MCMC（Markov Chain Monte Carlo）的理解与实践（Python）

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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)，否则不接受

repeated

import numpy as npimport scipy.special as ssimport matplotlib.pyplot as pltdef 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)
    
    

     
     
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