机器学习5 高斯判别法

如果
y ~ Bernoulli(ϕ)
x|y=1 ~ N(μ0,Σ)
x|y=0 ~ N(μ1,Σ)
则
p(y)=ϕy(1−ϕ)1−y
p(x|y=1)=1(2π)n2|Σ|12exp(−12(x−μ0)TΣ−1(x−μ0))=(1(2π)n2|Σ|12exp(−12(x−μ0)TΣ−1(x−μ0)))y
p(x|y=0)=1(2π)n2|Σ|12exp(−12(x−μ1)TΣ−1(x−μ1))=(1(2π)n2|Σ|12exp(−12(x−μ1)TΣ−1(x−μ1)))1−y
l(ϕ,μ0,μ1,Σ)=log∏i=1mp(x(i),y(i);ϕ,μ0,μ1,Σ)
=log∏i=1mp(x(i)|y(i);μ0,μ1,Σ)p(y(i);ϕ)
=log∏i=1m(1(2π)n2|Σ|12exp(−12(x(i)−μ0)TΣ−1(x(i)−μ0))ϕ)y(i)(1(2π)n2|Σ|12exp(−12(x(i)−μ1)TΣ−1(x(i)−μ1))(1−ϕ))1−y(i)
=∑i=1my(i)log(1(2π)n2|Σ|12exp(−12(x(i)−μ0)TΣ−1(x(i)−μ0))ϕ)+(1−y(i))log(1(2π)n2|Σ|12exp(−12(x(i)−μ1)TΣ−1(x(i)−μ1))(1−ϕ))
=∑i=1my(i)(−12(x(i)−μ0)TΣ−1(x(i)−μ0)+log(1(2π)n2|Σ|12)+log(ϕ))+(1−y(i))(−12(x(i)−μ1)TΣ−1(x(i)−μ1)+log(1(2π)n2|Σ|12)+log(1−ϕ))
=∑i=1mlog(1(2π)n2|Σ|12)+log(1−ϕ)+y(i)(log(ϕ)−log(1−ϕ))+y(i)(−12(x(i)−μ0)TΣ−1(x(i)−μ0))+(1−y(i))(−12(x(i)−μ1)TΣ−1(x(i)−μ1))
⇒
∂l∂ϕ=∑i=1m−11−ϕ+y(i)(1ϕ(1−ϕ))
=∑i=1m11−ϕ(−1+y(i)1ϕ)
=11−ϕ⎛⎝⎜−m+∑i=1m(y(i))ϕ⎞⎠⎟==0
⇒ϕ=∑i=1m(y(i))m
先证
d=(A−B)TC(A−B)
∇Bd=∇Btr(d)=∇Btr(−ATCB−BTCA+BTCB)=(−CTA−CA+CB+CTB)
fC==CT∇Bd=2CB−2CA

⇒∂l∂μ0=∑i=1my(i)Σ−1(−μ0+x(i))=Σ−1(∑i=1my(i)x(i)−∑i=1my(i)μ0)⇒μ0=∑i=1my(i)x(i)∑i=1my(i)∂l∂μ1=∑i=1m(1−y(i))Σ−1(−μ1+x(i))⇒μ1=∑i=1m(1−y(i))x(i)∑i=1m(1−y(i))

∇Σ−1l=∑i=1m12∣∣Σ−1∣∣∇Σ−1∣∣Σ−1∣∣+y(i)(−12(x(i)−μ0)(x(i)−μ0)T)+(1−y(i))(−12(x(i)−μ1)(x(i)−μ1)T)=mΣ2+∑i=1my(i)(−12(x(i)−μ0)(x(i)−μ0)T)+(1−y(i))(−12(x(i)−μ1)(x(i)−μ1)T)⇒Σ=∑i=1my(i)(−12(x(i)−μ0)(x(i)−μ0)T)+(1−y(i))(−12(x(i)−μ1)(x(i)−μ1)T)m