Ex. 10.7

Ex. 10.7

Derive expression (10.32).

Soln. 10.7

We are looking for the optimum of

γ^jm=argminγjmxiRjmeyifm1(xi)yiγjm=argminγjmxiRjmwi(m)eyiγjm,

where wi(m)=eyifm1(xi).

Let

F(γjm)=xiRjmwi(m)eyiγjm.

We have

Fγjm=xiRjmwi(m)eyiγjm(yi)=xiRjm,yi=1wi(m)eγjm+xiRjm,yi=1wi(m)eγjm.

Setting it to be zero and solve for γjm we obtain

γ^jm=12logxiRjmwi(m)1(yi=1)xiRjmwi(m)1(yi=1).