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n

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i

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n

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D

π

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i

D

π

π

(

i

)

D

D

π

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i

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i

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(

D

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1

(

D

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, !

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D

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[

i

]

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π

(

i

)

D

π

[

i

]

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d

π

(

i

)

,

" ;

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π

(1) = 3

, π

(2) =

4

, π

(3) = 2

, π

(4) = 1

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d1, d2, d3, d4

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d3, d4, d2, d1

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j

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j

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r

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r

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j

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8

77

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7&?"@8

7)

n

k

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n

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π

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j

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min

8

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(

j

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70

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d

min

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π

s

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( &

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n

1

j

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s

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(9)

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*"+",+"

"

s

! "

j

[1

, n

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D

π

s

[

j

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d

l

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0,

R

0,

· · ·

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A

s

1,

R

s

1

) = 1

/n,

*:+

l

[1

, n

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A

i

R

i

!

i

[0

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1]

! $ *$

i

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; / $"

j

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s

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j

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, n

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s

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π

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π

0

!

Pr(

D

π

0

[

j

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d

l

| ∅

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/n

1

l

n

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s

=

i

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D

π

i

[

j

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d

l

|A

0,

R

0,

· · ·

,

A

i

1,

R

i

1

) =

1

/n

s

=

i

+ 1

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D

π

i

+1

[

j

] =

d

l

|A

1,

R

1,

· · ·

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A

i

,

R

i

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/n,

*<+

l

[1

, n

]

(

D

π

i

+1

D

π

i

!!

Pr(

D

π

i

+1

[

j

] =

d

l

|A

1,

R

1,

· · ·

,

A

i

,

R

i

)

=

n

x

=1

Pr(

D

π

i

+1

[

j

] =

D

π

i

[

x

]

|A

1,

R

1,

· · ·

,

A

i

,

R

i

)

·

Pr(

D

π

i

[

x

] =

d

l

|A

1,

R

1,

· · ·

,

A

i

,

R

i

)

.

*>+

,

l

x

,

!

1

/n

!

! $

p

x

q

x

!

p

x

= Pr(

D

π

i

+1

[

j

] =

D

π

i

[

x

]

|A

1,

R

1,

· · ·

,

A

i

,

R

i

)

,

q

x

= Pr(

D

π

i

[

x

] =

d

l

|A

1,

R

1,

· · ·

,

A

i

,

R

i

)

.

,6

n

x

=1

(10)

$

X

=

{

x

|

x

[1

, n

]

, π

i

(

x

)

Γ

}

Y

=

{

x

|

x

[1

, n

]

, π

i

(

x

)

/

Γ

}

A

|

X

|

=

|

Γ

|

=

k

|

Y

|

=

n

k

X

Y

= [1

, n

]

,

n

x

=1

p

x

q

x

=

x

X

p

x

q

x

+

x

Y

p

x

q

x

.

*E+

! 96

D

π

i

%

D

π

i

X

!&

D

π

i

+1

8

D

π

i

Y

! ! &

4

D

π

i

D

!$ @ $

k

+ 1

D

π

i

+1

, !

x

X

p

x

= Pr(

π

i

+1

(

j

) =

π

i

(

x

)) = 1

/n

*F+

p

x

x

Y

2 !

x

Y

p

x

= 1

x

X

p

x

= (

n

k

)

/n.

*0+

,

q

x

! " !

, ;

C

i

&

1

l

n

5

2*/+

l

∈ C

5

x

X

q

x

= 1

!%

"! !

d

l

x

Y

q

x

= 0

,

n

x

=1

p

x

q

x

= 1

/n

+ 0 = 1

/n,

l

∈ C

.

*/7+

2*=+

l /

∈ C

5

x

X

q

x

=

δ

0

δ

1

x

Y

q

x

= 1

δ

A " 9 !%&

!, !

5

Pr(

D

π

i

[

x

] =

d

l

| A

0,

R

0,

· · ·

,

A

i

1,

R

i

1

) = 1

/n

l

[1

, n

]

q

x

=

1

δ

n

k

!

n

x

=1

p

x

q

x

=

δ/n

+ (1

δ

)

/n

= 1

/n.

*//+

$ "

x

0

, x

1

Y

q

x

0

=

q

x

1

Pr(

D

π

i

[

x

0

] =

d

l

)

= Pr(

D

π

i

[

x

1

] =

d

l

)

"

d /

∈ C

(11)

22/2=

n

x

=1

p

x

q

x

= 1

/n,

1

l

n,

;/@

,

, / !

)(

- $! $

A

=

(

a

1

, a

2

,

· · ·

, a

N

)

!

N >

0

! $ ."

#

q

!

j

[1

, n

]

!

Pr(

q

=

j

|A

) = Pr(

q

=

j

)

*/=+

Pr(

q

=

j

)

$$ #

q

" %

j

1

< t

N

Pr(

a

t

|

a1,

· · ·

, a

t

1

)

!

a

t

!

t

1

;

Pr(

a

t

|

a

1

,

· · ·

, a

t

1

;

q

=

j

)

!

" !

q

j

A

! !

q

t

& !

Pr(

a

t

|

a1,

· · ·

, a

t

1

) = Pr(

a

t

|

a1,

· · ·

, a

t

1

;

q

=

j

)

*/:+

!

a

t

D

π

s

s

&

25

/

a

t

∈ R

s

a

t

@ 5 8!

Pr(

a

t

|

a1,

· · ·

, a

t

1

) = Pr(

a

t

|

a1,

· · ·

, a

t

1

;

q

=

j

)

a

t

!!

π

s

π

s

+1

=

a

t

∈ A

s

a

t

!5; !

l

&

!

l

[1

, k

]

,

l

1

!

a

t

#/5

*+ , 5

a

t

!

,

Pr(

a

t

|

a1,

· · ·

, a

t

1

) =

1

n

(

l

1)

*+ , 5

a

t

D

π

s

π

s

# ; / !

a

t

1

n

(

l

1)

A !

q

=

j

!

!

a

t

*=+*=+ A

Pr(

a

t

|

a

1

,

· · ·

, a

t

1

)

1

n

(

l

1)

,

Pr(

a

t

|

a

1

,

· · ·

, a

t

1

) = Pr(

a

t

|

a

1

,

· · ·

, a

t

1

, q

=

(12)

( !

t

q

=

j

Pr(

a

t

|

a

1

,

· · ·

, a

t

1

) = Pr(

a

t

|

a

1

,

· · ·

, a

t

1

;

q

=

j

)

.

#

Pr(

A |

q

=

j

) = Pr(

a1,

· · ·

, a

N

|

q

=

j

)

= Pr(

a

N

|

a1,

· · ·

, a

N

1

;

q

=

j

)

·

Pr(

a1,

· · ·

, a

t

1

|

q

=

j

)

=

N

t

=1

Pr(

a

t

|

a

1

,

· · ·

, a

t

1

;

q

=

j

)

=

N

t

=1

Pr(

a

t

|

a1,

· · ·

, a

t

1

)

= Pr(

A

)

.

*/<+

,

Pr(

q

=

j

|A

) = Pr(

q

=

j,

A

)

/

Pr(

A

)

=

Pr(

A |

q

=

j

)

·

Pr(

q

=

j

)

Pr(

A

)

= Pr(

q

=

j

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References

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