )

◮ ■♥♥❡r ❡①♣❡❝t❛t✐♦♥✿ ❞r❛✇ L s❛♠♣❧❡s ♦❢ s′

n|tn✿

Et|s[Es′|tf (s′, t)]] =

K

• k

ω(k)

N N

• n

✶ L

L

• ℓ

❢ (s(ℓ,k)

n

, t(k)

n )

✭■♥ ♣r❛❝t✐❝❡✱ ✇❡ s❡t ✶✵ ❛♥❞ ✶✮

❈♦♠♣✉t✐♥❣ t❤❡ ❣r❛❞✐❡♥t ❢r♦♠ ❙■❙

◮ ❖✉t❡r ❡①♣❡❝t❛t✐♦♥✿

Et|s[f (s, t)] =

K

• k

ω(k)

N N

• n

❢ (sn, t(k)

n )

◮ ■♥♥❡r ❡①♣❡❝t❛t✐♦♥✿ ❞r❛✇ L s❛♠♣❧❡s ♦❢ s′

n|tn✿

Et|s[Es′|tf (s′, t)]] =

K

• k

ω(k)

N N

• n

✶ L

L

• ℓ

❢ (s(ℓ,k)

n

, t(k)

n )

✭■♥ ♣r❛❝t✐❝❡✱ ✇❡ s❡t K = ✶✵ ❛♥❞ L = ✶✮

❊①❛♠♣❧❡

sn ya ur website ✳ ✳ ✳ ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ t(k)

n

②♦✉ ②♦✉r ✳ ✳ ✳ ②❡❛❤ ②♦✉✬r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω(k)

n

②❡❛❤ ②❡❛❤ ✶ ②♦✉r ②❡❛❤ ②♦✉r ②❡❛❤ ✷ ✇❡❜s✐t❡ ②♦✉r ✇❡❜s✐t❡ ②♦✉r

✳ ✳ ✳ s(ℓ)

n

✳ ✳ ✳

❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ t ✳ ❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ s ✳ ❯♣❞❛t❡✿ ❢ ②❡❛❤ ❢ ②❡❛❤ ❢ ②♦✉r ❢ ②♦✉r

❊①❛♠♣❧❡

sn ya ur website ✳ ✳ ✳ ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ t(k)

n

②♦✉ ②♦✉r ✳ ✳ ✳ ②❡❛❤ ②♦✉✬r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω(k)

n

②❡❛❤ ②❡❛❤ ✶ ②♦✉r ②❡❛❤ ②♦✉r ②❡❛❤ ✷ ✇❡❜s✐t❡ ②♦✉r ✇❡❜s✐t❡ ②♦✉r

✳ ✳ ✳ s(ℓ)

n

✳ ✳ ✳

❙❛♠♣❧❡ t(k)

n

❛❝❝♦r❞✐♥❣ t♦ Q(t|◦, ya)✳ ❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ s ✳ ❯♣❞❛t❡✿ ❢ ②❡❛❤ ❢ ②❡❛❤ ❢ ②♦✉r ❢ ②♦✉r

❊①❛♠♣❧❡

sn ya ur website ✳ ✳ ✳ ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ t(k)

n

②♦✉ ②♦✉r ✳ ✳ ✳ ②❡❛❤ ②♦✉✬r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω(k)

n

P(②❡❛❤,ya|◦) Q(②❡❛❤|◦,ya) ✶ ②♦✉r ②❡❛❤ ②♦✉r ②❡❛❤ ✷ ✇❡❜s✐t❡ ②♦✉r ✇❡❜s✐t❡ ②♦✉r

✳ ✳ ✳ s(ℓ)

n

✳ ✳ ✳

❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ t ✳ ❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ s ✳ ❯♣❞❛t❡✿ ❢ ②❡❛❤ ❢ ②❡❛❤ ❢ ②♦✉r ❢ ②♦✉r

