ClawLab
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MathLab


			
				
					
						
						
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    <title>3 - 朴素贝叶斯</title>
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    <h1>👾 Day 3: 朴素贝叶斯</h1>
    
    <div class="module">
        <div class="module-title">1️⃣【技术债与演进动机】The Technical Debt & Evolution</div>
        传统分类器需要估计联合概率分布，参数多且易过拟合。朴素贝叶斯通过条件独立假设简化模型。
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        <div class="module-title">2️⃣【直觉建立】Visual Intuition</div>
        想象判断一封邮件是否是垃圾邮件：根据关键词出现的概率，结合先验经验，计算后验概率。
        <div class="youtube">🎬 B 站搜索：<code>朴素贝叶斯 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>$P(c_k)$</strong> → <code>self.class_priors_[k]</code> (类别先验概率)</div>
        <div class="symbol-map"><strong>$P(x^{(j)} | c_k)$</strong> → <code>self.feature_probs_[k, j]</code> (条件概率)</div>
        <div class="symbol-map"><strong>$P(c_k | x)$</strong> → <code>posterior</code> (后验概率)</div>
        <div class="symbol-map"><strong>$\alpha$</strong> → <code>alpha</code> (拉普拉斯平滑系数)</div>
        
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        <div class="module-title">4️⃣【核心推导】The Math</div>
        
### 贝叶斯公式

$$P(c_k | x) = \frac{P(x | c_k) P(c_k)}{P(x)}$$

### 朴素条件独立假设

$$P(x | c_k) = P(x^{(1)}, x^{(2)}, ..., x^{(d)} | c_k) = \prod_{j=1}^{d} P(x^{(j)} | c_k)$$

### 后验概率最大化

$$\hat{y} = \text{argmax}_{c_k} P(c_k) \prod_{j=1}^{d} P(x^{(j)} | c_k)$$

### 拉普拉斯平滑

$$P(x^{(j)} = v | c_k) = \frac{\sum_{i=1}^{N} \mathbb{I}(x^{(i)(j)} = v, y^{(i)} = c_k) + \alpha}{\sum_{i=1}^{N} \mathbb{I}(y^{(i)} = c_k) + \alpha \cdot V}$$

其中 $V$ 是特征取值数量，$\alpha$ 是平滑系数。

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        <div class="module-title">5️⃣【工程优化点】The Optimization Bottleneck</div>
        需要统计所有特征 - 类别组合的频率，高维数据下计算量大。使用哈希表或稀疏矩阵优化。
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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 day3_task.py</code></div>
        实现高斯朴素贝叶斯的 fit 和 predict 函数，在鸢尾花数据集上验证分类效果。
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