ClawLab
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MathLab


			
				
					
						
						
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    <title>4 - EM 算法</title>
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    <h1>👾 Day 4: EM 算法</h1>
    
    <div class="module">
        <div class="module-title">1️⃣【技术债与演进动机】The Technical Debt & Evolution</div>
        监督学习需要完整标注数据，但现实中很多数据缺失或隐变量未知。EM 算法通过迭代估计隐变量和参数。
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        <div class="module-title">2️⃣【直觉建立】Visual Intuition</div>
        想象混合高斯模型：E 步猜测每个点属于哪个高斯，M 步根据猜测更新高斯参数，循环迭代直到收敛。
        <div class="youtube">🎬 B 站搜索：<code>EM 算法 直观解释</code></div>
    </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>$\theta$</strong> → <code>self.params</code> (模型参数)</div>
        <div class="symbol-map"><strong>$z$</strong> → <code>latent_vars</code> (隐变量)</div>
        <div class="symbol-map"><strong>$Q(\theta | \theta^{(t)})$</strong> → <code>Q_function</code> (期望下界)</div>
        <div class="symbol-map"><strong>$\mathcal{L}(\theta)$</strong> → <code>log_likelihood</code> (对数似然)</div>
        
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        <div class="module-title">4️⃣【核心推导】The Math</div>
        
### 对数似然函数

$$\mathcal{L}(\theta) = \log P(X | \theta) = \log \sum_{Z} P(X, Z | \theta)$$

### E 步：构造下界

$$\mathcal{L}(\theta) \geq Q(\theta | \theta^{(t)}) = \sum_{Z} P(Z | X, \theta^{(t)}) \log \frac{P(X, Z | \theta)}{P(Z | X, \theta^{(t)})}$$

### M 步：最大化下界

$$\theta^{(t+1)} = \text{argmax}_{\theta} Q(\theta | \theta^{(t)})$$

### 迭代收敛

$$\mathcal{L}(\theta^{(t+1)}) \geq \mathcal{L}(\theta^{(t)})$$

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        <div class="module-title">5️⃣【工程优化点】The Optimization Bottleneck</div>
        每次迭代需要遍历所有样本计算期望，复杂度 $O(N \cdot K \cdot I)$，K 为隐变量数，I 为迭代次数。
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        <div class="module-title">6️⃣【今日靶机】The OJ Mission</div>
        <div class="warning">🎯 任务：<code>cd exercises/ && python3 day4_task.py</code></div>
        实现 EM 算法拟合混合高斯分布，在双峰数据上验证参数收敛过程。
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