❊①❛♠♣❧❡

sn ya ur website ✳ ✳ ✳ ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ t(k)

n

②♦✉ ②♦✉r ✳ ✳ ✳ ②❡❛❤ ②♦✉✬r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω(k)

n

P(②❡❛❤,ya|◦) Q(②❡❛❤|◦,ya) ✶ ②♦✉r ②❡❛❤ ②♦✉r ②❡❛❤ ✷ ✇❡❜s✐t❡ ②♦✉r ✇❡❜s✐t❡ ②♦✉r

✳ ✳ ✳ s(ℓ)

n

yea ✳ ✳ ✳

❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ t ✳ ❙❛♠♣❧❡ s(l)

n

❛❝❝♦r❞✐♥❣ t♦ P(s|yeah)✳ ❯♣❞❛t❡✿ ❢ ②❡❛❤ ❢ ②❡❛❤ ❢ ②♦✉r ❢ ②♦✉r

❊①❛♠♣❧❡

sn ya ur website ✳ ✳ ✳ ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ t(k)

n

②♦✉ ②♦✉r ✳ ✳ ✳ ②❡❛❤ ②♦✉✬r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω(k)

n

P(②❡❛❤,ya|◦) Q(②❡❛❤|◦,ya)

ω(k)

✶ P(②♦✉r,ur|②❡❛❤) Q(②♦✉r|②❡❛❤,ur) ✷ ✇❡❜s✐t❡ ②♦✉r ✇❡❜s✐t❡ ②♦✉r

✳ ✳ ✳ s(ℓ)

n

yea youu ✳ ✳ ✳

❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ t ✳ ❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ s ✳ ❯♣❞❛t❡✿ ❢ ②❡❛❤ ❢ ②❡❛❤ ❢ ②♦✉r ❢ ②♦✉r

❊①❛♠♣❧❡

sn ya ur website ✳ ✳ ✳ ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ t(k)

n

②♦✉ ②♦✉r ✳ ✳ ✳ ②❡❛❤ ②♦✉✬r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω(k)

n

P(②❡❛❤,ya|◦) Q(②❡❛❤|◦,ya)

ω(k)

✶ P(②♦✉r,ur|②❡❛❤) Q(②♦✉r|②❡❛❤,ur)

ω(k)

✷ Pt(✇❡❜s✐t❡,website|②♦✉r) Q(✇❡❜s✐t❡|②♦✉r,website)

✳ ✳ ✳ s(ℓ)

n

yea youu website ✳ ✳ ✳

❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ t ✳ ❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ s ✳ ❯♣❞❛t❡✿ ❢ ②❡❛❤ ❢ ②❡❛❤ ❢ ②♦✉r ❢ ②♦✉r

❊①❛♠♣❧❡

sn ya ur website ✳ ✳ ✳ ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ t(k)

n

②♦✉ ②♦✉r ✳ ✳ ✳ ②❡❛❤ ②♦✉✬r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω(k)

n

P(②❡❛❤,ya|◦) Q(②❡❛❤|◦,ya)

ω(k)

✶ P(②♦✉r,ur|②❡❛❤) Q(②♦✉r|②❡❛❤,ur)

ω(k)

✷ Pt(✇❡❜s✐t❡,website|②♦✉r) Q(✇❡❜s✐t❡|②♦✉r,website)

✳ ✳ ✳ s(ℓ)

n

yea youu website ✳ ✳ ✳

❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ t ✳ ❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ s ✳ ❯♣❞❛t❡✿ ω(k)

N (❢ (②❡❛❤, ya) − ❢ (②❡❛❤, yea)

+ ❢ (②♦✉r, ur) − ❢ (②♦✉r, youu) + . . .)

❊①❛♠♣❧❡

sn ya ur website ✳ ✳ ✳ ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ t(k)

n

②♦✉ ②♦✉r ✳ ✳ ✳ ②❡❛❤ ②♦✉✬r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω(k)

n

P(②❡❛❤,ya|◦) Q(②❡❛❤|◦,ya)

ω(k)

✶ P(②♦✉r,ur|②❡❛❤) Q(②♦✉r|②❡❛❤,ur)

ω(k)

✷ Pt(✇❡❜s✐t❡,website|②♦✉r) Q(✇❡❜s✐t❡|②♦✉r,website)

✳ ✳ ✳ s(ℓ)

n

yea youu website ✳ ✳ ✳

❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ t ✳ ❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ s ✳ ❯♣❞❛t❡✿ ω(k)

N (❢ (②❡❛❤, ya) − ❢ (②❡❛❤, yea)

+ ❢ (②♦✉r, ur) − ❢ (②♦✉r, youu) + . . .)

