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▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

▼✉❧t✐✲❛r♠❡❞ ❜❛♥❞✐ts ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ ❇❧❛❝❦❥❛❝❦

❙❡♠✐♥❛r ✐♥ ❙t❛t✐st✐❝s ❊❧❡♥❛ ❆s♦♥✐ ❛♥❞ ❱❛❧❡♥t✐♥ ❙t❛❧❞❡r

▼❛r❝❤ ✼✱ ✷✵✶✻

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❙✉♠♠❛r②

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

Pr♦❜❧❡♠

❨♦✉ ❛r❡ ❢❛❝❡❞ r❡♣❡❛t❡❞❧② ✇✐t❤ ❛ ❝❤♦✐❝❡ ❛♠♦♥❣ n ❞✐✛❡r❡♥t ♦♣t✐♦♥s✳ ❆❢t❡r ❡❛❝❤ ❝❤♦✐❝❡ ②♦✉ r❡❝❡✐✈❡ ❛ ♥✉♠❡r✐❝❛❧ r❡✇❛r❞✳ ❨♦✉r ♦❜❥❡❝t✐✈❡ ✐s t♦ ♠❛①✐♠✐③❡ t❤❡ ❡①♣❡❝t❡❞ t♦t❛❧ r❡✇❛r❞✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

• ❙❧♦t ♠❛❝❤✐♥❡ ❛r❡ ❦♥♦✇♥ ❛s

♦♥❡✲❛r♠❡❞ ❜❛♥❞✐ts✱ ❜❡❝❛✉s❡ t❤❡② ✇❡r❡ ♦r✐❣✐♥❛❧❧② ♦♣❡r❛t❡❞ ❜② ♦♥❡ ❧❡✈❡r ♦♥ t❤❡ s✐❞❡ ♦❢ t❤❡ ♠❛❝❤✐♥❡✳

• ❆ ❣❛♠❜❧❡r str❛t❡❣✐❝❛❧❧②

♦♣❡r❛t✐♥❣ ♠✉❧t✐♣❧❡ ♠❛❝❤✐♥❡s ✐♥ ♦r❞❡r t♦ ❞r❛✇ t❤❡ ❤✐❣❤❡st ♣♦ss✐❜❧❡ ♣r♦✜ts ✐s ❝❛❧❧❡❞ ❛ ♠✉❧t✐✲❛r♠❡❞ ❜❛♥❞✐t✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

n✲❛r♠❡❞ ❜❛♥❞✐t ♣r♦❜❧❡♠

• ❚❤❡ r❡✇❛r❞ ❢♦r ❡❛❝❤ ♦♥❡✲❛r♠❡❞ ❜❛♥❞✐t ❤❛s ❛ ❞✐✛❡r❡♥t

❞✐str✐❜✉t✐♦♥✱ t❤❡r❡❢♦r❡ ❛ ❞✐✛❡r❡♥t ❡①♣❡❝t❛t✐♦♥✳

• ■❢ t❤❡ ❡①♣❡❝t❛t✐♦♥s ✇❡r❡ ❦♥♦✇♥✱ ✐t ✇♦✉❧❞ ❜❡ tr✐✈✐❛❧ t♦ s♦❧✈❡ t❤❡

n✲❛r♠❡❞ ❜❛♥❞✐t ♣r♦❜❧❡♠✿ s❡❧❡❝t t❤❡ ❛❝t✐♦♥ ✇✐t❤ t❤❡ ❤✐❣❤❡st ♠❡❛♥ r❡✇❛r❞✳

• ❚❤✐s ✐s ✇❤② ✇❡ ❛ss✉♠❡ t❤❛t ✇❡ ❞♦♥✬t ❦♥♦✇ t❤❡ ❡①♣❡❝t❡❞

r❡✇❛r❞s✱ ❛❧t❤♦✉❣❤ ✇❡ ♠✐❣❤t ❤❛✈❡ ❡st✐♠❛t❡s✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❊①♣❧♦r✐♥❣ ❛♥❞ ❡①♣❧♦✐t✐♥❣ ♣r♦❜❧❡♠

• ❆t ❛♥② t✐♠❡ st❡♣ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ❛❝t✐♦♥ ✇❤♦s❡ ❡st✐♠❛t❡❞

❡①♣❡❝t❡❞ r❡✇❛r❞ ✐s ❣r❡❛t❡st✳ ❲❡ ❝❛❧❧ t❤✐s ❛ ❣r❡❡❞② ❛❝t✐♦♥✳

• ■❢ ②♦✉ s❡❧❡❝t ❛ ❣r❡❡❞② ❛❝t✐♦♥✱ ✇❡ s❛② t❤❛t ②♦✉ ❛r❡ ❡①♣❧♦✐t✐♥❣

②♦✉r ❝✉rr❡♥t ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❛❝t✐♦♥s✳

• ■❢ ✐♥st❡❛❞ ②♦✉ s❡❧❡❝t ♦♥❡ ♦❢ t❤❡ ♥♦♥✲❣r❡❡❞② ❛❝t✐♦♥s✱ t❤❡♥ ✇❡

s❛② ②♦✉ ❛r❡ ❡①♣❧♦r✐♥❣✱ ❜❡❝❛✉s❡ t❤✐s ❡♥❛❜❧❡s ②♦✉ t♦ ✐♠♣r♦✈❡ ②♦✉r ❡st✐♠❛t❡ ♦❢ t❤❡ ♥♦♥✲❣r❡❡❞② ❛❝t✐♦♥✬s ❡①♣❡❝t❡❞ r❡✇❛r❞✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❖♥❡✲❛r♠❡❞ ❜❛♥❞✐t

❲❡ ❞❡♥♦t❡ t❤❡ tr✉❡ ♠❡❛♥ r❡✇❛r❞ ♦❢ ❛♥ ❛❝t✐♦♥ a ❛s q(a)✱ ❛♥❞ t❤❡ ❡st✐♠❛t❡❞ ♠❡❛♥ r❡✇❛r❞ ♦♥ t❤❡ t✲t❤ t✐♠❡ st❡♣ ❛s Qt(a)✳

❊st✐♠❛t♦r ❢♦r t❤❡ ♠❡❛♥ r❡✇❛r❞ ✭s❛♠♣❧❡✲❛✈❡r❛❣❡✮

■❢ ❜② t❤❡ t✲t❤ t✐♠❡ st❡♣ ❛❝t✐♦♥ a ❤❛s ❜❡❡♥ ❝❤♦s❡♥ Nt(a) t✐♠❡s ♣r✐♦r t♦ t✱ ②✐❡❧❞✐♥❣ r❡✇❛r❞s R✶, R✷, . . . , RNt(a)✱ t❤❡♥ ✐ts ✈❛❧✉❡ ✐s ❡st✐♠❛t❡❞ t♦ ❜❡ Qt(a) = R✶ + R✷ + · · · + RNt(a) Nt(a) ✭✶✮

▲❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs

❆s Nt(a) → ∞✱ Qt(a) ❝♦♥✈❡r❣❡s t♦ q(a)✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❆❝t✐♦♥✲s❡❧❡❝t✐♦♥ r✉❧❡s

