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-/" &0+Ǯ"##

0&$*�Ǯ�), (

�%�1

0&$*�Ǯ� 1,/ 0&$*�Ǯ�), (

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sigma_blocksigma_actorbpcbpaa_block[6]a_block[5]a_block[4]a_block[3]a_block[2]a_block[1]a_actor[7]a_actor[6]a_actor[5]a_actor[4]a_actor[3]a_actor[2]a_actor[1]

-2 0 2 4 6Value

0 1 2 3 40.0

0.5

1.0

1.5

2.0

2.5

3.0

sigma

Density

actor

block

*ƾƿǑǂ�Ǯ� 1,/ �Ǯ�), (

�Ǯ� 1,/�Ǯ�), (�), (

� 1,/

,*-�/"ǯ*ƾƿǑǁǒ*ƾƿǑǂǰ

�� -�� !�� 4"&$%1 �� !��*ƾƿǑǁ ǂǀƾǑǂ ǅǑƾ ƽǑƽ ƽǑǃǂ ƾǆǑǂƽ ��*ƾƿǑǂ ǂǀƿǑǄ ƾƽǑǂ ƾǑƿ ƽǑǀǂ ƾǆǑǄǁ ƾǑǆǁ

-�� *ƾƿǑǂ*ƾƿǑǁ

0&$*�Ǯ�), (�Ǯ�), (

�Ǯ� 1,/

�Ǯ� 1,/ -��

*ƾƿǑǁ

�Ǯ�), (0&$*�Ǯ�), (

0&$*�

)&+( 0&*

!�1�ǯ %&*-�+7""0ǰ*ƾƿǑǁ

)&+(

%&*- ʆǦ ƿ!Ǒ-/"! ʆǦ )&01ǯ

-/,0, Ǯ)"#1 ʅ ǯƽǒƾǒƽǒƾǰǒ ȅ /&$%1ǵ)"#1ǵ/&$%1ǵ)"#1 ,+!&1&,+ ʅ ǯƽǒƽǒƾǒƾǰǒ ȅ ,+1/,)ǵ ,+1/,)ǵ-�/1+"/ǵ-�/1+"/� 1,/ ʅ /"-ǯ %&*-ǒǁǰ

ǰ)&+(Ǒ*ƾƿǑǁ ʆǦ )&+(ǯ *ƾƿǑǁ ǒ !�1�ʅ!Ǒ-/"! ǰ-/"!Ǒ- ʆǦ �--)6ǯ )&+(Ǒ*ƾƿǑǁ ǒ ƿ ǒ *"�+ ǰ-/"!Ǒ-Ǒ�� ʆǦ �--)6ǯ )&+(Ǒ*ƾƿǑǁ ǒ ƿ ǒ �� ǰ

)&+(

-,01 ʆǦ "51/� 1Ǒ0�*-)"0ǯ*ƾƿǑǁǰ01/ǯ-,01ǰ

�&01 ,# ǂɢ �Ǯ� 1,/ Ǔ +2* ǱƾǓǅƽƽƽǒ ƾǓǄǲ ǦƾǑǅǁƿ ǦƽǑƿƿǂ ǦƾǑǅƾƾ ǦƽǑǄǂǆ ǦƾǑǅǅƿ ǑǑǑɢ � Ǔ +2* ǱƾǓǅƽƽƽǯƾ!ǰǲ ƾǑƿǆƾ ǦƽǑǃǀƿ ƽǑƿǅǂ ǦƽǑƾƽǆ ƾǑƿƿǆ ǑǑǑɢ �- Ǔ +2* ǱƾǓǅƽƽƽǯƾ!ǰǲ ƾ ƾǑƽǃǁ ƾǑƽǅǄ ƽǑƿǂǁ ƽǑǆƽǅ ǑǑǑɢ �-� Ǔ +2* ǱƾǓǅƽƽƽǯƾ!ǰǲ ǦƽǑƿǄƿ ǦƽǑǂǀǆ ǦƽǑƿǆǂ ƽǑǀǄǂ ǦƽǑƿƾǅ ǑǑǑɢ 0&$*�Ǯ� 1,/Ǔ +2* ǱƾǓǅƽƽƽǯƾ!ǰǲ ƿǑƾǀ ƿǑǁǆ ƿǑǀƿ ƾǑǂƾ ǁǑƾƿ ǑǑǑ

