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Physical Fluctuomatics 6th Markov chain Monte Carlo methods Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University [email protected] http://www.smapip.is.tohoku.ac.jp/~kazu/ Physical Fluctuomatics (Tohoku University) 1 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 Fluctuomatics (Tohoku University) 2 What is an important point in computational complexity? How should we treat the calculation of the summation over 2N configuration? x1 T, F x2 T, F f x1, x2 ,, xN a 0; for( x1 T or F){ for( x2 T or F){ for( x N T or F){ a a f x1 , x2 , , xL ; x N T, F 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 Metod Belief Propagation Method Physical Fluctuomatics (Tohoku University) N fold loops This Talk Next Talk 3 Calculation of the ratio of the circumference of a circle to its diameter by using random numbers (Monte Carlo Method) 1 n0 m0 -1 nn+1 Generate uniform random numbers a and b in the interval [-1,1] a2+b2≤1 Yes mm+1 No 0 1 -1 Count the number of points inside of the unit circle after plotting points randomly the ratio of the circumference of a circle to its diameter S R 2 R 1 Physical Fluctuomatics (Tohoku University) S 4m n Accuracy is improved as the number of trials increases 4 Law of Large Numbers Let us suppose that random variables X1,X2,...,Xn are identical and mutual independent random variables with average . Then we have 1 Yn ( X 1 X 2 X n ) (n ) n Central Limit Theorem We consider a sequence of independent, identical distributed random variables, {X1,X2,...,Xn}, with average and variance s2. Then the distribution of the random variable Yn 1 ( X1 X 2 X n ) n tends to the Gauss distribution with average and variance s2/n as n+. Physical Fluctuomatics (Tohoku University) 5 Calculation of the ratio of the circumference of a circle to its diameter by using random numbers (Monte Carlo Method) 1 x- and y- coordinates of each random points is the average 0 and the variance ½. 0 -1 1 From the central limit theorem, the sample average and the sample -1 variance are 0 and 1/2n for n random points. Count the number of points m inside of the unit circle after The width of probability density plotting points n randomly function decreases by according to 1/n1/2 as the number of points, n, the ratio of the circumference of a increases. circle to its diameter S R Order of the error of the ratio of the circumference of a circle to its R 1 Physical Fluctuomatics (Tohoku diameter is O(1/n1/2) University) 2 S 4m n Accuracy is improved as the number of trials increases 6 Integral Computation by Monte Carlo Method 1 n0 m0 I 1 1 -1 -1 f ( x, y )dxdy nn+1 -1 Generate uniform random numbers a and b in the interval [-1,1] mm + f(a,b) 0 1 -1 Compute the value of f(x,y) at each point (x,y) after plotting points n inside of the green region randomly 4m I n Accuracy is O(1/n1/2) Physical Fluctuomatics (Tohoku University) 7 Marginal Probability Marginal Probability P1 x1 Px , x ,, x x2 0,1 x3 0,1 P12 x1 , x2 P13 x1 , x3 xL 0,1 1 2 L Px , x , x ,, x x3 0,1 x4 0,1 xL 0,1 1 2 3 L Px , x , x ,, x x2 0,1 x4 0,1 xL 0,1 1 Physical Fluctuomatics (Tohoku University) 2 3 L 8 Important Point of Computations 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 P x Px1 , x2 ,, x|V | Computational time generating one random numbers should be order of |V|. Physical Fluctuomatics (Tohoku University) 9 Fundamental Stochastic Process: Markov Process Transition Probability w(x|y)≥0 (x,y=0,1) wz x 1 For any initial distribution P0(x), Pt ( x) 1 w( x | z ) Pt -1 ( z ) z 0 ( x 0,1; t 1,2,) z 0 Pt (0) w(0 | 0) w(0 | 1) Pt -1 (0) Pt (1) w(1 | 0) w(1 | 1) Pt -1 (1) Transition Matrix Physical Fluctuomatics (Tohoku University) 10 Fundamental Stichastic Process: Markov Chain Pt ( x) 1 w( x | z[t - 1]) Pt -1 ( z[t - 1]) z[t -1] 0 1 1 z[t -1] 0 z[t - 2] 0 1 w( x | z[t - 1]) w( z[t - 1] | z[t - 2]) w( z[1] | z[0]) P0 ( z[0]) z[t -1] 0 Pt (0) P0 (0) t W Pt (1) P0 (1) Pt (0) 1 0 -1 P0 (0) U t U 0 Pt (1) P0 (1) Transition matrix can be diagonalized as 1 0 -1 U W U ( 1) 0 Pt (0) P(0) 1 0 -1 P0 (0) U lim U P(1) t Pt (1) 0 0 P0 (1) Limit Distribution Physical Fluctuomatics (Tohoku University) 11 Fundamental Stochastic Process: Markov Process Pt ( x) 1 w( x | z[t - 1]) Pt -1 ( z[t - 1]) z[t -1] 0 P(0) P(0) W P( x) w( x | z[t - 1]) P( z[t - 1]) P(1) P(1) z[t -1] 0 1 Stationary Distribution or Equilibrium Distribution In the Markov