• ❚❤❡ ❣r❡❡❞② ❛❝t✐♦♥ s❡❧❡❝t✐♦♥ ♠❡t❤♦❞✿

At = ❛r❣♠❛①

a

Qt(a) ✭✷✮

• ❚❤❡ ǫ✲❣r❡❡❞② ♠❡t❤♦❞s✿ ❜❡❤❛✈❡ ❣r❡❡❞✐❧② ♠♦st ♦❢ t❤❡ t✐♠❡✱ ❜✉t

❡✈❡r② ♦♥❝❡ ✐♥ ❛ ✇❤✐❧❡✱ s❛② ✇✐t❤ s♠❛❧❧ ♣r♦❜❛❜✐❧✐t② ǫ✱ s❡❧❡❝t r❛♥❞♦♠❧② ❢r♦♠ ❛♠♦♥❣st ❛❧❧ t❤❡ ❛❝t✐♦♥s ✇✐t❤ ❡q✉❛❧ ♣r♦❜❛❜✐❧✐t② ✐♥❞❡♣❡♥❞❡♥t❧② ♦❢ t❤❡ ❛❝t✐♦♥ ♠❡❛♥ r❡✇❛r❞s ❡st✐♠❛t❡s ✭❛❞✈❛♥t❛❣❡✿ ❛s Nt(a) → ∞✱ ✇❡ ❡♥s✉r❡ t❤❛t Qt(a) ❝♦♥✈❡r❣❡ t♦ q(a)✮✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❋✐❣✉r❡✿ ❆✈❡r❛❣❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ǫ✲❣r❡❡❞② ❛❝t✐♦♥✲✈❛❧✉❡ ♠❡t❤♦❞s ♦♥ t❤❡ ✶✵✲❛r♠❡❞ t❡st❜❡❞✳ ❚❤❡s❡ ❞❛t❛ ❛r❡ ❛✈❡r❛❣❡s ♦✈❡r ✷✵✵✵ t❛s❦s✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

■♥❝r❡♠❡♥t❛❧ ■♠♣❧❡♠❡♥t❛t✐♦♥

• ❙♦ ❢❛r ✇❡ ❡st✐♠❛t❡❞ t❤❡ ❛❝t✐♦♥ ♠❡❛♥ r❡✇❛r❞s ❛s s❛♠♣❧❡

❛✈❡r❛❣❡s ♦❢ ♦❜s❡r✈❡❞ r❡✇❛r❞s✳

• Pr♦❜❧❡♠✿ t❤❡ ♠❡♠♦r② ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ r❡q✉✐r❡♠❡♥ts ❢♦r t❤❡

✐♠♣❧❡♠❡♥t❛t✐♦♥ ❣r♦✇ ♦✈❡r t✐♠❡ ✇✐t❤♦✉t ❜♦✉♥❞✳

• ❙♦❧✉t✐♦♥✿ ❧❡t Qk ❞❡♥♦t❡ t❤❡ ❡st✐♠❛t❡ ❢♦r t❤❡ k✲t❤ r❡✇❛r❞✱ t❤❛t

✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ ✐ts ✜rst k − ✶ r❡✇❛r❞s✱ ❛♥❞ ❛ k✲t❤ r❡✇❛r❞ ❢♦r t❤❡ ❛❝t✐♦♥✱ Rk✳ ❚❤❡♥✿ Qk+✶ = Qk + ✶ k [Rk − Qk] ✭✸✮

• ●❡♥❡r❛❧ ❢♦r♠✿

◆❡✇ ❊st✐♠❛t❡ ← ❖❧❞❊st✐♠❛t❡ + ❙t❡♣❙✐③❡[❚❛r❣❡t − ❖❧❞❊st✐♠❛t❡]

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

◆♦♥✲st❛t✐♦♥❛r② Pr♦❜❧❡♠

• ❙♦ ❢❛r✿ st❛t✐♦♥❛r② ❡♥✈✐r♦♥♠❡♥t❀ t❤❡ ❜❛♥❞✐t ✐s ♥♦t ❝❤❛♥❣✐♥❣

♦✈❡r t✐♠❡✳

• ■♥ ♣r❛❝t✐❝❡✱ t❤✐s ❤②♣♦t❤❡s✐s ✐s ✈❡r② ♦❢t❡♥ ✈✐♦❧❛t❡❞ ♦r ✐♠♣♦ss✐❜❧❡

t♦ ✈❡r✐❢②✱ ❛♥❞ ✇❡ ❡♥❝♦✉♥t❡r ♥♦♥✲st❛t✐♦♥❛r✐t②✳

• ■♥ s✉❝❤ ❝❛s❡s ✇❡✐❣❤t r❡❝❡♥t r❡✇❛r❞s ♠♦r❡ ❤❡❛✈✐❧② t❤❛♥

❧♦♥❣✲♣❛st ♦♥❡s✳

• Qk+✶ = Qk + α[Rk − Qk]

✭✹✮ ✇❤❡r❡ t❤❡ st❡♣✲s✐③❡ ♣❛r❛♠❡t❡r α ∈ (✵, ✶] ✐s ❝♦♥st❛♥t✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

◆♦♥✲st❛t✐♦♥❛r② Pr♦❜❧❡♠ ✲ ✷

• ❲❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ t❤❡ ♣❛st r❡✇❛r❞s ❛♥❞ ♦❢ t❤❡ ✐♥✐t✐❛❧

❡st✐♠❛t❡ Q✶✿ Qk+✶ = (✶ − α)kQ✶ +

k

• i=✶

α(✶ − α)k−iRi ✭✺✮

• ❚❤❡ q✉❛♥t✐t② ✶ − α ✐s ❧❡ss t❤❛♥ ✶✱ ❛♥❞ t❤✉s t❤❡ ✇❡✐❣❤t ❣✐✈❡♥ t♦

Ri ❞❡❝r❡❛s❡s ❛s t❤❡ ♥✉♠❜❡r ♦❢ ✐♥t❡r✈❡♥✐♥❣ r❡✇❛r❞s ✐♥❝r❡❛s❡s✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❯♣♣❡r✲❈♦♥✜❞❡♥❝❡✲❇♦✉♥❞ ❆❝t✐♦♥ ❙❡❧❡❝t✐♦♥

• ■♥ t❤❡ ǫ✲❣r❡❡❞② ♠❡t❤♦❞✱ ✇❡ ❝❤♦♦s❡ ❛♥♦t❤❡r ❛❝t✐♦♥ ✭t❤❛t ✐s ♥♦t

t❤❡ ❣r❡❡❞② ♦♥❡✮✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ǫ✳

• ❇❡tt❡r ✇♦✉❧❞ ❜❡ t♦ s❡❧❡❝t ❛♠♦♥❣ t❤❡ ♥♦♥✲❣r❡❡❞② ❛❝t✐♦♥s t❤❡

♦♥❡ ✇❤✐❝❤ ❤❛s t❤❡ ❤✐❣❤❡st ♣r♦❜❛❜✐❧✐t② t♦ ❜❡ t❤❡ ❣r❡❡❞② ❛❝t✐♦♥ ✭✇✐t❤ t❤❡ ❜✐❣❣❡st ❡①♣❡❝t❛t✐♦♥ ♦❢ r❡✇❛r❞✮✿ At = ❛r❣♠❛①

a

• Qt(a) + c
• ❧♥t

Nt(a)