�Ǯ� 1,/

!"+0ǯ -,01ɢ�Ǯ� 1,/Ǳǒǂǲ ǰ

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� ʀ �Ǯ� 1,/Ǳǒ� 1,/ǲ ʀ ǯ�- ʀ �-� ǹ ,+!&1&,+ǰ ǹ -/,0, Ǯ)"#1ǰ/"12/+ǯ ),$&01& ǯ),$,!!0ǰ ǰ

Ǵ

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)&+(

�Ǯ� 1,/� 0&$*�Ǯ� 1,/

� α

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ȅ /"-)� " 3�/6&+$ &+1"/ "-1 0�*-)"0 4&1% 7"/,0ȅ ƾƽƽƽ 0�*-)"0 �6 Ǆ � 1,/0�Ǯ� 1,/Ǯ7"/,0 ʆǦ *�1/&5ǯƽǒƾƽƽƽǒǄǰ

)&+(/"-)� "

�Ǯ� 1,/

ȅ #&/" 2- )&+(ȅ +,1" 20" ,# /"-)� " )&01)&+(Ǒ*ƾƿǑǁ ʆǦ )&+(ǯ *ƾƿǑǁ ǒ +ʅƾƽƽƽ ǒ !�1�ʅ!Ǒ-/"! ǒ

/"-)� "ʅ)&01ǯ�Ǯ� 1,/ʅ�Ǯ� 1,/Ǯ7"/,0ǰ ǰ

ȅ 02**�/&7" �+! -),1-/"!Ǒ-Ǒ*"�+ ʆǦ �--)6ǯ )&+(Ǒ*ƾƿǑǁ ǒ ƿ ǒ *"�+ ǰ-/"!Ǒ-Ǒ�� ʆǦ �--)6ǯ )&+(Ǒ*ƾƿǑǁ ǒ ƿ ǒ �� ǒ -/,�ʅƽǑǅ ǰ-),1ǯ ƽ ǒ ƽ ǒ 16-"ʅǛ+Ǜ ǒ 5)��ʅǛ-/,0, Ǯ)"#1ǵ ,+!&1&,+Ǜ ǒ

6)��ʅǛ-/,-,/1&,+ -2))"! )"#1Ǜ ǒ 6)&*ʅ ǯƽǒƾǰ ǒ 5�51ʅǛ+Ǜ ǒ5)&*ʅ ǯƾǒǁǰ ǰ

�5&0ǯ ƾ ǒ �1ʅƾǓǁ ǒ )��")0ʅ ǯǛƽǵƽǛǒǛƾǵƽǛǒǛƽǵƾǛǒǛƾǵƾǛǰ ǰ)&+"0ǯ ƾǓǁ ǒ -/"!Ǒ-Ǒ*"�+ ǰ0%�!"ǯ -/"!Ǒ-Ǒ�� ǒ ƾǓǁ ǰ

-/,0, Ǯ)"#1

0&$*�Ǯ� 1,/)&+( /"-)� "

α ∼ ( ,σ )

σ/+,/*

ȅ /"-)� " 3�/6&+$ &+1"/ "-1 0�*-)"0 4&1% 0&*2)�1&,+0-,01 ʆǦ "51/� 1Ǒ0�*-)"0ǯ*ƾƿǑǁǰ�Ǯ� 1,/Ǯ0&*0 ʆǦ /+,/*ǯǄƽƽƽǒƽǒ-,01ɢ0&$*�Ǯ� 1,/ǰ�Ǯ� 1,/Ǯ0&*0 ʆǦ *�1/&5ǯ�Ǯ� 1,/Ǯ0&*0ǒƾƽƽƽǒǄǰ

)&+( /"-)� "

0.0

0.2

0.4

0.6

0.8

1.0

prosoc_left/condition

prop

ortio

n pu

lled

left

0/0 1/0 0/1 1/1

average actor

0.0

0.2

0.4

0.6

0.8

1.0


prop

ortio

n pu

lled

left

0/0 1/0 0/1 1/1

marginal of actor

0.0

0.2

0.4

0.6

0.8

1.0


prop

ortio

n pu

lled

left

0/0 1/0 0/1 1/1

50 simulated actors

*ƾƿǑǁ

�Ǯ� 1,/

0&$*�Ǯ� 1,/

)&+(Ǒ*ƾƿǑǁ ʆǦ )&+(ǯ *ƾƿǑǁ ǒ +ʅƾƽƽƽ ǒ !�1�ʅ!Ǒ-/"! ǒ/"-)� "ʅ)&01ǯ�Ǯ� 1,/ʅ�Ǯ� 1,/Ǯ0&*0ǰ ǰ

