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内容目录：MCMC（Markov Chain Monte Carlo）的理解与实践（Python）
Markov Chain Monte Carlo (MCMC) methods are a class of
algorithms for sampling from a probability distribution
based on constructing a Markov chain that has the desired
distribution as its stationary distribution. The state of
the chain after a number of steps is then used as a sample
of the desired distribution. The quality of the sample
improves as a function of the number of steps.
With MCMC, we draw samples from a (simple) proposal
distribution so that each draw depends only on the state of
the previous draw (i.e. the samples form a Markov chain,
θp=θ+Δθ,Δθ∼N(0,σ)θp=θ+Δθ,Δθ∼N(0,σ)).
MCMC（Markov Chain Monte Carlo）用MCMC采样算法实现对Beta 分布的采样
MCMC（Markov Chain Monte Carlo）
首先来看经典的MCMC采样算法：
用MCMC采样算法实现对Beta 分布的采样
关于Beta distribution更详尽的内容请参见 Beta函数与Gamma函数及其与Beta分布的关系。已知Beta distribution的概率密度函数（pdf）为：
import numpy as np
import scipy.special as ss
import matplotlib.pyplot as plt
def beta_s(x, a, b):
return x**(a-1)*(1-x)**(b-1)
def beta(x, a, b):
return beta_s(x, a, b)/ss.beta(a, b)
def plot_mcmc(a, b):
cur = np.random.rand()
states = [cur]
for i in range(10**5):
next, u = np.random.rand()
if u < np.min((beta_s(next, a, b)/beta_s(cur, a, b), 1)):
states.append(next)
cur = next
x = np.arange(0, 1, .01)
plt.figure(figsize=(10, 5))
plt.plot(x, beta(x, a, b), lw=2, label='real dist: a={}, b={}'.format(a, b))
plt.hist(states[-1000:], 25, normed=True, label='simu mcmc: a={}, b={}'.format(a, b))
plt.show()
if __name__ == '__main__':
plot_mcmc(0.1, 0.1)
plot_mcmc(1, 1)
plot_mcmc(2, 3)
