import numpy as np

迭代求平稳分布

转移概率

P = np.array([[1/2, 1/2, 1/4], [1/4, 0, 1/4], [1/4, 1/2, 1/2]])
P

array([[0.5 , 0.5 , 0.25],
       [0.25, 0.  , 0.25],
       [0.25, 0.5 , 0.5 ]])

初始化状态分布

pi = np.random.dirichlet(np.ones(3),size=1).flatten()
pi

array([0.38661374, 0.5182709 , 0.09511536])

迭代

遍历定理证明：通过N次迭代，状态分布趋于平稳

epochs = 15

print(0, pi)
for i in range(epochs):
    pi = np.dot(P, pi)
    print(i, pi)

0 [0.38661374 0.5182709  0.09511536]
0 [0.47622116 0.12043227 0.40334657]
1 [0.39916336 0.21989193 0.38094471]
2 [0.40476382 0.19502702 0.40020916]
3 [0.39994771 0.20124325 0.39880904]
4 [0.40029774 0.19968919 0.40001307]
5 [0.39999673 0.2000777  0.39992557]
6 [0.40001861 0.19998057 0.40000082]
7 [0.3999998  0.20000486 0.39999535]
8 [0.40000116 0.19999879 0.40000005]
9 [0.39999999 0.2000003  0.39999971]
10 [0.40000007 0.19999992 0.4       ]
11 [0.4        0.20000002 0.39999998]
12 [0.4 0.2 0.4]
13 [0.4 0.2 0.4]
14 [0.4 0.2 0.4]