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Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 1
Physical FluctuomaticsApplied Stochastic Process
8th Markov chain Monte Carlo methods
Kazuyuki TanakaGraduate School of Information Sciences, Tohoku University
kazu@smapip.is.tohoku.ac.jphttp://www.smapip.is.tohoku.ac.jp/~kazu/
Physical Fuctuomatics (Tohoku University) 2
Horizon of Computation in Probabilistic Information Processing
Compensation of expressing uncertainty using probability and statistics
It must be calculated by taking account of both events with high probability and events with low probability.
Computational Complexity
It is expected to break throw the computational complexity by introducing approximation algorithms.
Physical Fuctuomatics (Tohoku University) 3
What is an important point in What is an important point in computational complexity? computational complexity?
How should we treat the calculation of the summation over 2N configuration?
FT, FT, FT,
21
1 2
,,,x x x
N
N
xxxf
}
}
}
;,,,
F){or Tfor(
F){or Tfor(
F){or Tfor(
;0
21
2
1
L
N
xxxfaa
x
x
x
a
N fold loops
If it takes 1 second in the case of N=10, it takes 17 minutes in N=20, 12 days in N=30 and 34 years in N=40.
Markov Chain Monte Carlo MetodBelief Propagation Method
This Talk
Next Talk
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)
4
Calculation of the ratio of the circumference of a circle to its diameter by using random
numbers (Monte Carlo Method)
1
1
-1
-1
0
Generate uniform random numbers a and b in the interval [1,1] Count the number of points inside of the
unit circle after plotting points randomly
mm+1
a2+b2≤1
Yes
No
nn+1
n0 m0
2RS 1R n
mS
4
the ratio of the circumference of a circle to its diameter
Accuracy is improved as the number of trials increases
Physics Fluctuomatics/Applied Stochastic Process (Tohoku
University) 5
Law of Large Numbers
)( )(1
21 nXXXn
Y nn
Let us suppose that random variables X1,X2,...,Xn are identical and mutual independent random variables with average Then we have
Central Limit Theorem
)(1
21 nn XXXn
Y
tends to the Gauss distribution with average and variance 2/n as n+.
We consider a sequence of independent, identical distributed random variables, {X1,X2,...,Xn}, with average and variance Then the distribution of the random variable
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 6
Calculation of the ratio of the circumference of a circle to its diameter by using random
numbers (Monte Carlo Method)
1
1
-1
-1
0
Count the number of points m inside of the unit circle after plotting points n randomly
2RS
1Rn
mS
4
the ratio of the circumference of a circle to its diameter
Accuracy is improved as the number of trials increases
From the central limit theorem, the sample average and the sample variance are 0 and 1/2n for n random points.The width of probability density function decreases by according to 1/n1/2 as the number of points, n, increases.
x- and y- coordinates of each random points is the average 0 and the variance ½.
Order of the error of the ratio of the circumference of a circle to its diameter is O(1/n1/2)
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 7
Integral Computation by Monte Carlo Method
1
1
-1
-1
0
Generate uniform random numbers a and b in the
interval [1,1] Compute the value of f(x,y) at each point (x,y) after plotting points n
inside of the green region randomlymm + f(a,b)
nn+1
n0 m0
n
mI
4
1
1
1
1),( dxdyyxfI
Accuracy is O(1/n1/2)
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 8
Marginal Probability
Marginal Probability
1,0 1,0 1,0
2111
2 3
,,,x x x
L
L
xxxPxP
1,0 1,0 1,0
3212112
3 4
,,,,,x x x
L
L
xxxxPxxP
1,0 1,0 1,0
3213113
2 4
,,,,,x x x
L
L
xxxxPxxP
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 9
Important Point of Computations
||21 ,,, VxxxPP x
Computational time generating one random numbers should be order of |V|.
Can we make an algorithm to generate |V| random vectors (x1,x2,…,x|V|) which are independent of each other?
The random numbers should be according to
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 10
Fundamental Stochastic Process: Markov Process
),2,1 ;1,0( )()|()(1
01
txzPzxwxP
ztt
For any initial distribution P0(x),
Transition Probability w(x|y)≥0 (x,y=0,1)
)1(
)0(
)1|1()0|1(
)1|0()0|0(
)1(
)0(
1
1
t
t
t
t
P
P
ww
ww
P
P
Transition Matrix
10
z
xzw
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 11
Fundamental Stichastic Process: Markov Chain
1
0]1[
1
0]2[
1
0]1[0
1
0]1[1
])0[(])0[|]1[(])2[|]1[(])1[|(
])1[(])1[|()(
tz tz tz
tztt
zPzzwtztzwtzxw
tzPtzxwxP
)1(
)0(
)1(
)0(
0
0
P
PW
P
P t
t
t
1)( 0
01 1
UUW
)1(
)0(
0
01
)1(
)0(
0
01
P
PUU
P
Pt
t
t
Transition matrix can be diagonalized as
)1(
)0(
00
01
)1(
)0(lim
)1(
)0(
0
01
P
PUU
P
P
P
P
t
t
t
Limit Distribution
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 12
Fundamental Stochastic Process: Markov Process
1
0]1[1 ])1[(])1[|()(
tztt tzPtzxwxP
)1(
)0(
)1(
)0(
P
PW
P
P
1
0]1[
])1[(])1[|()(tz
tzPtzxwxP
Stationary Distribution or Equilibrium Distribution
)1(
)0(lim
)1(
)0(
t
t
t P
P
P
P
In the Markov process, if there exists one unique limiting distribution, it is an equilibrium distribution.
