## Question

###### Let n 2 2 hc an integer . DenoteSn = {(PL,. Pn-1T Ã¢â€šÂ¬R"-I : 0 < Di < for 1 < i<" -L,and 0 < 1 -Pi < 1}.Let (Pi , 'Pr-VT Ã¢â€šÂ¬ Sn be fixed vector. Consider the Cross Eutropy loss LcE defined OH Sn byLcE (q) =(+2 ) log (-2") 2" log %, where (41, 'n= 41) e Sn, aud the "log" is the natural logarithin (with base Xio(W)-1 ~ 2.71828 . . .) . (10 marks) Find q Ã¢â€šÂ¬ Sn such that ~LcE(q) = 0 for <isnm1 (10 marks) For i,j Ã¢â€šÂ¬ {1,2. let H;j (4) 8q.89

Let n 2 2 hc an integer . Denote Sn = {(PL,. Pn-1T Ã¢â€šÂ¬R"-I : 0 < Di < for 1 < i<" -L,and 0 < 1 - Pi < 1}. Let (Pi , 'Pr-VT Ã¢â€šÂ¬ Sn be fixed vector. Consider the Cross Eutropy loss LcE defined OH Sn by LcE (q) = (+2 ) log (-2") 2" log %, where (41, 'n= 41) e Sn, aud the "log" is the natural logarithin (with base Xio(W)-1 ~ 2.71828 . . .) . (10 marks) Find q Ã¢â€šÂ¬ Sn such that ~LcE(q) = 0 for <isnm1 (10 marks) For i,j Ã¢â€šÂ¬ {1,2. let H;j (4) 8q.89, LCE(q) . Find Hij(q). Let H(a) (His(a)) = be the (n _ 1) * (n 1) Hessian matrix of LCE- Provc that H(q) positive semni-definite That is, prove that for AHY C1, Gn-1 Ã¢â€šÂ¬R, GCjHis(q) > 0.

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