核心数学原理

贝叶斯定理

P(y|X) = [P(X|y)P(y)] / P(X)

高斯分布

P(x_i|y) = (1/√(2πσ²))e^(-(x_i-μ)²/(2σ²))

Python实现示例

              
# 高斯朴素贝叶斯分类器实现
import numpy as np

class GaussianNB:
    def fit(self, X, y):
        self.classes = np.unique(y)
        self.parameters = []
        for i in self.classes:
            X_class = X[y == i]
            self.parameters.append({
                'mean': X_class.mean(axis=0),
                'var': X_class.var(axis=0)
            })
    
    def _pdf(self, X, mean, var):
        eps = 1e-4
        return np.exp(-(X - mean)**2 / (2 * var + eps)) / np.sqrt(2 * np.pi * var + eps)
    
    def predict(self, X):
        posteriors = []
        for params in self.parameters:
            likelihood = np.sum(np.log(self._pdf(X, params['mean'], params['var'])), axis=1)
            posteriors.append(likelihood)
        return self.classes[np.argmax(posteriors, axis=0)]

分类边界可视化