贝叶斯准则——多元假设检验
多元假设检验
当判决结果有KKK种可能时，称为KKK元假设检验问题。通信中经常需要检测KKK个信号中哪个出现，模式识别问题中也经常遇到区分KKK种模式的问题。对于多元假设检验问题，通常采用最小错误概率准则或者贝叶斯准则，尽管纽曼 - 皮尔逊准则同样可应用，但实际中很少采用。
风险函数Lij=L(Ci,Dj)L_{ij}= L \left(C_i , D_j \right)Lij=L(Ci,Dj)
假定对KKK种可能假设{ C1,C2,⋯ ,CK}\{C_1,C_2,\cdots,C_{K}\}{C1,C2,⋯,CK}进行判决，CjC_jCj为真判CiC_iCi成立的代价用LijL_{ij}Lij表示，则平均代价为
R=∑i=1K∑j=1KLijP(Ci,Dj)=∑i=1K∑j=1KLijP(Dj∣Ci)P(Ci)条件概率的乘积公式 \begin{aligned} R=\sum\limits_{i = 1}^{K}\sum\limits_{j = 1}^{K}L_{ij}P\left({C}_i,D_j\right) &=\sum\limits_{i = 1}^{K}\sum\limits_{j = 1}^{K}L_{ij}P\left({D}_j\mid C_i\right)P\left(C_i\right) \text{条件概率的乘积公式}\\ \end{aligned}R=i=1∑Kj=1∑KLijP(Ci,Dj)=i=1∑Kj=1∑KLijP(Dj∣Ci)P(Ci)条件概率的乘积公式
由于
P(Dj∣Ci)=P(X∈Dj∣Ci)=∫Xjp(x∣Ci) dx \begin{aligned} P\left({D}_j \mid {C}_i\right) =P\left( {\boldsymbol X}\in { {\mathcal D}_j} \mid C_i \right)= \int\nolimits_{ {\mathscr X} _j} p\left({\boldsymbol x}\mid {C}_i\right) \, {\rm d}{\boldsymbol x} \end{aligned}P(Dj∣Ci)=P(X∈Dj∣Ci)=∫Xjp(x∣Ci)dx
代入贝叶斯风险RRR
R=∑i=1K∑j=1KLijP(X∈Xj∣Ci)P(Ci)=∑i=1K∑j=1KLij∫Xjp(x∣Ci)dxP(Ci)=∑j=1K∫Xj∑i=1KLijp(x∣Ci)P(Ci)dx（交换i和j求和的顺序）=∑j=1K∫Xj[∑i=1KLijP(Ci∣x)]p(x)dx（p(x∣Ci)P(Ci)=P(Ci∣x)p(x)） \begin{aligned} R&=\sum\limits_{i = 1}^{K}\sum\limits_{j = 1}^{K}L_{ij}P\left(\boldsymbol{X}\in{ {\mathscr X} _j} \mid C_i\right)P\left(C_i\right)\\ &=\sum\limits_{i = 1}^{K}\sum\limits_{j = 1}^{K}L_{ij} \int\nolimits_{ {\mathscr X} _j}p\left(\boldsymbol{x}\mid C_i\right){\rm d}\boldsymbol{x}P\left(C_i\right)\\ &=\sum\limits_{j = 1}^{K}\int\nolimits_{ {\mathscr X} _j}\sum\limits_{i = 1}^{K}L_{ij}p\left(\boldsymbol{x}\mid C_i\right)P\left(C_i\right){\rm d}\boldsymbol{x} &{\text{（交换}i\text{和}j\text{求和的顺序）}}\\ &=\sum\limits_{j = 1}^{K}\int\nolimits_{ {\mathscr X} _j} \left[ \sum\limits_{i = 1}^{K}L_{ij}P\left(C_i\mid \boldsymbol{x}\right) \right]p\left(\boldsymbol{x}\right){\rm d}\boldsymbol{x} &（p\left(\boldsymbol{x}\mid C_i\right)P\left(C_i\right)=P\left(C_i\mid \boldsymbol{x}\right)p\left(\boldsymbol{x}\right)） \end{aligned}R=i=1∑Kj=1∑KLijP(X∈Xj∣Ci)P(Ci)=i=1∑Kj=1∑KLij∫Xj