本篇文章对隐马尔可夫模型的前向和后向算法进行了Python实现，并且每种算法都给出了循环和递归两种方式的实现。

前向算法Python实现

循环方式

import numpy as np
def hmm_forward(Q, V, A, B, pi, T, O, p):
  """
  :param Q: 状态集合
  :param V: 观测集合
  :param A: 状态转移概率矩阵
  :param B: 观测概率矩阵
  :param pi: 初始概率分布
  :param T: 观测序列和状态序列的长度
  :param O: 观测序列
  :param p: 存储各个状态的前向概率的列表，初始为空
  """
  for t in range(T):
    # 计算初值
    if t == 0:
      for i in range(len(Q)):
        p.append(pi[i] * B[i, V[O[0]]])
    # 初值计算完毕后，进行下一时刻的递推运算
    else:
      alpha_t_ = 0
      alpha_t_t = []
      for i in range(len(Q)):
        for j in range(len(Q)):
          alpha_t_ += p[j] * A[j, i]
        alpha_t_t.append(alpha_t_ * B[i, V[O[t]]])
        alpha_t_ = 0
      p = alpha_t_t
  return sum(p)
# 《统计学习方法》书上例10.2
Q = [1, 2, 3]
V = {'红':0, '白':1}
A = np.array([[0.5, 0.2, 0.3], [0.3, 0.5, 0.2], [0.2, 0.3, 0.5]])
B = np.array([[0.5, 0.5], [0.4, 0.6], [0.7, 0.3]])
pi = [0.2, 0.4, 0.4]
T = 3
O = ['红', '白', '红']
p = []
print(hmm_forward(Q, V, A, B, pi, T, O, p)) # 0.130218

递归方式

import numpy as np
def hmm_forward_(Q, V, A, B, pi, T, O, p, T_final):
  """
  :param T_final:递归的终止条件
  """
  if T == 0:
    for i in range(len(Q)):
      p.append(pi[i] * B[i, V[O[0]]])
  else:
    alpha_t_ = 0
    alpha_t_t = []
    for i in range(len(Q)):
      for j in range(len(Q)):
        alpha_t_ += p[j] * A[j, i]
      alpha_t_t.append(alpha_t_ * B[i, V[O[T]]])
      alpha_t_ = 0
    p = alpha_t_t
  if T >= T_final:
    return sum(p)
  return hmm_forward_(Q, V, A, B, pi, T+1, O, p, T_final)

Q = [1, 2, 3]
V = {'红':0, '白':1}
A = np.array([[0.5, 0.2, 0.3], [0.3, 0.5, 0.2], [0.2, 0.3, 0.5]])
B = np.array([[0.5, 0.5], [0.4, 0.6], [0.7, 0.3]])
pi = [0.2, 0.4, 0.4]
T = 0
O = ['红', '白', '红']
p = []
T_final = 2 # T的长度是3，T的取值是(0时刻, 1时刻, 2时刻)
print(hmm_forward_(Q, V, A, B, pi, T, O, p, T_final))

后向算法Python实现

循环方式

import numpy as np
def hmm_backward(Q, V, A, B, pi, T, O, beta_t, T_final):
  for t in range(T, -1, -1):
    if t == T_final:
      beta_t = beta_t
    else:
      beta_t_ = 0
      beta_t_t = []
      for i in range(len(Q)):
        for j in range(len(Q)):
          beta_t_ += A[i, j] * B[j, V[O[t + 1]]] * beta_t[j]
        beta_t_t.append(beta_t_)
        beta_t_ = 0
      beta_t = beta_t_t
    if t == 0:
      p=[]
      for i in range(len(Q)):
        p.append(pi[i] * B[i, V[O[0]]] * beta_t[i])
      beta_t = p
  return sum(beta_t)
# 《统计学习方法》课后题10.1
Q = [1, 2, 3]
V = {'红':0, '白':1}
A = np.array([[0.5, 0.2, 0.3], [0.3, 0.5, 0.2], [0.2, 0.3, 0.5]])
B = np.array([[0.5, 0.5], [0.4, 0.6], [0.7, 0.3]])
pi = [0.2, 0.4, 0.4]
T = 3
O = ['红', '白', '红', '白']
beta_t = [1, 1, 1]
T_final = 3
print(hmm_backward_(Q, V, A, B, pi, T, O, beta_t, T_final)) # 0.06009