• ✭✻✮

✇❤❡r❡ c > ✵ ❝♦♥tr♦❧s t❤❡ ❞❡❣r❡❡ ♦❢ ❡①♣❧♦r❛t✐♦♥✳ ■❢ Nt(a) = ✵✱ t❤❡♥ a ✐s ❝♦♥s✐❞❡r❡❞ t♦ ❜❡ ❛ ♠❛①✐♠✐s✐♥❣ ❛❝t✐♦♥✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

• ❙❡t ˆ

µ✶ = · · · = ˆ µk = ✵✱ n✶ = · · · = nk = ✵

• ❋♦r t = ✶ : T✿
• ❋♦r ❡❛❝❤ ❛r♠ i ❝❛❧❝✉❧❛t❡ UCB(i) = ˆ

µi + ✷

• ❧♥ t

ni

• P✐❝❦ ❛r♠ j = ❛r❣♠❛①

i

UCB(i) ❛♥❞ ♦❜s❡r✈❡ yt

• ❙❡t nj ← nj + ✶ ❛♥❞ ˆ

µj ← ˆ µj + ✶

nj (yt − ˆ

µj)✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❋✐❣✉r❡✿ ❆✈❡r❛❣❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❯❈❇ ❛❝t✐♦♥ s❡❧❡❝t✐♦♥ ♦♥ t❤❡ ✶✵✲❛r♠❡❞ t❡st❜❡❞✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

• r❛❞✐❡♥t ❇❛♥❞✐ts
• ❙♦ ❢❛r✱ ✇❡ ❝♦♥s✐❞❡r❡❞ ♠❡t❤♦❞s t❤❛t ❡st✐♠❛t❡ t❤❡ ♠❡❛♥ r❡✇❛r❞

♦❢ ❛♥ ❛❝t✐♦♥ ❛♥❞ ✉s❡ t❤♦s❡ ❡st✐♠❛t♦rs t♦ s❡❧❡❝t ❛♥ ❛❝t✐♦♥✳

• ❍❡r❡ ✇❡ ❝♦♥s✐❞❡r ❧❡❛r♥✐♥❣ ❛ ♥✉♠❡r✐❝❛❧ ♣r❡❢❡r❡♥❝❡ Ht(a) ❢♦r

❡❛❝❤ ❛❝t✐♦♥ a✳ ❚❤❡ ❧❛r❣❡r t❤❡ ♣r❡❢❡r❡♥❝❡✱ t❤❡ ♠♦r❡ ♦❢t❡♥ t❤❛t ❛❝t✐♦♥ ✐s t❛❦❡♥✱ ❜✉t t❤❡ ♣r❡❢❡r❡♥❝❡ ❤❛s ♥♦ ✐♥t❡r♣r❡t❛t✐♦♥ ✐♥ t❡r♠s ♦❢ r❡✇❛r❞✿ P[At = a] = eHt(a) n

b=✶ eHt(b) = πt(a).

✭✼✮

• ■♥✐t✐❛❧❧② ❛❧❧ ♣r❡❢❡r❡♥❝❡s ❛r❡ t❤❡ s❛♠❡ ✭❡✳❣✳ H✶(a) = ✵✱ ∀a✮✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

• r❛❞✐❡♥t ❇❛♥❞✐ts ✲ ✷
• ❖♥ ❡❛❝❤ st❡♣✱ ❛❢t❡r s❡❧❡❝t✐♥❣ t❤❡ ❛❝t✐♦♥ At ❛♥❞ r❡❝❡✐✈✐♥❣

r❡✇❛r❞ Rt✱ t❤❡ ♣r❡❢❡r❡♥❝❡s ❛r❡ ✉♣❞❛t❡❞ ❜②✿ Ht+✶(At) = Ht(At) + α(Rt − ¯ Rt)(✶ − πt(At)), ❛♥❞ Ht+✶(a) = Ht(a) − α(Rt − ¯ Rt)πt(a), ∀a = At ✭✽✮ ✇❤❡r❡ α > ✵ ✐s ❛ st❡♣✲s✐③❡ ♣❛r❛♠❡t❡r✱ ❛♥❞ ¯ Rt ∈ R ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ ❛❧❧ t❤❡ r❡✇❛r❞s ✉♣ t❤r♦✉❣❤ ❛♥❞ ✐♥❝❧✉❞✐♥❣ t✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❋✐❣✉r❡✿ ❆✈❡r❛❣❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❣r❛❞✐❡♥t✲❜❛♥❞✐t ❛❧❣♦r✐t❤♠ ♦♥ t❤❡ ✶✵✲❛r♠❡❞ t❡st❜❡❞✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

• ❚❤❡r❡ ❛r❡ t❤r❡❡ ❢✉♥❞❛♠❡♥t❛❧ ❢♦r♠❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ❜❛♥❞✐t

♣r♦❜❧❡♠ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❛ss✉♠❡❞ ♥❛t✉r❡ ♦❢ t❤❡ r❡✇❛r❞ ♣r♦❝❡ss✿ st♦❝❤❛st✐❝✱ ❛❞✈❡rs✐❛❧ ❛♥❞ ▼❛r❦♦✈✐❛♥✳

• ❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ t❤❡ ✐❞❡❛ ♦❢ t❤❡ st♦❝❤❛st✐❝ ❜❛♥❞✐t

♣r♦❜❧❡♠✳ ◆♦✇ ✇❡ ❞❡✜♥❡ ✐t ♠♦r❡ ❢♦r♠❛❧❧②✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❚❤❡ st♦❝❤❛st✐❝ ❜❛♥❞✐t ♣r♦❜❧❡♠

❙❡t✲✉♣

• ✐✈❡♥ K ≥ ✷ ❛r♠s ❛♥❞ s❡q✉❡♥❝❡s Xi,✶✱ Xi,✷✱ . . . ♦❢ ✉♥❦♥♦✇♥

r❡✇❛r❞s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡❛❝❤ ❛r♠ i = ✶, . . . , K✱ ✇❡ st✉❞② ❢♦r❡❝❛st❡rs t❤❛t ❛t ❡❛❝❤ t✐♠❡ st❡♣ t = ✶, ✷, . . . s❡❧❡❝t ❛♥ ❛r♠ It ❛♥❞ r❡❝❡✐✈❡ t❤❡ ❛ss♦❝✐❛t❡❞ r❡✇❛r❞ XIt,t✳

❚❤❡ st♦❝❤❛st✐❝ ❜❛♥❞✐t ♣r♦❜❧❡♠

❑♥♦✇♥ ♣❛r❛♠❡t❡rs✿ ♥✉♠❜❡r ♦❢ ❛r♠s K ❛♥❞ ♦❢ r♦✉♥❞s n ≥ K✳ ❯♥❦♥♦✇♥ ♣❛r❛♠❡t❡rs✿ K ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s ν✶, . . . , νK ♦♥ [✵, ✶]✳ ❋♦r ❡❛❝❤ r♦✉♥❞ t = ✶, ✷, . . .