-/,0, Ǯ)"#1

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-,01ɢ�Ǳ&ǲ ʀ0&*Ǯ�Ǯ� 1,/ ʀǯ-,01ɢ�-Ǳ&ǲ ʀ -,01ɢ�-�Ǳ&ǲǹ�ǰǹ�

ǰ/"12/+ǯ-ǰ

Ǵ

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0&$*�Ǯ� 1,/ -� �

-

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6)��ʅǛ-/,-,/1&,+ -2))"! )"#1Ǜ ǒ 6)&*ʅ ǯƽǒƾǰ ǒ 5�51ʅǛ+Ǜ ǒ 5)&*ʅ ǯƾǒǁǰ ǰ�5&0ǯ ƾ ǒ �1ʅƾǓǁ ǒ )��")0ʅ ǯǛƽǵƽǛǒǛƾǵƽǛǒǛƽǵƾǛǒǛƾǵƾǛǰ ǰ

ȅ -),1 ǂƽ 0&*2)�1"! � 1,/0#,/ ǯ & &+ ƾǓǂƽ ǰ )&+"0ǯ ƾǓǁ ǒ 0&*Ǒ� 1,/ǯ&ǰ ǒ ,)ʅ ,)Ǒ�)-%�ǯǛ�)� (ǛǒƽǑǂǰ ǰ

0&$*�Ǯ� 1,/

α σ *ƾƿǑǁ

∼ (µ )

(µ ) = α+ α [ ] + β

α ∼ ( , )

β ∼ ( , )

α ∼ ( ,σ )

σ ∼ ( , )

1,1�)Ǯ1,,)0 -,-2)�1&,+σ

α

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ȅ #&1 *,!")*ƾƿǑǃ ʆǦ *�-ƿ01�+ǯ

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ǰǒ!�1�ʅ! ǒ&1"/ʅǁƽƽƽ ǒ %�&+0ʅǀ ǰ

-,01 %" (ǯ*ƾƿǑǃǰ�Ǯ0, &"16

α σ

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ǰ�Ǯ0, &"16Ǯ0&*0 ʆǦ /+,/*ǯƿƽƽƽƽǒƽǒ-,01ɢ0&$*�Ǯ0, &"16ǰ�Ǯ0, &"16Ǯ0&*0 ʆǦ *�1/&5ǯ�Ǯ0, &"16Ǯ0&*0ǒƿƽƽƽǒƾƽǰ)&+(Ǒ*ƾƿǑǃ ʆǦ )&+(ǯ *ƾƿǑǃ ǒ +ʅƿƽƽƽ ǒ !�1�ʅ!Ǒ-/"! ǒ

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ȅ -),1 /�4 !�1�-),1ǯ !ɢ),$-,- ǒ !ɢ1,1�)Ǯ1,,)0 ǒ ,)ʅ/�+$&ƿ ǒ - %ʅƾǃ ǒ

5)��ʅǛ),$ -,-2)�1&,+Ǜ ǒ 6)��ʅǛ1,1�) 1,,)0Ǜ ǰ

ȅ -),1 -,01"/&,/ *"!&�+*2Ǒ*"!&�+ ʆǦ �--)6ǯ )&+(Ǒ*ƾƿǑǃ ǒ ƿ ǒ *"!&�+ ǰ)&+"0ǯ !Ǒ-/"!ɢ),$-,- ǒ *2Ǒ*"!&�+ ǰ

ȅ -),1 ǆǄɵǒ ǅǆɵǒ �+! ǃǄɵ &+1"/3�)0 ǯ�)) -/&*" +2*�"/0ǰ*2Ǒ�� ʆǦ �--)6ǯ )&+(Ǒ*ƾƿǑǃ ǒ ƿ ǒ �� ǒ -/,�ʅƽǑǆǄ ǰ0%�!"ǯ *2Ǒ�� ǒ !Ǒ-/"!ɢ),$-,- ǰ

7 8 9 10 11 12

2030

4050

6070

log population

tota

l too

ls *ƾƿǑǃ

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α ∼( , ) α ∼ ( , )

∼ ( , )

( ) = α [ ] + β

α ∼ ( , )

β ∼ ( , )

∼ (µ ,σ)

µ = α [ ] + β

α ∼ ( , )

β ∼ ( , )

σ ∼ ( , )

!�1�ǯ/""!#/,$0ǰ -/"!�1&,+ 0&7"

∼ ( , )

( ) = α [ ]

α ∼ (α,σ)

α ∼ ( , )

σ ∼ ( , )

α

%&*-�+7""0

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α ∼ (α,σ )

α ∼ (γ,σ )

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