随机采样系列4：MCMC

参考资料：

1、http://www.52nlp.cn/lda-math-mcmc-%e5%92%8c-gibbs-sampling2

2、http://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm

3、http://www.quantiphile.com/2010/11/01/metropolis-hastings/

python脚本如下：

import math
import time
import numpy as np
import matplotlib.pylab as plt
import random
class Samples:
    def __init__(self):
        pass
    def mh(self, epsilon_0, num_iteration, fpdf):
        #Metropolis–Hastings algorithm
        normal_randoms = np.zeros(num_iteration)
        uniform_randoms = np.zeros(num_iteration)
        for i in range(0, num_iteration):
            uniform_randoms[i] = random.uniform(0, 1)
            normal_randoms[i] = random.normalvariate(0, 1)
        #fig = plt.figure()
        #ax = fig.add_subplot(211)
        #ax.plot(normal_randoms, '.')
        #ax1 = fig.add_subplot(212)
        #ax1.plot(uniform_randoms, '.')
        #plt.show()
        
        epsilon = np.zeros(num_iteration)
        previous_epsilon = epsilon_0
        for i in range(0, num_iteration):
            epsilon_tilde = previous_epsilon + normal_randoms[i]
            if(fpdf(epsilon_tilde) > fpdf(previous_epsilon)):
                epsilon[i] = epsilon_tilde
            else:
                if(uniform_randoms[i] <= fpdf(epsilon_tilde) / fpdf(previous_epsilon)):
                    epsilon[i] = epsilon_tilde
                else:
                    epsilon[i] = previous_epsilon
            previous_epsilon = epsilon[i]
        return epsilon
    
    def mh1(self, epsilon_0, num_iteration, fpdf):
        #Metropolis–Hastings algorithm
        normal_randoms = np.zeros(num_iteration)
        uniform_randoms = np.zeros(num_iteration)
        for i in range(0, num_iteration):
            uniform_randoms[i] = random.uniform(0, 1)
            normal_randoms[i] = random.normalvariate(0, 1)
        
        epsilon = np.zeros(num_iteration)
        previous_epsilon = epsilon_0
        for i in range(0, num_iteration):
            epsilon_tilde = previous_epsilon + normal_randoms[i]
            rate = fpdf(epsilon_tilde) / fpdf(previous_epsilon)
            alfa = min(rate, 1.0)
            if(uniform_randoms[i] < alfa):
                epsilon[i] = epsilon_tilde
            else:
                epsilon[i] = previous_epsilon
            previous_epsilon = epsilon[i]
        return epsilon
def nor(x):
    return (1.0/np.sqrt(2.0*np.pi))*np.exp(-np.power(x,2)/2)
if __name__=='__main__':
    s = Samples()
    x0 = s.mh(0, 5000, nor)
    x = s.mh1(0, 5000, nor)
    fig = plt.figure()
    ax = fig.add_subplot(211)
    ax.hist(x0,200)
    ax = fig.add_subplot(212)
    ax.hist(x, 200)
    plt.show()
    s.acceptanceRejection('normal')

样本的直方图为：