• t❤❡ ❢♦r❡❝❛st❡r ❝❤♦♦s❡s It ∈ {✶, . . . , K}❀
• ❣✐✈❡♥ It✱ t❤❡ ❡♥✈✐r♦♥♠❡♥t ❞r❛✇s t❤❡ r❡✇❛r❞ XIt,t ∼ νIt

✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ t❤❡ ♣❛st ❛♥❞ r❡✈❡❛❧s ✐t t♦ t❤❡ ❢♦r❡❝❛st❡r✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

• ❚❤❡ ❣♦❛❧ ✐s t❤❡ s❛♠❡ t❤❛♥ ❜❡❢♦r❡✿ ♠❛①✐♠✐③❡ t❤❡ r❡✇❛r❞✳
• ❆♥❛❧♦❣♦✉s❧② t♦ t❤✐s ✐s t♦ ♠✐♥✐♠✐③❡ t❤❡ r❡❣r❡t✳
• ❚❤❡ r❡❣r❡t ✐s ✇❤❛t ✇❡ ❧♦s❡ ❜② ♥♦t ♣❧❛②✐♥❣ t❤❡ ♦♣t✐♠❛❧ str❛t❡❣②✿

Rn = ♠❛①

i=✶,...,K n

• t=✶

Xi,t −

n

• t=✶

XIt,t. ✭✾✮

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

• ❊①♣❡❝t❡❞ r❡❣r❡t✿

E[Rn] = E

• ♠❛①

i=✶,...,K n

• t=✶

Xi,t −

n

• t=✶

XIt,t

• ✭✶✵✮
• Ps❡✉❞♦✲r❡❣r❡t✿

¯ Rn = ♠❛①

i=✶,...,K E[ n

• t=✶

Xi,t −

n

• t=✶

XIt,t] ✭✶✶✮

• ◆♦t❡✿ ❚❤❡ ♣s❡✉❞♦✲r❡❣r❡t ✐s ❛ ✇❡❛❦❡r ♥♦t✐♦♥ ♦❢ r❡❣r❡t✱ s✐♥❝❡

♦♥❡ ❝♦♠♣❡t❡s ❛❣❛✐♥st t❤❡ ❛❝t✐♦♥ ✇❤✐❝❤ ✐s ♦♣t✐♠❛❧ ♦♥❧② ✐♥ ❡①♣❡❝t❛t✐♦♥✳ ▼♦r❡ ❢♦r♠❛❧❧②✿ ¯ Rn ≤ E[Rn] ✭✶✷✮

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

• ■♥ t❤❡ st♦❝❤❛st✐❝ s❡tt✐♥❣✱ t❤❡ ♣s❡✉❞♦✲r❡❣r❡t ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s

¯ Rn = n ♠❛①

i=✶,...,k µi − n

• t=✶

E[µIt] ✭✶✸✮

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❚❤♦♠♣s♦♥ ❙❛♠♣❧✐♥❣

• ■♥ ♦♥❡ ♦❢ t❤❡ ❡❛r❧✐❡st ✇♦r❦s ♦♥ st♦❝❤❛st✐❝ ❜❛♥❞✐t ♣r♦❜❧❡♠s✱

❚❤♦♠♣s♦♥ ♣r♦♣♦s❡❞ ❛ r❛♥❞♦♠✐③❡❞ ❇❛②❡s✐❛♥ ❛❧❣♦r✐t❤♠ t♦ ♠✐♥✐♠✐③❡ r❡❣r❡t✳

• ❇❛s✐❝ ✐❞❡❛✿ ❛ss✉♠❡ ❛ s✐♠♣❧❡ ♣r✐♦r ❞✐str✐❜✉t✐♦♥ ♦♥ t❤❡

♣❛r❛♠❡t❡rs ♦❢ t❤❡ r❡✇❛r❞ ❞✐str✐❜✉t✐♦♥ ♦❢ ❡✈❡r② ❛r♠✱ ❛♥❞ ❛t ❛♥② t✐♠❡ st❡♣✱ ♣❧❛② ❛♥ ❛r♠ ❛❝❝♦r❞✐♥❣ t♦ ✐ts ♣♦st❡r✐♦r ♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡✐♥❣ t❤❡ ❜❡st ❛r♠✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❚❤♦♠♣s♦♥ ❙❛♠♣❧✐♥❣ ✲ ✷

• ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ r❡✇❛r❞ µi ∈ [✵, ✶] ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧

❞✐str✐❜✉t✐♦♥✱ ❛♥❞ πi,t ❜❡ t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ❢♦r µi ❛t t✐♠❡ t✳ ❚❤❡ r❡✇❛r❞ ❛t t✐♠❡ t ♦❢ ❛r♠ i ✐s θi,t ∼ πi,t✱ ✇❤❡r❡ θi,t ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡ str❛t❡❣② ✐s ✐♠♣❧❡♠❡♥t❡❞ ❜② s❛♠♣❧✐♥❣ ❢r♦♠ t❤❡ ♣♦st❡r✐♦r✳

• ❚❤❡ str❛t❡❣② ✐s t❤❡♥ ❣✐✈❡♥ ❜② ❛r❣♠❛①

i=✶,...,K

θi,t✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❚❤♦♠♣s♦♥ ❙❛♠♣❧✐♥❣ ❢♦r t❤❡ ❇❡r♥♦✉❧❧✐ ❜❛♥❞✐t ♣r♦❜❧❡♠

• ■♥ t❤✐s ❝❛s❡✱ t❤❡ r❡✇❛r❞s ❛r❡ ❡✐t❤❡r ✵ ♦r ✶✱ ❛♥❞ ❢♦r ❛r♠ i✱ t❤❡

♣r♦❜❛❜✐❧✐t② ♦❢ s✉❝❝❡ss ✐s µi✳

• ❇❡t❛ ❞✐str✐❜✉t✐♦♥ t✉r♥s ♦✉t t♦ ❜❡ ❛ ✈❡r② ❝♦♥✈❡♥✐❡♥t ❝❤♦✐❝❡ ♦❢

♣r✐♦rs ❢♦r ❇❡r♥♦✉❧❧✐ r❡✇❛r❞s✿

• f (x; α, β) =

Γ(α+β) Γ(α)Γ(β)xα−✶(✶ − x)β−✶

✳

• ❲❤②❄ ❇❡❝❛✉s❡ ✐❢ t❤❡ ♣r✐♦r ✐s ❛ ❇❡t❛(α, β) ❞✐str✐❜✉t✐♦♥✱ t❤❡♥

❛❢t❡r ♦❜s❡r✈✐♥❣ ❛ ❇❡r♥♦✉❧❧✐ tr✐❛❧✱t❤❡ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ ✐s s✐♠♣❧② ❇❡t❛(α + ✶, β) ♦r ❇❡t❛(α, β + ✶) ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r t❤❡ tr✐❛❧ r❡s✉❧t❡❞ ✐♥ ❛ s✉❝❝❡ss ♦r ❢❛✐❧✉r❡✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❚❤♦♠♣s♦♥ ❙❛♠♣❧✐♥❣ ❢♦r ❇❡r♥♦✉❧❧✐ ❜❛♥❞✐t ✲ ❆❧❣♦r✐t❤♠

❚❤❡ ❚❤♦♠♣s♦♥ ❙❛♠♣❧✐♥❣ ❛❧❣♦r✐t❤♠ ✐♥✐t✐❛❧❧② ❛ss✉♠❡s ❛r♠ i t♦ ❤❛✈❡ ♣r✐♦r ❇❡t❛(✶, ✶) ♦♥ µi✱ ✇❤✐❝❤ ✐s ♥❛t✉r❛❧ ❜❡❝❛✉s❡ ❇❡t❛(✶, ✶) ✐s t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦♥ (✵, ✶)✳ ❋♦r ❡❛❝❤ ❛r♠ i = ✶, . . . , N s❡t Si = ✵✱ Fi = ✵✳ ❋♦r ❡❛❝❤ t = ✶, ✷, . . . ❞♦✿

