▼❛❥♦r✐t② ❱♦t✐♥❣ ◆♦t❡s ❖❝t♦❜❡r ✶✹✱ ✷✵✶✵ ▼❛❥♦r✐t② ❱♦t✐♥❣ • ❚❤❡r❡ ❛r❡ t❤r❡❡ t②♣❡s ♦❢ ♠❛❥♦r✐t② ✈♦t✐♥❣ ✕ ✉♥❛♥✐♠✐t② ✲ ❛❧❧ ❝❧❛ss✐✜❡rs ❛❣r❡❡ ✕ s✐♠♣❧❡ ♠❛❥♦r✐t② ✲ ✺✵✪ ✰ ✶ ❝❧❛ss✐✜❡rs ❛❣r❡❡ ✕ ♣❧✉r❛❧✐t② ✲ ✇❤✐❝❤❡✈❡r ❝❧❛ss✐✜❡r ❤❛s t❤❡ ♠♦st ✈♦t❡s • ❲✐t❤ ❙✐♠♣❧❡ ♠❛❥♦r✐t② ✈♦t✐♥❣ ②♦✉ ♥❡❡❞ ❛t ❧❡❛st ⌊ L 2 ⌋ + 1 ❝❧❛ss✐✜❡rs t♦ ❣✐✈❡ t❤❡ ❝♦rr❡❝t ❛♥s✇❡r L � L � � p m (1 − p ) L − m ✕ ❚❤❡r❡❢♦r❡✱ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❡♥s❡♠❜❧❡ ✐s P maj = m m = ⌊ L/ 2 ⌋ +1 ❛ss✉♠✐♥❣ t❤❛t t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ s✉❝❝❡ss p ✐s t❤❡ s❛♠❡ ❢♦r ❛❧❧ ❝❧❛ss✐✜❡rs • ❈❧❛ss✐✜❡rs✱ ✇❤✐❧❡ ❤❛✈✐♥❣ t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② ♦❢ s✉❝❝❡ss ❝❛♥ st✐❧❧ ❤❛✈❡ ♠❛r❦❡❞❧② ❞✐✛❡r❡♥t ❞✐str✐❜✉t✐♦♥s ♦❢ ❡rr♦r ♦✈❡r t❤❡ ❝❧❛ss ❧❛❜❡❧s✳ • ❆ ♣❛tt❡r♥ ♦❢ s✉❝❝❡ss ✐s ❛ ❞✐st✐❜✉t✐♦♥ ♦❢ t❤❡ ❝❧❛ss✐✜❡r ♦✉t♣✉ts s✉❝❤ t❤❛t ✕ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ⌊ L 2 ⌋ +1 ❝♦rr❡❝t ❛♥❞ ⌊ L 2 ⌋ ✐♥❝♦rr❡❝t ✐s α ✳ ✕ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❧❧ L ✈♦t❡s ❜❡✐♥❣ ✐♥❝♦rr❡❝t ✐s γ ✳ ✕ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥② ♦t❤❡r ❝♦♠❜✐♥❛t✐♦♥ ✐s ③❡r♦✳ � L � • ❇❡❝❛✉s❡ t❤❡r❡ ❛r❡ ✇❛②s t♦ ❝❤♦♦s❡ l + 1 ❝♦rr❡❝t ♦✉t ♦❢ L ❝❧❛ss✐✜❡rs l +1 � L � p maj = α ✇❤✐❝❤ ✐s t❤❡ ✉♣♣❡r ❜♦✉♥❞ ♦♥ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥ ❡♥s❡♠❜❧❡ l +1 ♦❢ ▲ ❝❧❛ss✐✜❡rs ✇✐t❤ ♣r♦❜❛❜✐❧✐t② α • ❚❤❡ ♣❛tt❡r♥ ♦❢ ❢❛✐❧✉r❡ ❛❧t❡r♥❛t✐✈❡❧② ✐s ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ L ❝❧❛ss✐✜❡r ♦✉t♣✉ts s✉❝❤ t❤❛t ✕ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥② ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ⌊ L 2 ⌋ ❝♦rr❡❝t ❛♥❞ ⌊ L 2 ⌋ + 1 ✐♥✲ ❝♦rr❡❝t ✈♦t❡s ✐s β ✕ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛❧❧ L ✈♦t❡s ❜❡✐♥❣ ❝♦rr❡❝t ✐s δ ✳ ✶