❊①❛♠♣❧❡

sn ya ur website ✳ ✳ ✳ ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ t(k)

n

②♦✉ ②♦✉r ✳ ✳ ✳ ②❡❛❤ ②♦✉✬r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω(k)

n

P(②❡❛❤,ya|◦) Q(②❡❛❤|◦,ya)

ω(k)

✶ P(②♦✉r,ur|②❡❛❤) Q(②♦✉r|②❡❛❤,ur)

ω(k)

✷ Pt(✇❡❜s✐t❡,website|②♦✉r) Q(✇❡❜s✐t❡|②♦✉r,website)

✳ ✳ ✳ s(ℓ)

n

yea youu website ✳ ✳ ✳

❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ t ✳ ❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ s ✳ ❯♣❞❛t❡✿ ω(k)

N (❢ (②❡❛❤, ya) − ❢ (②❡❛❤, yea)

+ ❢ (②♦✉r, ur) − ❢ (②♦✉r, youu) + . . .)

❊①❛♠♣❧❡

sn ya ur website ✳ ✳ ✳ ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ t(k)

n

②♦✉ ②♦✉r ✳ ✳ ✳ ②❡❛❤ ②♦✉✬r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω(k)

n

P(②❡❛❤,ya|◦) Q(②❡❛❤|◦,ya)

ω(k)

✶ P(②♦✉r,ur|②❡❛❤) Q(②♦✉r|②❡❛❤,ur)

ω(k)

✷ Pt(✇❡❜s✐t❡,website|②♦✉r) Q(✇❡❜s✐t❡|②♦✉r,website)

✳ ✳ ✳ s(ℓ)

n

yea youu website ✳ ✳ ✳

❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ t ✳ ❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ s ✳ ❯♣❞❛t❡✿ ω(k)

N (❢ (②❡❛❤, ya) − ❢ (②❡❛❤, yea)

+ ❢ (②♦✉r, ur) − ❢ (②♦✉r, youu) + . . .)

❊①❛♠♣❧❡

sn ya ur website ✳ ✳ ✳ ②❛♠ ✉r♥ ✇❡❜s✐t❡ ✳ ✳ ✳ t(k)

n

②♦✉ ②♦✉r ✳ ✳ ✳ ②❡❛❤ ②♦✉✬r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ω(k)

n

P(②❡❛❤,ya|◦) Q(②❡❛❤|◦,ya)

ω(k)

✶ P(②♦✉r,ur|②❡❛❤) Q(②♦✉r|②❡❛❤,ur)

ω(k)

✷ Pt(✇❡❜s✐t❡,website|②♦✉r) Q(✇❡❜s✐t❡|②♦✉r,website)

✳ ✳ ✳ s(ℓ)

n

yea youu website ✳ ✳ ✳

❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ t ✳ ❙❛♠♣❧❡ ❛❝❝♦r❞✐♥❣ t♦ s ✳ ❯♣❞❛t❡✿ ω(k)

N (❢ (②❡❛❤, ya) − ❢ (②❡❛❤, yea)

+ ❢ (②♦✉r, ur) − ❢ (②♦✉r, youu) + . . .)