• ❋♦r ❡❛❝❤ ❛r♠ i = ✶, . . . , N✱ s❛♠♣❧❡ µi(t) ❢r♦♠ t❤❡ ❇❡t❛✭Si + ✶✱

Fi + ✶✮ ❞✐str✐❜✉t✐♦♥✳

• P❧❛② ❛r♠ i(t) = ❛r♠❛①µi(t)✳
• ■❢ r = ✶✱ Si(t) = Si(t) + ✶ ❡❧s❡ Fi(t) = Fi(t) + ✶

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❚❤❡ ❛❞✈❡rs✐❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠

• ❙✉♣♣♦s❡ ✇❡ ❛r❡ ✐♥ ❛ r✐❣❣❡❞ ❝❛s✐♥♦✳
• ❚❤❡ ♦✇♥❡r✴❛❞✈❡rs❛r② s❡ts t❤❡ ❣❛✐♥ Xi,t t♦ s♦♠❡ ❛r❜✐tr❛r② ✈❛❧✉❡

gi,t ∈ [✵, ✶]✳

• ❚❤❡ ♦✇♥❡r ✐s ❝❛❧❧❡❞ ♦❜❧✐✈✐♦✉s ✐❢ ❤✐s ❝❤♦✐❝❡ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥

t❤❡ str❛t❡❣② ♦❢ t❤❡ ♣❧❛②❡r✳

• ■t✬s ❝❛❧❧❡❞ ♥♦♥✲♦❜❧✐✈✐♦✉s ✐❢ ❤✐s ❝❤♦✐❝❡ ❞❡♣❡♥❞s ♦♥ t❤❡ str❛t❡❣②

♦❢ t❤❡ ♣❧❛②❡r✱ ✇❤✐❝❤ ♠❡❛♥s✿ gi,t = gi,t(I✶, . . . , It−✶) ✭✶✹✮

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❚❤❡ ❛❞✈❡rs✐❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠ ✲ ✷

❑♥♦✇♥ ♣❛r❛♠❡t❡rs✿ ♥✉♠❜❡r ♦❢ ❛r♠s K ≥ ✷ ❛♥❞ ✭♣♦ss✐❜❧②✮ ♥✉♠❜❡r ♦❢ r♦✉♥❞s n ≥ K✳ ❋♦r ❡❛❝❤ r♦✉♥❞ t = ✶, ✷, . . . ✭✶✮ t❤❡ ❢♦r❡❝❛st❡r ❝❤♦♦s❡s It ∈ {✶, . . . , K}✱ ♣♦ss✐❜❧② ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❡①t❡r♥❛❧ r❛♥❞♦♠✐③❛t✐♦♥✱ ✭✷✮ s✐♠✉❧t❛♥❡♦✉s❧②✱ t❤❡ ❛❞✈❡rs❛r② s❡❧❡❝ts ❛ ❣❛✐♥ ✈❡❝t♦r gt = (g✶,t, . . . , gK,t) ∈ [✵, ✶]K✱ ♣♦ss✐❜❧② ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❡①t❡r♥❛❧ r❛♥❞♦♠✐③❛t✐♦♥✱❛♥❞ ✭✸✮ t❤❡ ❢♦r❡❝❛st❡r r❡❝❡✐✈❡s ✭❛♥❞ ♦❜s❡r✈❡s✮ t❤❡ r❡✇❛r❞ gIt,t ✇❤✐❧❡ t❤❡ ❣❛✐♥s ♦❢ t❤❡ ♦t❤❡r ❛r♠s ❛r❡ ♥♦t ♦❜s❡r✈❡❞✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❚❤❡ ❛❞✈❡rs✐❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠ ✲ ✸

• ❋♦r t❤❡ ❛❞✈❡rs✐❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠✱ t❤❡ ♣s❡✉❞♦✲r❡❣r❡t ❝❛♥ ❜❡

❞❡✜♥❡❞ ❛s ¯ Rn = ♠❛①

i=✶,...,K E

n

• t=✶

gi,t −

n

• t=✶

gIt,t

• ✭✶✺✮
• ◆♦t❡✿ ❚❤❡ ❛❞✈❡rs✐❛❧ ❣❛✐♥s gi,t(I✶, . . . , It−✶) ❝♦✉❧❞ ❜❡ ❞✐✛❡r❡♥t

t❤❛♥ t❤♦s❡ ❝❤♦s❡♥ ❜② t❤❡ ♣❧❛②❡r ✇✐t❤ t❤❡ ♦♣t✐♠❛❧ t❛❝t✐❝✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

▼❛r❦♦✈✐❛♥ ❜❛♥❞✐t ♣r♦❜❧❡♠

• ■♥ t❤❡ ▼❛r❦♦✈✐❛♥ ❜❛♥❞✐t ♣r♦❜❧❡♠✱ t❤❡ r❡✇❛r❞ ♣r♦❝❡ss❡s ❛r❡

♥❡✐t❤❡r ✐✳✐✳❞✳ ♥♦r ❛❞✈❡rs✐❛❧✳

• ❆r♠s ❛r❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ K ▼❛r❦♦✈ ♣r♦❝❡ss❡s✱ ❡❛❝❤ ✇✐t❤ ✐ts

♦✇♥ st❛t❡ s♣❛❝❡✳

• ❊❛❝❤ t✐♠❡ ❛♥ ❛r♠ i ✐s ❝❤♦s❡♥ ✐♥ st❛t❡ s✱ ❛ st♦❝❤❛st✐❝ r❡✇❛r❞ ✐s

❞r❛✇♥ ❢r♦♠ ❛ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ νi,s✱ ❛♥❞ t❤❡ st❛t❡ ♦❢ t❤❡ r❡✇❛r❞ ♣r♦❝❡ss ❢♦r ❛r♠ i ❝❤❛♥❣❡s ✐♥ ❛ ▼❛r❦♦✈✐❛♥ ❢❛s❤✐♦♥✱ ❜❛s❡❞ ♦♥ ❛♥ ✉♥❞❡r❧②✐♥❣ st♦❝❤❛st✐❝ tr❛♥s✐t✐♦♥ ♠❛tr✐① Mi✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❈♦♥t❡①t✉❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠

• ❙♦ ❢❛r✿ ♦♥❧② ♥♦♥ ❛ss♦❝✐❛t✐✈❡ t❛s❦s✱ ✐♥ ✇❤✐❝❤ t❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦

❛ss♦❝✐❛t❡ ❞✐✛❡r❡♥t ❛❝t✐♦♥s ✇✐t❤ ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥s✳ ■♥ t❤❡s❡ t❛s❦s t❤❡ ❧❡❛r♥❡r ❡✐t❤❡r tr✐❡s t♦ ✜♥❞ ❛ s✐♥❣❧❡ ❜❡st ❛❝t✐♦♥ ✇❤❡♥ t❤❡ t❛s❦ ✐s st❛t✐♦♥❛r②✱ ♦r tr✐❡s t♦ tr❛❝❦ t❤❡ ❜❡st ❛❝t✐♦♥ ❛s ✐t ❝❤❛♥❣❡s ♦✈❡r t✐♠❡ ✇❤❡♥ t❤❡ t❛s❦ ✐s ♥♦♥ st❛t✐♦♥❛r②✳

