1. Brief Paper Review
B.Balaji (2009)
“Nonlinear filtering and quantum physics
A Feynman path integral perspective”
Kohta Ishikawa
Oct 30. 2011 3
1

2. 2

3. xk = f (xk 1 , tk 1 )xk 1 + e(xk 1 , tk 1 ) System Model
yk = h(xk , tk ) + wk Observation Model
dx(t) = f (x(t), t)dt + e(x(t), t)dw(t) System Model(SDE)
y(tk ) = h(x(tk ), tk ) + wk Observation Model
dx(t) = f (x(t), t)dt + e(x(t), t)dv(t) System Model(SDE)
dy(t) = h(x(t), t)dt + dw(t) Observation Model(SDE)
3

4. m
X
d (t, x) = LY (t, x)dt + hi (x) (t, x)dyi (t)
i=1
n
X @ n
X @ 2 (t, x) m
X
1 1
LY (t, x) = (fi (x) (t, x)) + h2 (x) (t, x)
i
i=1
@xi 2 i=1 @x2 i 2 i=1
(0, x) = 0 (x)
dy(t)
4

5. dy(t)
m
!
X
u(t, x) = exp hi (x)yi (t) (t, x)
i=1
Then, DMZ equation translates to
0 1
n
X @ 2 u(t, x) n
X m
X
@u(t, x) 1 @ fi (x) + @hj (x) A @u(t, x)
= + yj (t)
@t 2 i=1
@x2
i i=1 j=1
@xi @xi
✓X
n m
X m
X
@fi (x) 1 1
+ h2 (x)
i yi (t) hi (x)
i=1
@xi 2 i=1
2i=1
m n
XX m
X X n ◆
@hi (x) 1 @hi (x) @hj (x)
+ yi (t)fj (x) yi (t)yj (t) u(t, x)
i=1 j=1
@xj 2 i,j=1 @xk @xk
k=1
u(0, x) = 0 (x)
5

6. ⇢
y(⌧l ) post-measurement form
y(t) !
y(⌧l 1 ) pre-measurement form
6

7. ⌧l 1  t  ⌧l
{⌧0 , ⌧1 , · · · }
0 1
n n m
@ul (t, x) 1 X @ 2 ul (t, x) X @ X @hj (x) A @ul (t, x)
= 2 + fi (x) + yj (⌧l )
@t 2 i=1 @xi i=1 j=1
@xi @xi
✓X
n m
X m
X
@fi (x) 1 1
+ h2 (x)
i yi (⌧l ) hi (x)
i=1
@xi 2 i=1
2i=1
m n
XX m
X X n ◆
@hi (x) 1 @hi (x) @hj (x)
+ yi (⌧l )fj (x) yi (⌧l )yj (⌧l ) ul (t, x)
i=1 j=1
@xj 2 i,j=1 @xk @xk
k=1
ul (⌧l , x) = ul 1 (⌧l 1 , x)
7

8. m
!
X (t, x)
ul (t, x) = exp
˜ yi (⌧l )hi (x) ul (t, x)
i=1
n
X @ 2 ul (t, x) n
X
@ ul (t, x)
˜ 1 ˜ @ ul (t, x)
˜
= fi (x)
@t 2 i=1
@x2
i i=1
@xi
n m
!
X @fi (x) 1 X
+ h2 (x) ul (t, x)
i ˜
i=1
@xi 2 i=1
m
!
X
ul (⌧l
˜ 1 , x) = exp yi (⌧l )hi (x) ul 1 (⌧l 1 , x)
i=1
yi (⌧l 1)
8

9. 9

10. Stochastic Process
probability density of state path measure,
functional differentiation
Fokker-Planck equation Path Integral representation
green’s function
10

11. SDE for continuous filtering
xi (t) = fi (x(t)) + vi (t)
˙
yi (t) = hi (x(t)) + wi (t)
˙
Probability density
Z
P (t, x, y|t0 , x0 .y0 ) = [d⇢(v(t))][d⇢(w(t))]
(x(t)
˙ f (x(t)) v(t)) n
⇥ [dx(t)] (x(t)
˙ f (x(t)) v(t))
x(t)
(y(t)
˙ h(x(t)) w(t)) m
⇥ [dy(t)] (y(t)
˙ h(y(t)) w(t))
y(t)
n m
⇥ (x(t) x)|x(t0 )=x0 (y(t) y)|y(t0 ) = y0
11