❚❤❡ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥

❆ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ❝❛♥ ❣✉✐❞❡ s❛♠♣❧✐♥❣ ❢♦❝✉s❡s ♦♥ ❤✐❣❤✲♣r♦❜❛❜✐❧✐t② r❡❣✐♦♥✳

✶✳

✶ ✶ ✶

❆❝❝✉r❛t❡ ✭♦♣t✐♠❛❧✦✮ ◆♦t ❢❛st✿ ❝♦♠♣✉t✐♥❣ ❝♦sts ✷✳ ❖✉r ♣r♦♣♦s❛❧

✶ ✶

❡①♣ ❢

✶

❋❛st✿ t♦ ❝♦♠♣✉t❡ ❋❛✐r❧② ❛❝❝✉r❛t❡✿ ❜✐❛s❡❞ ❜② ❛ ❢❛❝t♦r ♦❢

❚❤❡ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥

❆ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ❝❛♥ ❣✉✐❞❡ s❛♠♣❧✐♥❣ ❢♦❝✉s❡s ♦♥ ❤✐❣❤✲♣r♦❜❛❜✐❧✐t② r❡❣✐♦♥✳

✶✳ Q(tn|sn, tn−✶) = P(tn|sn, tn−✶) ∝ P(sn|tn)P(tn|tn−✶)

◮ ❆❝❝✉r❛t❡ ✭♦♣t✐♠❛❧✦✮ ◮ ◆♦t ❢❛st✿ ❝♦♠♣✉t✐♥❣ P(sn|tn) ❝♦sts O(VTVS)

✷✳ ❖✉r ♣r♦♣♦s❛❧

✶ ✶

❡①♣ ❢

✶

❋❛st✿ t♦ ❝♦♠♣✉t❡ ❋❛✐r❧② ❛❝❝✉r❛t❡✿ ❜✐❛s❡❞ ❜② ❛ ❢❛❝t♦r ♦❢

❚❤❡ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥

❆ ♣r♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ❝❛♥ ❣✉✐❞❡ s❛♠♣❧✐♥❣ ❢♦❝✉s❡s ♦♥ ❤✐❣❤✲♣r♦❜❛❜✐❧✐t② r❡❣✐♦♥✳

✶✳ Q(tn|sn, tn−✶) = P(tn|sn, tn−✶) ∝ P(sn|tn)P(tn|tn−✶)

◮ ❆❝❝✉r❛t❡ ✭♦♣t✐♠❛❧✦✮ ◮ ◆♦t ❢❛st✿ ❝♦♠♣✉t✐♥❣ P(sn|tn) ❝♦sts O(VTVS)

✷✳ ❖✉r ♣r♦♣♦s❛❧ Q(tn|sn, tn−✶) ∝ P(sn|tn)Z(tn)P(tn|tn−✶) = ❡①♣ (θ′❢ (sn, tn)) P(tn|tn−✶)

◮ ❋❛st✿ O(VS + VT) t♦ ❝♦♠♣✉t❡ ω(k)

n

◮ ❋❛✐r❧② ❛❝❝✉r❛t❡✿ ❜✐❛s❡❞ ❜② ❛ ❢❛❝t♦r ♦❢ Z(tn)

■♠♣❧❡♠❡♥t❛t✐♦♥ ❞❡t❛✐❧s

✉♥▲❖▲✿ ✉♥s✉♣❡r✈✐s❡❞ ♥♦r♠❛❧✐③❛t✐♦♥ ✐♥ ❛ ▲❖❣✲▲✐♥❡❛r ♠♦❞❡❧

◮ ❉❡❝♦❞✐♥❣✿ ♣r♦♣♦s❡ K s❡q✉❡♥❝❡s t(k)✱ ♣❡r❢♦r♠

❱✐t❡r❜✐ ✇✐t❤✐♥ t❤✐s ❧✐♠✐t❡❞ s❡t✳

◮ ◆♦r♠❛❧✐③❛t✐♦♥ t❛r❣❡ts✿

◮ ❚r❛✐♥✐♥❣✿ ❛❧❧ ❛❧♣❤❛♥✉♠❡r✐❝ str✐♥❣s ♥♦t ✐♥ ❛s♣❡❧❧ ◮ ❚❡st✿ ♥♦r♠❛❧✐③❛t✐♦♥ t❛r❣❡ts ❛r❡ ❣✐✈❡♥

