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**Graduate School of Information Sciences, Tohoku University**

Physical Fluctuomatics Applied Stochastic Process 8th Markov chain Monte Carlo methods Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**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)

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**What is an important point in computational complexity?**

How should we treat the calculation of the summation over 2N configuration? 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. N fold loops Markov Chain Monte Carlo Metod Belief Propagation Method This Talk Next Talk Physical Fuctuomatics (Tohoku University)

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**Generate uniform random numbers a and b in the interval [-1,1]**

Calculation of the ratio of the circumference of a circle to its diameter by using random numbers (Monte Carlo Method) 1 n0 m0 -1 1 nn+1 -1 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 a2+b2≤1 No the ratio of the circumference of a circle to its diameter Yes mm+1 Accuracy is improved as the number of trials increases Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Physics Fluctuomatics/Applied Stochastic Process (Tohoku University)**

Law of Large Numbers Let us suppose that random variables X1,X2,...,Xn are identical and mutual independent random variables with average m. Then we have Central Limit Theorem We consider a sequence of independent, identical distributed random variables, {X1,X2,...,Xn}, with average m and variance s2. Then the distribution of the random variable tends to the Gauss distribution with average m and variance s2/n as n+. Physics Fluctuomatics/Applied Stochastic Process (Tohoku University)

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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. -1 1 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. -1 Count the number of points m inside of the unit circle after plotting points n randomly the ratio of the circumference of a circle to its diameter Order of the error of the ratio of the circumference of a circle to its diameter is O(1/n1/2) Accuracy is improved as the number of trials increases Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Integral Computation by Monte Carlo Method**

1 n0 m0 nn+1 -1 1 Generate uniform random numbers a and b in the interval [-1,1] -1 Compute the value of f(x,y) at each point (x,y) after plotting points n inside of the green region randomly mm + f(a,b) Accuracy is O(1/n1/2) Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Marginal Probability Marginal Probability**

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**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 Computational time generating one random numbers should be order of |V|. Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Fundamental Stochastic Process: Markov Process**

Transition Probability w(x|y)≥0 (x,y=0,1) For any initial distribution P0(x), Transition Matrix Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Fundamental Stichastic Process: Markov Chain**

Transition matrix can be diagonalized as Limit Distribution Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Fundamental Stochastic Process: Markov Process**

Stationary Distribution or Equilibrium Distribution 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 The stationary distribution is Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Stationary Process and Detailed Balance in Markov Process**

where P1(x), P2(x), P3(x),…: Markov Chain 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). Stationary Distribution of Markov Process Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Markov Chain Monte Carlo Method**

Let us consider a joint probability distribution P(x1,x2,…,xL) 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)

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**Markov Chain Monte Carlo Method**

Randomly generated For sufficient large t, x[t], x[2t], x[3t], …, x[Nt] are independent of each other Reject How large number t ? t: relaxation time They can be regarded as samples from the given probability distribution P(x). Accuracy O(1/t1/2) Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Markov Chain Monte Carlo Method**

Histgram Xi Marginal Probability Distribution Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Markov Chain Monte Carlo Method**

V：Set of all the nodes E：Set of all the neighbouring pairs of nodes Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Markov Chain Monte Carlo Method**

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 / Applied Stochastic Process (Tohoku University)

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**Markov Chain Monte Carlo Method**

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Markov Chain Monte Carlo Method**

x True False xi = ○ or ● x’ Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Sampling by Markov Chain Monte Carlo Method**

Small p Large p Sampling by Markov Chain Monte Carlo Method Disordered State Ordered State More is different. Near Critical Point of p Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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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 / Applied Stochastic Process (Tohoku University)

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Practice 8-1 When the probability distribution P(x) and the transition probability w(x’|x) satisfy the detailed balance where , prove that Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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**Find the limit distribution defined by , where**

Practice 8-2 Let us consider that the transition matrix of the present stochastic process is given as Find the limit distribution defined by , where Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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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 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 Example of generated random vector in the case of Q=2, a=2 L=Lx×Ly Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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