✕ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥② ♦t❤❡r ❝♦♠❜✐♥❛t✐♦♥ ✐s ③❡r♦✳ • ❚❤✐s r❡s✉❧ts ✐♥ ❛ ❧♦✇❡r ❜♦✉♥❞ ♦♥ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❡♥s❡♠❜❧❡ ♦❢ p maj = pL − 1 ✇❤❡r❡ p ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ tr✉❡ ♣♦s✐t✐✈❡s ❢♦r ❛♥② ❝❧❛ss✐✜❡r l +1 • ✐❢ ❡❛❝❤ ❝❧❛ss✐✜❡r ❤❛s ❛ ❞✐✛❡r❡♥t ♣r♦❜❛❜✐❧✐t② t❤❡♥ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ♦♥ t❤❡ ♠❛❥♦r✐t② ✈♦t❡ ❛❝❝✉r❛❝② ❛ ♠❛①✐♠✉♠ ❜♦✉♥❞ ♠❛② ❜❡ ♦❜t❛✐♥❡❞ ❜② ✕ ♦r❞❡r✐♥❣ t❤❡ ❝❧❛ss✐✜❡rs ❜② t❤❡✐r ♣r♦❜❛❜✐❧✐t② s✉❝❤ t❤❛t p 1 ≤ p 2 ≤ ... ≤ p l ✕ ▲❡t k = L +1 2 • ❚❤❡♥ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ✐s max P maj = min { 1 , Σ( k ) , Σ( k − 1) , ..., Σ(1) } L − k + m ✇❤❡r❡ Σ( m ) = 1 � p i ✳ m i =1 • ❚❤❡ r❡s✉❧t✐♥❣ ❧♦✇❡r ❜♦✉♥❞ ✐s s✐♠✐❧❛r❧② min P maj = max { 0 , ξ ( k ) , ξ ( k − L p i − L − k � 1) , ..., ξ (1) } ✇❤❡r❡ ξ ( m ) = 1 ✳ m m i = k − m +1 ❲❡✐❣❤t❡❞ ▼❛❥♦r✐t② ❱♦t✐♥❣ • ◆♦t ❛❧❧ ❝❧❛ss✐✜❡rs ❛r❡ ❝r❡❛t❡❞ ❡q✉❛❧ ✕ ❯s❡ ✇❡✐❣❤t❡❞ ♠❛❥♦r✐t② ✈♦t❡s ✇❤❡♥ ❝❧❛ss✐✜❡rs ❞♦ ♥♦t ❛❧❧ ❤❛✈❡ t❤❡ s❛♠❡ ❛❝❝✉r❛❝② r❛t❡s ✕ ❚❤✐s ❣✐✈❡s ❜❡tt❡r ❝❧❛ss✐✜❡rs ♠♦r❡ ♣♦✇❡r ✐♥ t❤❡ ❞❡❝✐s✐♦♥✲♠❛❦✐♥❣ ♣r♦✲ ❝❡ss • ▲❛❜❡❧ ♦✉t♣✉ts ❧✐❦❡ s♦ � 1 ✐❢ D i ❧❛❜❡❧s ① ✐♥ w j d i,j = 0 ♦t❤❡r✇✐s❡ • ❖❜t❛✐♥ ❞✐s❝r✐♠✐♥❛♥t ❢✉♥❝t✐♦♥ ❢♦r ❝❧❛ss w j ✕ ▲❡t b i ❜❡ t❤❡ ❝♦❡✣❝✐❡♥t ❢♦r ❝❧❛ss✐✜❡r D i L � ✕ g j ( x ) = b i d i,j i =1 ✕ ■♥ ❊♥❣❧✐s❤✿ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❞✐s❝r✐♠✐♥❛♥t ❢✉♥❝t✐♦♥ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ❢♦r t❤♦s❡ ♠❡♠❜❡rs ✇❤♦s❡ ♦✉t♣✉t ❛t ① ✐s w j ✷
• ❍♦✇ ❞♦ ✇❡ ♥♦r♠❛❧✐③❡ ❝♦❡✣❝✐❡♥ts❄ ✕ ❆s ❝♦♥✈❡♥✐❡♥t❧② ❛s ♣♦ss✐❜❧❡ ✕ ◆♦r♠❛❧✐③❡ ❝♦❡✣❝✐❡♥ts t♦ ✶ c � ✕ b j = 1 i =1 • ▼❛①✐♠✐③✐♥❣ ❛❝❝✉r❛❝② ♦❢ P w maj ❜② ❛ss✐❣♥✐♥❣ ✇❡✐❣❤ts ❧✐❦❡ s♦ p i ✕ b i ∝ log 1 − p i c p i � • ❚❤❡ ❞✐s❝r✐♠✐♥❛♥t ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦✿ g i ( x ) = log P ( w j )+ d i,j log 1 − p i i =1 ✸
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