✭st❛♥❞❛r❞ ✐♥ t❤✐s t❛s❦✮

◮ ▲❛♥❣✉❛❣❡ ♠♦❞❡❧✿ ❑♥❡s❡r✲◆❡② s♠♦♦t❤❡❞

tr✐❣r❛♠ ♠♦❞❡❧✱ ❢r♦♠ t✇❡❡ts ✇✐t❤ ♥♦ ❖❖❱ ✇♦r❞s

◮ ❚r❛✐♥✐♥❣ ❞❛t❛✿ ❋♦r ❡❛❝❤ ❖❖❱ ✇♦r❞ ✐♥ t❤❡ t❡st

s❡t✱ ❞r❛✇ ✺✵ t✇❡❡ts ❢r♦♠ ❊❞✐♥❜✉r❣❤ ❝♦r♣✉s ❬✶✵❪ ❛♥❞ ❚✇✐tt❡r ❆P■✳

❊✈❛❧✉❛t✐♦♥

❉❛t❛s❡ts

◮ ▲❲❲▲✶✶ ❬✾❪✿ ✸✽✵✷ ✐s♦❧❛t❡❞ ✇♦r❞s ◮ ▲❡①◆♦r♠✶✳✶ ❬✺❪✿ ✺✹✾ t✇❡❡ts ✇✐t❤ ✶✶✽✹

♥♦♥st❛♥❞❛r❞ t♦❦❡♥s

◮ ▲❡①◆♦r♠✶✳✷✿ ✶✼✷ ♠❛♥✉❛❧ ❝♦rr❡❝t✐♦♥s t♦

▲❡①◆♦r♠✶✳✶ ✭✇✇✇✳❝❝✳❣❛t❡❝❤✳❡❞✉✴ ⑦❥❡✐s❡♥st✴❧❡①♥♦r♠✳✈✶✳✷✳t❣③✮

❘❡s✉❧ts

❆❞❞✐♥❣ ▲✶ ❘❡❣✉❧❛r✐③❛t✐♦♥

✼✾ ✽✵ ✽✶ ✽✷ ✽✸ ✵ ✶✵✵✵✵✵ ✷✵✵✵✵✵ ✸✵✵✵✵✵ ✹✵✵✵✵✵ ♥✉♠❜❡r ♦❢ ❢❡❛t✉r❡s ❋✲♠❡❛s✉r❡

• λ = ✶e − ✵✹

λ = ✺e − ✵✺ λ = ✶e − ✵✺ λ = ✺e − ✵✻ λ = ✶e − ✵✻ λ = ✵

❆♥❛❧②s✐s

❈❛♥ ♥♦r♠❛❧✐③❛t✐♦♥ ❤❡❧♣ ✐❞❡♥t✐❢② ♦rt❤♦❣r❛♣❤✐❝ st②❧❡s❄

◮ ❯s❡ ✉♥▲❖▲ t♦ ❛✉t♦♠❛t✐❝❛❧❧② ❧❛❜❡❧ ❧♦ts ♦❢ t✇❡❡ts ◮ ❯s❡ ▲❡✈❡♥s❤t❡✐♥ ❛❧✐❣♥♠❡♥t ❜❡t✇❡❡♥ ♦r✐❣✐♥❛❧ ❛♥❞

♥♦r♠❛❧✐③❡❞ t✇❡❡ts t♦ ✜♥❞ s✉❜st✐t✉t✐♦♥ ♣❛tt❡r♥s✱ ❡✳❣✳✱ ♥❣✴n_

◮ ❯s❡ ♠❛tr✐① ❢❛❝t♦r✐③❛t✐♦♥ t♦ ✜♥❞ s❡ts ♦❢ ♣❛tt❡r♥s

t❤❛t ❛r❡ ✉s❡❞ ❜② t❤❡ s❛♠❡ s❡t ♦❢ ❛✉t❤♦rs✳

tweets authors

tweets authors unLOL normalized tweets

tweets authors unLOL normalized tweets aligner character patterns e.g., t/_, ng/n_

tweets authors unLOL normalized tweets aligner author- pattern matrix character patterns

tweets authors unLOL normalized tweets aligner author- pattern matrix author loadings style dictionaries NMF character patterns

