ClawLab
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MathLab


			
				
					
						
						
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    <title>5 - 隐马尔可夫模型</title>
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    <h1>👾 Day 5: 隐马尔可夫模型</h1>
    
    <div class="module">
        <div class="module-title">1️⃣【技术债与演进动机】The Technical Debt & Evolution</div>
        传统序列模型无法处理隐藏状态。HMM 引入不可观测的状态序列，通过观测序列推断隐藏状态。
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        <div class="module-title">2️⃣【直觉建立】Visual Intuition</div>
        想象通过天气（晴/雨）推断草地干湿状态：天气是隐藏状态，草地干湿是观测，转移概率决定状态变化。
        <div class="youtube">🎬 B 站搜索：<code>隐马尔可夫模型 直观解释</code></div>
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        <div class="module-title">3️⃣【符号解码字典】The Symbol Decoder</div>
        
        <div class="symbol-map"><strong>$\pi$</strong> → <code>self.initial_probs</code> (初始状态概率)</div>
        <div class="symbol-map"><strong>$A$</strong> → <code>self.transition_probs</code> (状态转移矩阵)</div>
        <div class="symbol-map"><strong>$B$</strong> → <code>self.emission_probs</code> (观测发射矩阵)</div>
        <div class="symbol-map"><strong>$O$</strong> → <code>observations</code> (观测序列)</div>
        <div class="symbol-map"><strong>$Q$</strong> → <code>states</code> (隐藏状态序列)</div>
        
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        <div class="module-title">4️⃣【核心推导】The Math</div>
        
### 三大基本问题

**1. 概率计算问题**（前向算法）

$$\alpha_t(i) = P(o_1, o_2, ..., o_t, q_t = S_i | \lambda)$$

递推公式：
$$\alpha_{t+1}(i) = \left[ \sum_{j=1}^{N} \alpha_t(j) a_{ji} \right] b_i(o_{t+1})$$

**2. 学习问题**（Baum-Welch 算法）

$$\gamma_t(i) = P(q_t = S_i | O, \lambda)$$

$$\xi_t(i, j) = P(q_t = S_i, q_{t+1} = S_j | O, \lambda)$$

**3. 预测问题**（Viterbi 算法）

$$\delta_t(i) = \max_{q_1, ..., q_{t-1}} P(q_1, ..., q_t=i, o_1, ..., o_t | \lambda)$$

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        <div class="module-title">5️⃣【工程优化点】The Optimization Bottleneck</div>
        前向 - 后向算法需要 $O(N^2 \cdot T)$ 计算，Viterbi 解码需要 $O(N^2 \cdot T)$ 动态规划。
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    <div class="module">
        <div class="module-title">6️⃣【今日靶机】The OJ Mission</div>
        <div class="warning">🎯 任务：<code>cd exercises/ && python3 day5_task.py</code></div>
        实现 HMM 的前向算法计算观测概率，在中文分词数据集上验证 Viterbi 解码效果。
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