多元正态分布下的贝叶斯决策——判别分析分类器的训练

判别分析分类器的训练设xi(k){\boldsymbol x}_i^{\left(k\right)}xi(k)​从第kkk个高斯成分生成的样本NkN_kNk​是从第kkk个高斯成分生成的样本个数。第kkk个多元正态分布参数的极大似然估计μ^k1Nk∑i1Nkxi(k) \hat{\boldsymbol {\mu}}_k \dfrac{1}{N_k} \sum\limits_{i1}^{N_k} {\boldsymbol x}_i^{\left(k\right)}μ^​k​Nk​1​i1∑Nk​​xi(k)​Σ^k1Nk∑i1Nk(xi(k)−μk)(xi(k)−μk)⊤ \hat{\boldsymbol{\varSigma}}_k \dfrac{1}{N_k} \sum\limits_{i1}^{N_k} \left({\boldsymbol x}_i^{\left(k\right)} - {\boldsymbol {\mu}}_k\right)\left({\boldsymbol x}_i^{\left(k\right)} - {\boldsymbol {\mu}}_k\right)^\topΣ^k​Nk​1​i1∑Nk​​(xi(k)​−μk​)(xi(k)​−μk​)⊤式中μk{\boldsymbol {\mu}}_kμk​为第kkk类的样本均值Σk{{\boldsymbol \varSigma}}_kΣk​为第kkk类的样本协方差矩阵。多项分布的极大似然估计P^(Ck)NkN \hat P \left( C_k \right) \dfrac{N_k}{N}P^(Ck​)NNk​​P(Ck)P \left( C_k \right)P(Ck​)的估计是第kkk类样本的频率。