❖❜s❡r✈❛t✐♦♥s

❙♦♠❡ st②❧❡s ♠✐rr♦r ♣❤♦♥♦❧♦❣✐❝❛❧ ✈❛r✐❛❜❧❡s✿ ❣✲❞r♦♣♣✐♥❣✱ ✭❚❍✮✲st♦♣♣✐♥❣✱ t✲❞❡❧❡t✐♦♥✳✳✳ ❬✹❪

st②❧❡ r✉❧❡s ❡①❛♠♣❧❡s ❣✲❞r♦♣♣✐♥❣ ❣✯✴_* ♥❣✴n_ ❣✴_ goin✱ talkin✱ watchin✱ feelin✱ makin t✲❞❡❧❡t✐♦♥ t✯✴_* st✴s_ t✴_ jus✱ bc✱ shh✱ wha✱ gota✱ wea✱ mus✱ firts✱ jes✱ subsistutes t❤✲st♦♣♣✐♥❣ ❤✴_ ✯t✴*d t❤✴d_ t✴d dat✱ de✱ skool✱ fone✱ dese✱ dha✱ shid✱ dhat✱ dat’s

❖❜s❡r✈❛t✐♦♥s

❖t❤❡rs ❛r❡ ❦♥♦✇♥ ✏♥❡ts♣❡❛❦✑ ♣❤❡♥♦♠❡♥❛✱ ❧✐❦❡ ❡①♣r❡ss✐✈❡ ❧❡♥❣t❤❡♥✐♥❣ ❬✷❪

st②❧❡ r✉❧❡s ❡①❛♠♣❧❡s ✭❦❞✮✲❛❞❞✐♥❣ ✐❴✴id ❴✴k ❴✴d ❴✯✴k* idk✱ fuckk✱ okk✱ backk✱ workk✱ badd✱ andd✱ goodd✱ bedd✱ elidgible✱ pidgeon ♦✲❛❞❞✐♥❣ ♦❴✴oo ❴✯✴o* ❴✴o soo✱ noo✱ doo✱ oohh✱ loove✱ thoo✱ helloo ❡✲❛❞❞✐♥❣ ❴✴i ❡❴✴ee ❴✴e ❴✯✴e* mee✱ ive✱ retweet✱ bestie✱ lovee✱ nicee✱ heey✱ likee✱ iphone✱ homie✱ ii✱ damnit

❖❜s❡r✈❛t✐♦♥s

❲❡ ❣❡t ♠❡❛♥✐♥❣❢✉❧ st②❧❡s ❡✈❡♥ ❢r♦♠ ♠✐st❛❦❡♥ ♥♦r♠❛❧✐③❛t✐♦♥s✦ ✭❡✳❣✳✱ ✐✬♠✴ima✱ ♦✉t✴outta✮

st②❧❡ r✉❧❡s ❡①❛♠♣❧❡s ❛✲❛❞❞✐♥❣ ❴✴a ❴❴✴ma ❴✴m ❴✯✴a* ima✱ outta✱ needa✱ shoulda✱ woulda✱ mm✱ comming✱ tomm✱ boutt✱ ppreci- ate

❙✉♠♠❛r②

◮ ✉♥▲❖▲✿ ❥♦✐♥t✱ ✉♥s✉♣❡r✈✐s❡❞✱ ❛♥❞ ❧♦❣✲❧✐♥❡❛r

◮ ❋❡❛t✉r❡s ❜❛❧❛♥❝❡ ❜❡t✇❡❡♥ ❡❞✐t ❞✐st❛♥❝❡ ❛♥❞

❝♦♥✈❡♥t✐♦♥❛❧✐③❡❞ ✇♦r❞ ♣❛✐rs✳

◮ ▼❛✐♥ ❝❤❛❧❧❡♥❣❡✿ ♠❛ss✐✈❡ ❧❛❜❡❧ s♣❛❝❡✱ ✐♥ ✇❤✐❝❤

q✉❛❞r❛t✐❝ ❛❧❣♦r✐t❤♠s ❛r❡ ♥♦t ♣r❛❝t✐❝❛❧✳

◮ ❙♦❧✉t✐♦♥✿ ❛♣♣r♦①✐♠❛t❡ ❣r❛❞✐❡♥t ✇✐t❤ ❙▼❈ ◮ Pr♦♣♦s❛❧ ❞✐str✐❜✉t✐♦♥ ❢♦❝✉s❡s t❤❡ s❛♠♣❧❡s ♦♥

