Metropolis-Hasting Algorithm

one-dimension.py

# coding:utf-8
# one-dimensional Metropolis-Hasting Algorithm
from __future__ import division

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.mlab as mlab

def q(x):
    return mlab.normpdf(x, 0, 2)

N = 100000
s = 10
x = 0
p = q(x)                                # p(0)
samples = np.zeros(N // s)
for i in range(N):
    xn = x + np.random.normal()         # get j
    pn = q(xn)                          # get p(j)
    if pn >= p:                         # it means p(j) >= p(i),
        p = pn                          # also means alpha >= 1
        x = xn
    else:                               # it means p(j) < p(i), also means alpha < 1
        u = np.random.rand()            # u ~ Uniform(0,1)
        if u < pn/p:                    # u < alpha, accept change
            p = pn
            x = xn
    if i % s == 0:
        samples[i // s] = x

plt.hist(samples, bins='auto', normed=True)

dx = 0.01
x = np.arange(np.min(samples), np.max(samples), dx)
y = q(x)
plt.plot(x, y)
plt.show()

two-dimension.py

# coding:utf-8
# two-dimensional Metropolis-Hasting Algorithm
from __future__ import division

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.mlab as mlab

def q(x, y):
    g1 = mlab.bivariate_normal(x, y, 1.0, 1.0, -1, -1, -0.8)
    g2 = mlab.bivariate_normal(x, y, 1.5, 0.8, 1, 2, 0.6)
    return 0.6*g1+28.4*g2/(0.6+28.4)

N = 100000
s = 10
r = np.zeros(2)
p = q(r[0], r[1])                       # p(0)
samples = np.zeros(shape=(N // s, 2))
for i in range(N):
    rn = r + np.random.normal(size=2)   # get j
    pn = q(rn[0], rn[1])                # get p(j)
    if pn >= p:                         # it means p(j) >= p(i), also means alpha >= 1
        p = pn
        r = rn
    else:                               # it means p(j) < p(i), also means alpha < 1
        u = np.random.rand()            # u ~ Uniform(0,1)
        if u < pn/p:                    # u < alpha, accept change
            p = pn
            r = rn
    if i % s == 0:
        samples[i // s, :] = r

plt.scatter(samples[:, 0], samples[:, 1], alpha=0.5, s=1)

dx = 0.01
x = np.arange(np.min(samples), np.max(samples), dx)
y = np.arange(np.min(samples), np.max(samples), dx)
X, Y = np.meshgrid(x, y)
Z = q(X, Y)
CS = plt.contour(X, Y, Z, 10)
plt.clabel(CS, inline=1, fontsize=10)
plt.show()

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