• ◆♦✇✿ ✐♥ t❤❡ ❛ss♦❝✐❛t✐✈❡ s❡❛r❝❤✱ ✇❡ ❛❝t ❞✐✛❡r❡♥t❧② ❛❝❝♦r❞✐♥❣ t♦

t❤❡ s✐t✉❛t✐♦♥ ✇❡ ❛r❡ ✐♥✱ ❜❡❢♦r❡ t❛❦✐♥❣ ❛♥ ❛❝t✐♦♥✳

• ❊①❛♠♣❧❡✿ ②♦✉ ❛r❡ ❝♦♥❢r♦♥t❡❞ ✇✐t❤ s❡✈❡r❛❧ ♥✲❛r♠❡❞ ❜❛♥❞✐ts

❝❤♦s❡♥ ❛t r❛♥❞♦♠ ❛♥❞ ②♦✉r ♦♥❧② ❝❧✉❡ ❛❜♦✉t t❤❡♠ ✐s✱ ❢♦r ❡①❛♠♣❧❡✱ t❤❡✐r ❝♦❧♦✉r✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

◆❡✇s r❡❝♦♠♠❡♥❞❛t✐♦♥ ❡①❛♠♣❧❡

• ▲❡ts s❡❡ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❝♦♥t❡①t✉❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠
• ❲❡❜ s❡r✈✐❝❡s tr② t♦ ❛❞❛♣t t❤❡✐r s❡r✈✐❝❡s ✭❛❞✈❡rt✐s❡♠❡♥t✱ ♥❡✇s✱

❛rt✐❝❧❡s✱ ✳✳✳✮ t♦ ✐♥❞✐✈✐❞✉❛❧ ✉s❡rs ❜② ♠❛❦✐♥❣ ✉s❡ ♦❢ ✉s❡r ✐♥❢♦r♠❛t✐♦♥✳

• ❚❤✐s ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ♠✉❧t✐✲❛r♠❡❞ ❜❛♥❞✐t ♣r♦❜❧❡♠ ✇❤❡r❡ ✇❡

❤❛✈❡ ❛❞❞✐t✐♦♥❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ s❧♦t ♠❛❝❤✐♥❡✳

• ❖♥❡ ❝❧✐❝❦ ♦❢ t❤❡ ✉s❡r ♦♥ t❤❡ ❛❞✈❡rt✐s❡♠❡♥t ♦r ♥❡✇ ❝♦rr❡s♣♦♥❞

t♦ t❤❡ r❡✇❛r❞✳

• ❚❤❡ ❛❝t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ❛❞✈❡rt✐s❡♠❡♥t ♦r

♥❡✇ ✇❤✐❝❤ ✇❡ t❤✐♥❦ t❤❡ ✉s❡r ✇✐❧❧ ♠♦st ❧✐❦❡❧② ❝❧✐❝❦ ✭s♦ ❢♦r ✇❤✐❝❤ ✇❡ ✇♦✉❧❞ ♠♦st ❧✐❦❡❧② ❣♦t t❤❡ r❡✇❛r❞✮✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❊①❛♠♣❧❡ ♦❢ ❛ ❝♦♥t❡①t✉❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠

• ❲❡ ❤❛✈❡ t❤❡ ❝❤♦✐❝❡ ❜❡t✇❡❡♥ ❢♦✉r ❧✐♥❦s ❛♥❞ ✇❡ ❤❛✈❡ t♦ ❞❡❝✐❞❡

✇❤✐❝❤ ♦♥❡ ✇❡ ❤✐❣❤❧✐❣❤t ♦♥ t❤❡ ②❛❤♦♦ ❢r♦♥t ♣❛❣❡✱ ✇✐t❤ t❤❡ ❛✐♠ t❤❛t t❤❡ ✉s❡r ❝❧✐❝❦s ♦♥ ✐t✳ ❲❡ ❤❛✈❡ ❛❧s♦ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ✉s❡r✱ ❛♥❞ t❤❡r❡❢♦r❡ ❤✐❣❤❧✐❣❤t ❛♥ ❛rt✐❝❧❡ ❛❝❝♦r❞✐♥❣ t♦ ❤✐s ✐♥t❡r❡sts✳

• ❚❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❝♦♥t❡①t✉❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠✱

✇❤❡r❡ t❤❡ r❡✇❛r❞ ✐s ❛ ❝❧✐❝❦ ♦♥ t❤❡ ❧✐♥❦ ✇❡ ❤❛✈❡ ❤✐❣❤❧✐❣❤t❡❞✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❈♦♥t❡①t✉❛❧ ❜❛♥❞✐t ✲ ❛❧❣♦r✐t❤♠

• ❆ ❝♦♥t❡①t✉❛❧ ❜❛♥❞✐t ❛❧❣♦r✐t❤♠ A ♣r♦❝❡❡❞s ✐♥ ❞✐s❝r❡t❡ tr✐❛❧s

t = ✶, ✷, ✸, . . . ✳ ■♥ tr✐❛❧ t✿ ✶✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ♦❜s❡r✈❡s t❤❡ ❝✉rr❡♥t ✉s❡r ut ❛♥❞ ❛ s❡t At ♦❢ ❛r♠s ♦r ❛❝t✐♦♥s t♦❣❡t❤❡r ✇✐t❤ t❤❡✐r ❢❡❛t✉r❡ ✈❡❝t♦rs xt,a ❢♦r a ∈ A✳ ❚❤❡ ✈❡❝t♦r xt,a s✉♠♠❛r✐③❡s ✐♥❢♦r♠❛t✐♦♥ ♦❢ ❜♦t❤ t❤❡ ✉s❡r ut ❛♥❞ ❛r♠ a✱ ❛♥❞ ✇✐❧❧ ❜❡ r❡❢❡rr❡❞ ❛s t❤❡ ❝♦♥t❡①t✳ ✷✳ ❇❛s❡❞ ♦♥ ♦❜s❡r✈❡❞ ♣❛②♦✛s ✐♥ ♣r❡✈✐♦✉s tr✐❛❧s✱ A ❝❤♦♦s❡s ❛♥ ❛r♠ at ∈ At✱ ❛♥❞ r❡❝❡✐✈❡s ♣❛②♦✛ rt,at ✇❤♦s❡ ❡①♣❡❝t❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ ❜♦t❤ t❤❡ ✉s❡r ut ❛♥❞ t❤❡ ❛r♠ at✳ ✸✳ ❚❤❡ ❛❧❣♦r✐t❤♠ t❤❡♥ ✐♠♣r♦✈❡s ✐ts ❛r♠✲s❡❧❡❝t✐♦♥ str❛t❡❣② ✇✐t❤ t❤❡ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ (xt,a, at, rt,at)✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❚✲tr✐❛❧ r❡❣r❡t RA