t❤❡ ❤✐❣❤✲❧✐❦❡❧✐❤♦♦❞ ♣❛rt ♦❢ t❤❡ s❡❛r❝❤ s♣❛❝❡✳

◮ ◆♦r♠❛❧✐③❛t✐♦♥ ❡♥❛❜❧❡s t❤❡ st✉❞② ♦❢ s②st❡♠❛t✐❝

♦rt❤♦❣r❛♣❤✐❝ ✈❛r✐❛t✐♦♥✳

❘❡❢❡r❡♥❝❡s ■

❚✳ ❇❡r❣✲❑✐r❦♣❛tr✐❝❦✱ ❆✳ ❇♦✉❝❤❛r❞✲❈ôté✱ ❏✳ ❉❡◆❡r♦✱ ❛♥❞ ❉✳ ❑❧❡✐♥✳ P❛✐♥❧❡ss ✉♥s✉♣❡r✈✐s❡❞ ❧❡❛r♥✐♥❣ ✇✐t❤ ❢❡❛t✉r❡s✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ ◆❆❆❈▲✱ ♣❛❣❡s ✺✽✷✕✺✾✵✱ ✷✵✶✵✳ ❙✳ ❇r♦❞② ❛♥❞ ◆✳ ❉✐❛❦♦♣♦✉❧♦s✳ ❈♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦❧❧❧❧❧❧❧❧❧❧❧❧❧❧✦✦✦✦✦✦✦✦✦✦✦✦✦✦✿ ✉s✐♥❣ ✇♦r❞ ❧❡♥❣t❤❡♥✐♥❣ t♦ ❞❡t❡❝t s❡♥t✐♠❡♥t ✐♥ ♠✐❝r♦❜❧♦❣s✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ ❊▼◆▲P✱ ✷✵✶✶✳ ❉✳ ❈♦♥tr❛❝t♦r✱ ❚✳ ❆✳ ❋❛r✉q✉✐❡✱ ❛♥❞ ▲✳ ❱✳ ❙✉❜r❛♠❛♥✐❛♠✳ ❯♥s✉♣❡r✈✐s❡❞ ❝❧❡❛♥s✐♥❣ ♦❢ ♥♦✐s② t❡①t✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ ❈❖▲■◆●✱ ♣❛❣❡s ✶✽✾✕✶✾✻✱ ✷✵✶✵✳ ❏✳ ❊✐s❡♥st❡✐♥✳ P❤♦♥♦❧♦❣✐❝❛❧ ❢❛❝t♦rs ✐♥ s♦❝✐❛❧ ♠❡❞✐❛ ✇r✐t✐♥❣✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ◆❆❆❈▲ ❲♦r❦s❤♦♣ ♦♥ ▲❛♥❣✉❛❣❡ ❆♥❛❧②s✐s ✐♥ ❙♦❝✐❛❧ ▼❡❞✐❛✱ ✷✵✶✸✳

❘❡❢❡r❡♥❝❡s ■■

❇✳ ❍❛♥ ❛♥❞ ❚✳ ❇❛❧❞✇✐♥✳ ▲❡①✐❝❛❧ ♥♦r♠❛❧✐s❛t✐♦♥ ♦❢ s❤♦rt t❡①t ♠❡ss❛❣❡s✿ ♠❛❦♥ s❡♥s ❛ ★t✇✐tt❡r✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ ❆❈▲✱ ♣❛❣❡s ✸✻✽✕✸✼✽✱ ✷✵✶✶✳ ❇✳ ❍❛♥✱ P✳ ❈♦♦❦✱ ❛♥❞ ❚✳ ❇❛❧❞✇✐♥✳ ▲❡①✐❝❛❧ ♥♦r♠❛❧✐③❛t✐♦♥ ❢♦r s♦❝✐❛❧ ♠❡❞✐❛ t❡①t✳ ❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ ■♥t❡❧❧✐❣❡♥t ❙②st❡♠s ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ✹✭✶✮✿✺✱ ✷✵✶✸✳ ❍✳ ❍❛ss❛♥ ❛♥❞ ❆✳ ▼❡♥❡③❡s✳ ❙♦❝✐❛❧ t❡①t ♥♦r♠❛❧✐③❛t✐♦♥ ✉s✐♥❣ ❝♦♥t❡①t✉❛❧ ❣r❛♣❤ r❛♥❞♦♠ ✇❛❧❦s✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ ❆❈▲✱ ✷✵✶✸✳