process, if there exists one unique limiting distribution, it is an equilibrium distribution. Pt (0) P(0) lim P(1) t Pt (1) Even if there exists one equilibrium distribution, it is not always a limiting distribution. Example 0 1 1 1 1 1 0 1 1 The stationary distribution is W P(0) 1 / 2 1 0 2 1 - 1 0 - 11 - 1 P ( 1 ) 1 / 2 Physical Fluctuomatics (Tohoku University) 12 Stationary Process and Detailed Balance in Markov Process Pt x 1 wx y Pt -1 ( y) 1 where y 0 wy x 1 y 0 P1(x), P2(x), P3(x),…: Markov Chain Detailed Balance wy xP( x) wx y P( y) 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). P x 1 Stationary Distribution of w x y P y Markov Process y 0 Physical Fluctuomatics (Tohoku University) 13 Markov Chain Monte Carlo Method Let us consider a joint probability distribution P(x1,x2,…,xL) P( x) P( x1, x2 ,, xL ) x ( x1 , x2 ,, xL )T How to find the transition probability w(x|y) so as to satisfy lim Pt ( x ) P( x ) t where Pt ( x ) wx y Pt -1 ( y) (t 1,2,3,) y P1(x), P2(x), P3(x),…: Markov Process Physical Fluctuomatics (Tohoku University) 14 Markov Chain Monte Carlo Method xt - 1 wxt xt - 1 x t Reject x[1] x[2] x[t 1] x[t 2] x[( N - 1)t 1] x[( N - 2)t 1] Accuracy 1/2 O(1/t ) x[t ] x[2t ] x[ Nt 1] They can be regarded as samples from the given probability distribution P(x). Physical Fluctuomatics (Tohoku University) Randomly generated For sufficient large t, x[t], x[2t], x[3t], …, x[Nt] are independent of each other How large number t ? t: relaxation time 15 Markov Chain Monte Carlo Method x[1] x[2] x[t 1] x[t 2] x[( N - 1)t 1] x[( N - 1)t 2] P1 X 1 x[t ] x[2t ] x[ Nt ] P X 1 , X 2 , X 3 , , X L X2 X3 N XL Histgram 1 x1 nt , X 1 N n 1 Marginal Probability Distribution Physical Fluctuomatics (Tohoku University) Xi 16 Markov Chain Monte Carlo Method P( x ) P( x1 , x2 ,, xL ) ij ( xi , x j ) {i , j }E P x1 , x2 , , xi -1 , xi , xi 1 ,, xL Pxi x1 , x2 , , xi -1 , xi 1 ,, xL Px1 , x2 , , xi -1 , xi 1 ,, xL Px1 , x2 ,, xi -1 , xi 1 ,, xL Px1 , x2 ,, xi -1 , zi , xi 1 ,, xL zi V:Set of all the nodes E:Set of all the neighbouring pairs of nodes Physical Fluctuomatics (Tohoku University) 17 Markov Chain Monte Carlo Method P( x ) P( x1 , x2 ,, xL ) {i , j } ( xi , x j ) {i , j }E Pxi x1 , x2 ,, xi -1 , xi 1 ,, xL x , x Px x z , x {i , j } j ji i {i , j } zi i j j i j ji (V , E ) L | V | i (V , E ) Markov Random Field E:Set of all the neighbouring pairs of nodes ∂i:Set of all the neighbouring nodes of the node i Physical Fluctuomatics (Tohoku University) 18 Markov Chain Monte Carlo Method P( x ) P( x1 , x2 ,, xL ) {i , j } ( xi , x j ) {i , j }E wx1, x2 ,, xL x1 , x2 ,, xL xk xk k i (V , E ) x, x ( x , x ) z , x {i , j } i j ji iV k k kV / i {i , j } zi i j ji wx ' x Px wx x' Px' xt - 1 wxt xt - 1 Physical Fluctuomatics (Tohoku University) xt 19 Markov Chain Monte Carlo Method wxt xt - 1 x[t - 1] xt True xi = ○ or V x False ● V x’ Physical Fluctuomatics (Tohoku University) 20 Sampling by Markov Chain Monte Carlo Method p p Small p Large p Disordered State Ordered State Sampling by Markov Chain Monte Carlo Method More is different. Near Critical Point of p Physical Fluctuomatics (Tohoku University) 21 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 Talks 9th Belief propagation 10th Probabilistic image processing by means of physical models 11th Bayesian network and belief propagation in statistical inference Physical Fluctuomatics (Tohoku University) 22 Practice 8-1 When the probability distribution P(x) and the transition probability w(x’|x) satisfy the detailed balance wx' x P x wx x' P x' where wx x 1 , prove that P x wx x' P x' x x' Physical Fluctuomatics (Tohoku University) 23 Practice 8-2 Let us consider that the transition matrix of the present stochastic process is given as 1 2 1 Pt (0) Pt -1 (0) W , where W 3 1 2 Pt (1) Pt -1 (1) Find the limit distribution defined by Pt (0) P(0) lim P(1) t Pt (1) Physical Fluctuomatics (Tohoku University) 24 Practice 8-3 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,…,Q-1} . The joint probability distribution of the provability vector F=(F1,F2,…,FL)T is given as 1 2 PrF1 f1 , F2 f 2 ,, FL f L exp - a fi - f j Z Prior 2 {i , j}E 1 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 a, give numerical experiments. Example of generated random vector in the case of Q=256, a=0.0005 Ly L=Lx×Ly Example of generated random vector in the case of Q=2, a=2 Lx Physical Fluctuomatics (Tohoku University) 25