用Python实现em算法并回溯最优状态序列的代码

下面是使用Python实现EM算法并回溯最优状态序列的示例代码：

import numpy as np

def forward_backward(obs, states, start_p, trans_p, emit_p):
    # Forward probability
    forward_prob = [{}]
    for state in states:
        forward_prob[0][state] = start_p[state] * emit_p[state][obs[0]]
    for t in range(1, len(obs)):
        forward_prob.append({})
        for state in states:
            forward_prob[t][state] = sum(forward_prob[t-1][prev_state] * trans_p[prev_state][state] * emit_p[state][obs[t]] for prev_state in states)
    
    # Backward probability
    backward_prob = [{} for _ in obs]
    for state in states:
        backward_prob[-1][state] = 1
    for t in range(len(obs)-2, -1, -1):
        for state in states:
            backward_prob[t][state] = sum(trans_p[state][next_state] * emit_p[next_state][obs[t+1]] * backward_prob[t+1][next_state] for next_state in states)
    
    # Posterior probability
    posterior_prob = [{} for _ in obs]
    total_prob = sum(forward_prob[-1].values())
    for t in range(len(obs)):
        for state in states:
            posterior_prob[t][state] = forward_prob[t][state] * backward_prob[t][state] / total_prob
    
    return forward_prob, backward_prob, posterior_prob

def viterbi(obs, states, start_p, trans_p, emit_p):
    # Viterbi algorithm
    viterbi_prob = [{}]
    path = {}
    for state in states:
        viterbi_prob[0][state] = start_p[state] * emit_p[state][obs[0]]
        path[state] = [state]
    for t in range(1, len(obs)):
        viterbi_prob.append({})
        new_path = {}
        for state in states:
            (prob, state_) = max((viterbi_prob[t-1][prev_state] * trans_p[prev_state][state] * emit_p[state][obs[t]], prev_state) for prev_state in states)
            viterbi_prob[t][state] = prob
            new_path[state] = path[state_] + [state]
        path = new_path
    
    # Find the optimal path
    (prob, state) = max((viterbi_prob[-1][state], state) for state in states)
    return prob, path[state]

def em_algorithm(obs, states, start_p, trans_p_init, emit_p_init, epsilon=0.001, max_iter=100):
    trans_p = trans_p_init.copy()
    emit_p = emit_p_init.copy()
    for i in range(max_iter):
        # E step
        forward_prob, backward_prob, posterior_prob = forward_backward(obs, states, start_p, trans_p, emit_p)
        
        # M step
        start_p = {state: posterior_prob[0][state] for state in states}
        for state in states:
            for next_state in states:
                trans_p[state][next_state] = sum(posterior_prob[t][state] * trans_p[state][next_state] * emit_p[next_state][obs[t+1]] * backward_prob[t+1][next_state] for t in range(len(obs)-1)) / sum(posterior_prob[t][state] * backward_prob[t][state] for t in range(len(obs)-1))
            for observation in set(obs):
                emit_p[state][observation] = sum(posterior_prob[t][state] for t in range(len(obs)) if obs[t] == observation) / sum(posterior_prob[t][state] for t in range(len(obs))))
        
        # Check for convergence
        log_likelihoods = [np.log(np.sum([forward_prob[t][state] * backward_prob[t][state] for state in states])) for t in range(len(obs))]
        log_likelihood = np.sum(log_likelihoods)
        if i > 0 and abs(log_likelihood - prev_log_likelihood) < epsilon:
            break
        prev_log_likelihood = log_likelihood
    
    # Find the optimal state sequence
    prob, state_seq = viterbi(obs, states, start_p, trans_p, emit_p)
    return prob, state_seq

其中，`obs`表示观测序列，`states`表示状态集合，`start_p`表示状态初始概率分布，`trans_p`表示状态转移概率矩阵，`emit_p`表示状态发射概率矩阵。`forward_backward`函数实现前向-后向算法，返回前向概率、后向概率和后验概率；`viterbi`函数实现维特比算法，返回最大概率和最优状态序列；`em_algorithm`函数实现EM算法，返回最大似然和最优状态序列。