❘❡❢❡r❡♥❝❡s ■■■

❲✳ ▲✐♥❣✱ ❈✳ ❉②❡r✱ ■✳ ❚r❛♥❝♦s♦✱ ❛♥❞ ❆✳ ❇❧❛❝❦✳ P❛r❛♣❤r❛s✐♥❣ ✹ ♠✐❝r♦❜❧♦❣ ♥♦r♠❛❧✐③❛t✐♦♥✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✷✵✶✸ ❏♦✐♥t ❈♦♥❢❡r❡♥❝❡ ♦♥ ❊♠♣✐r✐❝❛❧ ▼❡t❤♦❞s ✐♥ ◆❛t✉r❛❧ ▲❛♥❣✉❛❣❡ Pr♦❝❡ss✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥❛❧ ◆❛t✉r❛❧ ▲❛♥❣✉❛❣❡ ▲❡❛r♥✐♥❣✱ ❙❡❛tt❧❡✱ ❯❙❆✱ ❖❝t♦❜❡r ✷✵✶✸✳ ❆ss♦❝✐❛t✐♦♥ ❢♦r ❈♦♠♣✉t❛t✐♦♥❛❧ ▲✐♥❣✉✐st✐❝s✳ ❋✳ ▲✐✉✱ ❋✳ ❲❡♥❣✱ ❇✳ ❲❛♥❣✱ ❛♥❞ ❨✳ ▲✐✉✳ ■♥s❡rt✐♦♥✱ ❞❡❧❡t✐♦♥✱ ♦r s✉❜st✐t✉t✐♦♥❄✿ ♥♦r♠❛❧✐③✐♥❣ t❡①t ♠❡ss❛❣❡s ✇✐t❤♦✉t ♣r❡✲❝❛t❡❣♦r✐③❛t✐♦♥ ♥♦r s✉♣❡r✈✐s✐♦♥✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ ❆❈▲✱ ♣❛❣❡s ✼✶✕✼✻✱ ✷✵✶✶✳ ❙✳ P❡tr♦✈✐➣✱ ▼✳ ❖s❜♦r♥❡✱ ❛♥❞ ❱✳ ▲❛✈r❡♥❦♦✳ ❚❤❡ ❡❞✐♥❜✉r❣❤ t✇✐tt❡r ❝♦r♣✉s✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ◆❆❆❈▲ ❍▲❚ ❲♦r❦s❤♦♣ ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ▲✐♥❣✉✐st✐❝s ✐♥ ❛ ❲♦r❧❞ ♦❢ ❙♦❝✐❛❧ ▼❡❞✐❛✱ ♣❛❣❡s ✷✺✕✷✻✱ ✷✵✶✵✳

❘❡❢❡r❡♥❝❡s ■❱

❈✳ ❩❤❛♥❣✱ ❚✳ ❇❛❧❞✇✐♥✱ ❍✳ ❍♦✱ ❇✳ ❑✐♠❡❧❢❡❧❞✱ ❛♥❞ ❨✳ ▲✐✳ ❆❞❛♣t✐✈❡ ♣❛rs❡r✲❝❡♥tr✐❝ t❡①t ♥♦r♠❛❧✐③❛t✐♦♥✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ ❆❈▲✱ ♣❛❣❡s ✶✶✺✾✕✶✶✻✽✱ ✷✵✶✸✳