Even if there exists one equilibrium distribution, it is not always a limiting distribution.Example
2/1
2/1
)1(
)0(
P
PThe stationary distribution is
11
11
10
01
11
11
2
101
10W
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 13
Stationary Process and Detailed Balance in Markov Process
1
01 )(
ytt yPyxwxP
1
0y
yPyxwxP Stationary Distribution of Markov Process
P1(x), P2(x), P3(x),…: Markov Chain
)()( yPyxwxPxyw Detailed Balance
When the transition probability w(x|y) is chosen so as to satisfy the detailed balance, the Markov process provide us a stationary distribution P(x).
11
0
y
xywwhere
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 14
Markov Chain Monte Carlo Method
),3,2,1( )()( 1
tyPyxwxP
ytt
)()(lim xPxPtt
),,,()( 21 LxxxPxP
Let us consider a joint probability distribution P(x1,x2,…,xL)
T21 ),,,( Lxxxx
How to find the transition probability w(x|y) so as to satisfy
where
P1(x), P2(x), P3(x),…: Markov Process
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 15
Markov Chain Monte Carlo Method
They can be regarded as samples from the given probability distribution P(x).
For sufficient large x[], x[2], x[3], …, x[N] are independent of each other
Randomly generated
Reject
How large number : relaxation time
]1[]1)2[(]1)1[(
]2[]2[]1[
][]2[]1[
NxNxNx
xxx
xxx
Accuracy O(1/1/2)
1txtxw
1tx tx
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 16
N
nXnx
X X XL
N
XXXXPXP
L
1,
32111
11
2 3
1
,,,,
Markov Chain Monte Carlo Method
][]2)1[(]1)1[(
]2[]2[]1[
][]2[]1[
NxNxNx
xxx
xxx
Histgram
XiMarginal Probability Distribution
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 17
Lii
LiiiLiii xxxxxP
xxxxxxPxxxxxxP
,,,,,,
,,,,,,,,,,,,,
1121
11211121
Markov Chain Monte Carlo Method
Eji
jiijL xxxxxPxP},{
21 ),(),,,()(
E : Set of all the neighbouring pairs of nodes
V : Set of all the nodes
iz
LiiiLii xxzxxxPxxxxxP ,,,,,,,,,,,,, 11211121
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 18
Markov Chain Monte Carlo Method
Eji
jijiL xxxxxPxP},{
},{21 ),(),,,()(
ijxxP
xz
xx
xxxxxxP ji
z ijjiji
ijjiji
Liii
i
,
,
,,,,,, },{
},{
1121
Markov Random Field
E : Set of all the neighbouring pairs of nodes
∂i : Set of all the neighbouring nodes of the node i
),( EV),( EV
||VL
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 19
Markov Chain Monte Carlo Method
Vi
z ijjiji
ijjiji
iVkkkLL
i
xz
xx
xxxxxxxxw,
,
),(,,,,,,},{
},{
/2121
'xP'xxwxPxxw
'),( EV
ikxx kk
Eji
jijiL xxxxxPxP},{
},{21 ),(),,,()(
1txtxw
1tx tx
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 20
Markov Chain Monte Carlo Method
x
x’
xi = ○ or ●
V
V
True False
1txtxw
]1[ tx tx
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 21
Sampling by Markov Chain Monte Carlo Method
Disordered State Ordered State
Sampling by Markov Chain Monte Carlo Method
Near Critical Point of p
Small p Large p
p p
More is different.
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 22
Summary
Calculation of the ratio of the circumference of a circle to its diameter by using random numbers
Law of Large Numbers and Central Limit Theorem
Markov Chain Monte Carlo Method
Future Talks9th Belief propagation10th Probabilistic image processing by means of physical models 11th Bayesian network and belief propagation in statistical inference
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 23
Practice 8-1
x'
x'x'xx PwP
x'x'xxxx' PwPw
When the probability distribution P(x) and the transition probability w(x’|x) satisfy the detailed balance
1x
xxwwhere , prove that
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 24
Practice 8-2
21
12
3
1W
Let us consider that the transition matrix of the present stochastic process is given as
)1(
)0(
)1(
)0(
1
1
t
t
t
t
P
PW
P
P
)1(
)0(lim
)1(
)0(
t
t
t P
P
P
PFind the limit distribution defined by
, where
Physical Fluctuomatics / Applied Stochastic Process (Tohoku University) 25
Practice 8-3
yL
xL
Example of generated random vector in the case of Q=2, =2
L=Lx×Ly
Let us consider an undirected square grid graph with L=Lx×Ly nodes. The set of all the nodes is denoted by V={1,2,…,L} and the set of all the neighbouring pairs of nodes is denoted by E. A random variable Fi is assigned at each node i and takes every integer in the set {0,1,2,…,Q1} . The joint probability distribution of the provability vector F=(F1,F2,…,FL)T is given as
Make a program which generate N mutual independent random vectors (f1,f2,…,fL)T randomly from the above joint probability distribution Pr{F1=f1,F2=f2,…,FL=fL} . For various values of positive numbers , give numerical experiments.
EjijiLL ff
ZfFfFfF
},{
2
Prior2211 2
1exp
1,,,Pr
Example of generated random vector in the case of Q=256, =0.0005
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