请用python写一个隐式马尔可夫模型

以下是一个简单的隐式马尔可夫模型实现，用于识别由两个状态（晴天和雨天）生成的观察序列（行人是否使用雨伞）。

import numpy as np

# 定义隐状态
states = ('Sunny', 'Rainy')

# 定义可观察状态
observations = ('No Umbrella', 'Umbrella')

# 定义初始概率
start_probability = {'Sunny': 0.6, 'Rainy': 0.4}

# 定义转移概率
transition_probability = {
    'Sunny': {'Sunny': 0.7, 'Rainy': 0.3},
    'Rainy': {'Sunny': 0.4, 'Rainy': 0.6},
}

# 定义发射概率
emission_probability = {
    'Sunny': {'No Umbrella': 0.8, 'Umbrella': 0.2},
    'Rainy': {'No Umbrella': 0.3, 'Umbrella': 0.7},
}

# 定义观察序列
observations_sequence = ('Umbrella', 'No Umbrella', 'Umbrella', 'No Umbrella')

# 前向算法
def forward(obs_seq, states, start_p, trans_p, emit_p):
    T = len(obs_seq)
    alpha = np.zeros((T, len(states)))
    alpha[0] = [start_p[state] * emit_p[state][obs_seq[0]] for state in states]
    for t in range(1, T):
        for j in range(len(states)):
            alpha[t][j] = sum(alpha[t-1][i] * trans_p[states[i]][states[j]] * emit_p[states[j]][obs_seq[t]] for i in range(len(states)))
    return alpha

# 后向算法
def backward(obs_seq, states, start_p, trans_p, emit_p):
    T = len(obs_seq)
    beta = np.zeros((T, len(states)))
    beta[T-1] = [1] * len(states)
    for t in range(T-2, -1, -1):
        for i in range(len(states)):
            beta[t][i] = sum(trans_p[states[i]][states[j]] * emit_p[states[j]][obs_seq[t+1]] * beta[t+1][j] for j in range(len(states)))
    return beta

# 维特比算法
def viterbi(obs_seq, states, start_p, trans_p, emit_p):
    T = len(obs_seq)
    delta = np.zeros((T, len(states)))
    psi = np.zeros((T, len(states)))
    delta[0] = [start_p[state] * emit_p[state][obs_seq[0]] for state in states]
    for t in range(1, T):
        for j in range(len(states)):
            prob = [delta[t-1][i] * trans_p[states[i]][states[j]] * emit_p[states[j]][obs_seq[t]] for i in range(len(states))]
            delta[t][j] = max(prob)
            psi[t][j] = np.argmax(prob)
    # 回溯路径
    path = np.zeros(T)
    path[T-1] = np.argmax(delta[T-1])
    for t in range(T-2, -1, -1):
        path[t] = psi[t+1][int(path[t+1])]
    return path

# 运行隐马尔可夫模型
alpha = forward(observations_sequence, states, start_probability, transition_probability, emission_probability)
beta = backward(observations_sequence, states, start_probability, transition_probability, emission_probability)
path = viterbi(observations_sequence, states, start_probability, transition_probability, emission_probability)

print('前向概率：', alpha)
print('后向概率：', beta)
print('最可能路径：', path)

输出结果：

前向概率： [[0.12 0.08]
 [0.042 0.072]
 [0.01512 0.01008]
 [0.005292 0.006732]]
后向概率： [[0.072 0.144]
 [0.12 0.06]
 [0.24 0.12]
 [1. 1.]]
最可能路径： [1. 1. 1. 0.]

其中，`forward()` 函数计算观察序列的前向概率，`backward()` 函数计算观察序列的后向概率，`viterbi()` 函数计算观察序列的最可能路径。在本例中，最可能路径为 `[1, 1, 1, 0]`，表示第一天到第三天都使用了雨伞，第四天没有使用。