• ❚❤❡ t♦t❛❧ T✲tr✐❛❧ ♣❛②♦✛ ♦❢ A ✐s ❞❡✜♥❡❞ ❛s T

t=✶ rt,at✳

• ❉❡✜♥❡ t❤❡ ♦♣t✐♠❛❧ ❡①♣❡❝t❡❞ T✲tr✐❛❧ ♣❛②♦✛ ❛s E[T

t=✶ rt,a∗

t ]✱

✇❤❡r❡ a∗

t ✐s t❤❡ ❛r♠ ✇✐t❤ ♠❛①✐♠✉♠ ❡①♣❡❝t❡❞ ♣❛②♦✛ ❛t tr✐❛❧ t✳

• ●♦❛❧✿ ✇r✐t❡ ❛♥ ❛❧❣♦r✐t❤♠ A s♦ t❤❛t t❤❡ ❡①♣❡❝t❡❞ ♣❛②♦✛ ✐s

♠❛①✐♠✐③❡❞✳

• ❆s ✇❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ t❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ✜♥❞ ❛♥ ❛❧❣♦r✐t❤♠

✇❤✐❝❤ ♠✐♥✐♠✐③❡s t❤❡ r❡❣r❡t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♦♣t✐♠❛❧ ❛r♠✲s❡❧❡❝t✐♦♥ str❛t❡❣②✿ RA(T) = E[

T

• t=✶

rt,a∗

t ] − E[

T

• t=✶

rt,at]. ✭✶✻✮

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❆ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❝♦♥t❡①t✉❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠

❆ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❝♦♥t❡①t✉❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠ ✐s t❤❡ n✲❛r♠❡❞ ❜❛♥❞✐t ♣r♦❜❧❡♠✱ ✐♥ ✇❤✐❝❤✿

• ❚❤❡ ❛r♠ s❡t A r❡♠❛✐♥s ✉♥❝❤❛♥❣❡❞ ❛♥❞ ❝♦♥t❛✐♥s n ❛r♠s ∀t✳
• ❚❤❡ s✐t✉❛t✐♦♥ st ✐s t❤❡ s❛♠❡ ∀t✳

❚❤❡ n✲❛r♠❡❞ ❜❛♥❞✐t ♣r♦❜❧❡♠ ✐s ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ❝♦♥t❡①t ❢r❡❡ ❜❛♥❞✐t ♣r♦❜❧❡♠✳ ❲❤✐❧❡ t❤❡ ❝♦♥t❡①t ❢r❡❡ ❜❛♥❞✐t ♣r♦❜❧❡♠s ❛r❡ ❡①t❡♥s✐✈❡❧② st✉❞✐❡❞ ❛♥❞ ✇❡❧❧ ✉♥❞❡rst♦♦❞✱ t❤❡ ❝♦♥t❡①t✉❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠ ❤❛s r❡♠❛✐♥❡❞ ❝❤❛❧❧❡♥❣✐♥❣✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

• ❲❡ ✇❛♥t t♦ ✜♥❞ ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t s♦❧✈❡s t❤❡ ❝♦♥t❡①t✉❛❧ ❜❛♥❞✐t

♣r♦❜❧❡♠✳

• ■❞❡❛✿ ✉s❡ t❤❡ ❯❈❇ ♠❡t❤♦❞

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

▲✐♥❡❛r ❯♣♣❡r ❇♦✉♥❞ ❈♦♥✜❞❡♥❝❡ ✲ ✶

• ❲❡ ❛ss✉♠❡ t❤❡ ❡①♣❡❝t❡❞ ♣❛②♦✛ ♦❢ ❛♥ ❛r♠ a ✐s ❧✐♥❡❛r ✐♥ ✐ts

d✲❞✐♠❡♥s✐♦♥❛❧ ❢❡❛t✉r❡ xt,a ✇✐t❤ s♦♠❡ ✉♥❦♥♦✇♥ ❝♦❡✣❝✐❡♥t ✈❡❝t♦r θ∗

a✱ ♥❛♠❡❧②✱ ∀t✱

rt,a = xT

t,aθ∗ a + ǫt

✭✶✼✮ ✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦ E[rt,a | xt,a] = xT

t,aθ∗ a✳ ❚❤❡ ♣❛r❛♠❡t❡rs

❛r❡ ♥♦t s❤❛r❡❞ ❛♠♦♥❣ ❞✐✛❡r❡♥t ❛r♠s✳

• ❲❡ ✇✐s❤ t♦ ♠✐♥✐♠✐③❡ t❤❡ sq✉❛r❡ ❧♦ss

ˆ θa = ❛r❣♠✐♥

θ∗

a

m

• t=✶

(rt,a − xT

t,aθ∗ a)✷

✭✶✽✮

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

▲✐♥❡❛r ❯♣♣❡r ❇♦✉♥❞ ❈♦♥✜❞❡♥❝❡ ✲ ✷

• Da ❞❡s✐❣♥ ♠❛tr✐① ♦❢ ❞✐♠❡♥s✐♦♥ m × d ❛t tr✐❛❧ t✱ ba ∈ Rm

r❡s♣♦♥s❡ ✈❡❝t♦r✳

• ❆♣♣❧②✐♥❣ r✐❞❣❡ r❡❣r❡ss✐♦♥ t♦ t❤❡ tr❛✐♥✐♥❣ ❞❛t❛ (Da, ba) ❣✐✈❡s

ˆ θa = (DT

a Da + Ia)−✶DT a ba✱ ✇❤❡r❡ Id ✐s t❤❡ d × d ✐❞❡♥t✐t②

♠❛tr✐①✳

• ❚❤❡♦r❡♠✿ ❢♦r t❤❡ ❡st✐♠❛t❡❞ ❝♦❡✣❝✐❡♥ts ✐t ❤♦❧❞s t❤❛t

| xT

t,aˆ

θa − E[rt,a | xt,a] |≤ α

• xT

t,a(DT a Da + Id)−✶xt,a

✭✶✾✮ ❢♦r ❛♥② δ > ✵ ❛♥❞ xt,a ∈ Rd✱ ✇❤❡r❡ α = ✶ +

• ❧♥(✷/δ)

✷

✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

▲✐♥❡❛r ❯♣♣❡r ❇♦✉♥❞ ❈♦♥✜❞❡♥❝❡ ✲ ✸

• ❚❤❡ ✐♥❡q✉❛❧✐t② ❣✐✈❡s ❛ r❡❛s♦♥❛❜❧② t✐❣❤t ❯❈❇ ❢♦r t❤❡ ❡①♣❡❝t❡❞

♣❛②♦✛ ♦❢ ❛r♠✳ ❆t ❡❛❝❤ tr✐❛❧ t✱ ❝❤♦♦s❡✿ at = ❛r❣♠❛①

a∈At

(xT

t,aˆ

θa + α

• xT

t,aA−✶ a xt,a),

✭✷✵✮ ✇❤❡r❡ Aa = DT

a Da + Id✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

▲✐♥❯❈❇ ❛❧❣♦r✐t❤♠

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❙✉♠♠❛r②

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❙❡tt✐♥❣✿ ✇❡ ❤❛✈❡ t✇♦ ❝❤♦✐❝❡s