12. !
1 XZ t
n
[d⇢(v(t))] = [Dv(t)] exp
2~v
vi (t)2 dt ~v , ~w
i=1 t0
!
1 XZ t
m
[d⇢(w(t))] = [Dw(t)] exp vi (t)2 dt
2~w i=1 t0
Z y(t)=y Z x(t)=x
P (t, x, y|t0 , x0 .y0 ) = [Dv(t)][Dw(t)][Dx(t)][Dy(t)]
y(t0 )=y0 x(t0 )=x0
!
1 XZ t
n XZ t
n
@fi (x(t)) 1 XZ t
m
2 2
⇥ exp vi (t)dt dt wi dt
2~v i=1 t0 i=1 t0 @xi 2~w i=1 t0
n
⇥ (x(t) f (x(t)) v(t)) m (y(t) h(y(t)) w(t))
˙ ˙
Z y(t)=y Z x(t)=x
= [Dx(t)][Dy(t)] exp ( S)
y(t0 )=y0 x(t0 )=x0
Z " n n m
#
1 t
1 X X @fi (x(t)) 1 X
2
S= dt (xi (t)
˙ fi (x(t))) + + ( yi
˙ hi (x(t)))2
2 t0 ~v i=1 i=1
@xi ~w i=1
12

13. measurement term in the action S
Z ti m
X Z ti m
X⇥
1 1 ⇤
dt ( yi
˙ hi (x(t))) = dt ˙2
yi (t) + h2 (x(t))
i 2hi (x(t))yi (t)
˙
2~w ti 1 i=1
2~w ti 1 i=1
relevant term
Z ti m
X ⇢ 1
Pm
1 ~w j=1 hj ([x(ti ) + x(ti
Pm 1 )]/2)[yj (tj ) yj (tj 1 )]
dt hj (x(t))yj (t) ⇠
˙ 1
~w ti 1 j=1 ~w j=1 hj ([x(ti ) + x(ti 1 )]/2)[yj (tj 1 ) yj (tj 2 )]
13

14. leads to approximated probability density
P (ti , xi , yi |ti 1 , xi 1 , yi 1 )
˜
⇠ P (ti , xi |ti 1 , xi 1 )
⇣ Pm ⌘
⇢ 1
exp j=1 hj ([x(ti ) + x(ti 1 )]/2)[yj (tj ) yj (tj 1 )]
⇥ ⇣ ~w Pm ⌘
1
exp ~w j=1 hj ([x(ti ) + x(ti 1 )]/2)[yj (tj 1 ) yj (tj 2 )]
Z x(ti )=xi
˜
P (ti , xi |ti 1 , xi 1 ) = [Dx(t)] exp( S(ti 1 , ti ))
x(ti 1 )=xi 1
Z " n n m
#
1 ti
1 X X @fi (x(t)) 1 X
2
S(ti 1 , ti ) = dt (xi (t)
˙ fi (x(t))) + + h2 (x(t))
i
2 ti 1
~v i=1 i=1
@xi ~w i=1
˜
P (t, x|ti 1 , xi 1 )
14

15. Z x(t)=x ✓ ◆
˜ 1
P (t, x|t0 , x0 ) = [Dx(t)] exp S
x(t0 )=x0 ~v
Z " n n m
#
1 t X @fi (x(t)) ~v X X
S= dt + [x2 (t)
˙i ˙ fi2 (x)
2xi (t)fi (x(t))] + ~v + h2 (x(t))
i
2 t0 i=1 i=1
@xi ~w i=1
Z t "X n n
X @fi (x(t)) m
#
1 ~v X 2
= dt [x2 (t) + fi2 (x)] + ~v
˙i + hi (x(t))
2 t0 i=1 i=1
@xi ~w i=1
X Z x(t)
n
dxi (t)fi (x(t))
i=1 x(t0 )
Z t XZ
n x(t) L=T V
1
⌘ dtL dxi (t)fi (x(t)) Z t n
X
2 1
t0 i=1 x(t0 )
T = dt x2 (t)
˙i
2 t0 i=1
" #
1
Z t X
n
@fi (x(t)) ~v
m
X
V = dt fi2 (x) + ~v + h2 (x(t))
i
2 t0 i=1
@xi ~w i=1
15

16. 16

17. 17

18. dx = (b(x, t) + Bu)dt + dv
⌧ Z tf ✓ ◆
1
C(xint , tint , u) = (x(tf )) + dt u(t)T Ru(t) + V (x(t), t)
tint 2 xint
J(x, t) = min C(x, t, u(t ! tf ))
u(t!tf )
18

19.  T 
@J 1 @J @J @J 1 @2J
= BR 1 B T + V + bT + Tr ⌫ 2
@t 2 @x @x @x 2 @x
Z ✓ ◆
S
J(x, t) = log [Dx(t)] exp
x(t)=x
Z tf ✓ ◆
1
S = (x(tf )) + d⌧ (x(⌧ )
˙ b(x(⌧ ), ⌧ ))T R(x(⌧ )
˙ b(x(⌧ ), ⌧ )) + V (x(⌧ ), ⌧ )
t 2
V (x(t), t)
19

20. 20