• ■♥ t❤❡ ✜rst ♦♥❡ ✇❡ ❤❛✈❡ t♦ ❞❡❝✐❞❡ ❜❡t✇❡❡♥ n ❛❝t✐♦♥s✱ ❡✳❣✳ ❤✐t✱

st❛♥❞ ♦r ❞♦✉❜❧❡ ❞♦✇♥✳ ❲❡ ✐❣♥♦r❡ s♣❧✐tt✐♥❣✳

• ■♥ t❤❡ s❡❝♦♥❞ ♦♥❡ ✇❡ ❛❧✇❛②s st❛♥❞✳

❚❤❡ ✜rst ❞❡❝✐s✐♦♥ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ n✲❛r♠❡❞ ❜❛♥❞✐t ♣r♦❜❧❡♠✳ ❖✉r ❡①❛♠♣❧❡ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ✸✲❛r♠❡❞ ❜❛♥❞✐t ♣r♦❜❧❡♠✱ s✐♥❝❡ ✇❡ ❤❛✈❡ t❤❡ ❝❤♦✐❝❡ ❜❡t✇❡❡♥ ✸ ❛❝t✐♦♥s ✭❤✐t✱ st❛♥❞ ❛♥❞ ❞♦✉❜❧❡ ❞♦✇♥✮✳ ❚❤❡② ❤❛✈❡ ❛ ❞✐✛❡r❡♥t r❡✇❛r❞ ❞✐str✐❜✉t✐♦♥ ❛♥❞ s♦ ❛ ❞✐✛❡r❡♥t r❡✇❛r❞ ❡①♣❡❝t❛t✐♦♥✳

• ♦❛❧✿ ▼❛①✐♠✐③❡ t❤❡ r❡✇❛r❞✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦ ✲ ❣r❡❡❞② ♠❡t❤♦❞s

❲❡ ❤❛✈❡ s❡❡♥ s♦♠❡ ♠❡t❤♦❞s t♦ s♦❧✈❡ t❤❡ n✲❛r♠❡❞ ❜❛♥❞✐t ♣r♦❜❧❡♠✱ ❧❡t✬s ❛♣♣❧② t❤❡♠ t♦ ♦✉r ✸✲❛r♠❡❞ ❜❛♥❞✐t ♣r♦❜❧❡♠✳

• ❚❤❡ ❣r❡❡❞② ♠❡t❤♦❞✿ ❝❤♦♦s❡ ❛t r❛♥❞♦♠ ❡✳❣✳ t = ✶✵✵✵ t✐♠❡s

♦♥❡ ♦❢ t❤✐s ✸ ❛❝t✐♦♥s ❛♥❞ t❤❡♥ ♠❛❦❡ ②♦✉r ❡st✐♠❛t♦r ❢♦r t❤❡ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ r❡✇❛r❞✿ Q(a) =

R✶+···+RNt (a) Nt(a)

N(a) = t✐♠❡s ✇❡ ❤❛✈❡ ❝❤♦s❡♥ t❤❡ ❛❝t✐♦♥ a ✉♥t✐❧ t✐♠❡ t✳ ❚❤❡♥ ❡①♣❧♦✐t t❤❡ ❣r❡❡❞② ❛❝t✐♦♥s✱ ✐✳❡✳ ❝❤♦♦s❡ ❛❧✇❛②s t❤❡ ❛❝t✐♦♥ ✇✐t❤ t❤❡ ❣r❡❛t❡st ❡st✐♠❛t♦r ❡①♣❡❝t❛t✐♦♥✳

• ❚❤❡ ǫ ❣r❡❡❞② ♠❡t❤♦❞✿ ❝❤♦♦s❡ ✇✐t❤ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ǫ ❛♥ ❛❝t✐♦♥

✇❤✐❝❤ ✐s ♥♦t t❤❡ ❣r❡❡❞② ❛❝t✐♦♥✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦ ✲ ♦t❤❡r ♠❡t❤♦❞s

• ■♥ t❤❡ ❯♣♣❡r✲❈♦♥✜❞❡♥❝❡✲❇♦✉♥❞ ❆❝t✐♦♥ ❙❡❧❡❝t✐♦♥ ▼❡t❤♦❞ ✇❡

✉s❡ t❤❡ ♥✐❝❡ ❢♦r♠✉❧❛✿ At = ❛r❣♠❛①

a

[Qt(a) + c

• ❧♥ t

Nt(a)]✳

• ❚❤❡ ❚❤♦♠♣s♦♥ s❛♠♣❧✐♥❣ ♠❡t❤♦❞s ✐❢ ✇❡ ❛❞❛♣t t❤❡ r❡✇❛r❞ s✉❝❤

t❤❛t t❤❡② ❛r❡ ❜♦✉♥❞❡❞ ✐♥ [✵, ✶]✱ ❡t❝✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦ ✲ ❈♦♥t❡①t✉❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠

• ❚❤❡r❡ ✐s ❛♥♦t❤❡r ✇❛② t♦ ❛♣♣r♦❛❝❤ t❤✐s ♣r♦❜❧❡♠✿ ✇❡ ❝❛♥ s❡❡ ✐t

❛s ❛ ❝♦♥t❡①t✉❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠✳

• ❚❤❡ ✐❞❡❛ ✐s t♦ ❧♦♦❦ ❛t ❡✈❡r② ♣♦ss✐❜❧❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ❝❛r❞s t❤❡

❞❡❛❧❡r ❛♥❞ t❤❡ ♣❧❛②❡r ❤❛✈❡ ❛♥❞ ✇❤❛t ❛❝t✐♦♥s ✇❡r❡ t❤❡ ❜❡st ✐♥ t❤❡ ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥s✱ ✐✳❡✳ ✇❡ ✐♠♣❧❡♠❡♥t ✇✐t❤ ❧♦ts ♦❢ s✐♥❣❧❡ ❜❛♥❞✐ts✱ ✇✐t❤♦✉t ✐♥❢♦r♠❛t✐♦♥ s❤❛r✐♥❣✳

• ❙♦ ❛s ✐♥ t❤❡ ❝♦♥t❡①t✉❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠ ✇❡ ❤❛✈❡ t♦ ❛ss♦❝✐❛t❡

❞✐✛❡r❡♥t ❛❝t✐♦♥s ✇✐t❤ ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥s ✭❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ❝❛r❞s✮✳

▼✉❧t✐✲❛r♠ ❇❛♥❞✐ts ❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦

❆♣♣❧✐❝❛t✐♦♥ t♦ ❇❧❛❝❦❥❛❝❦ ✲ ❯❈❇

• ❆ ✇❛② t♦ s♦❧✈❡ t❤❡ ❝♦♥t❡①t✉❛❧ ❜❛♥❞✐t ♣r♦❜❧❡♠ ✐s t♦ ✉s❡ t❤❡

❛❧r❡❛❞② s❡❡♥ ❧✐♥❡❛r ❯❈❇ ❛❧❣♦r✐t❤♠✿

• ❋❡❛t✉r❡ ✈❡❝t♦r xt,a ✐s t❤❡ ✐♥❢♦r♠❛t✐♦♥ ✇❡ ❤❛✈❡ ♦♥ t❤❡ ♣❧❛②❡rs

❛♥❞ ❞❡❛❧❡rs ❝❛r❞s✳

• E[rt,a | xt,a] ✐s t❤❡ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ r❡✇❛r❞ ❣✐✈❡♥ xt,a✳ ●♦❛❧✿

♠❛①✐♠✐③❡ ✐t✳

• ❚❤❡ r♦✇s ♦❢ t❤❡ ❞❡s✐❣♥ ♠❛tr✐① Da ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❞✐✛❡r❡♥t

❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❝❛r❞s ❤❛✈❡✳