2.Decision Theory of Dependence
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1 .Deciio Theoy of Depedece Theoy :I et of vecto if thee i uet which i liely depedet the whole et i liely depedet too. Coolly :If the et i liely idepedet y oepty uet of it i liely idepedet. Theoy : Give di vecto d i oly if the thee followig vecto tix e tifie <. i liely depedet deote i the i k of i if. d
2 . Deteie the depedece of e.g.. Solutio: < depedet. e liely e liely depedet 5 the vecto if Wht i e.g.. λ? λ Solutio: λ 5 λ
3 λ λ Fo < λ depedet. e liely Pove Theoy. ": " depedet e liely vecto y othe c e liely epeeted we kow By theoy k k j j j j k k k The: Eleety opetio o
4 < ": " if > <. - ode udeteit The D Thik out the -ode udeteit of.
5 D Ufold D j j j j y the lt colu the D j j j j jd j d j D D Q e liely depedet. So e liely depedet. Coolly :Whe > -di vecto e liely depedet.
6 : di vecto e liely idepedet if d oly if the tix de y vecto tifie : di vecto e liely idepedet if d oly if i ot zeo o. : -di vecto e liely depedet if d oly if i zeo o <. Theoy 5:If -di vecto i i i i i e liely idepedetthe the coepodig -di vecto i e liely idepedet. Tht i i i i i i iely idepedet vecto e till idepedet eve if oe vecto e dded. Cutio:fte ddig - vecto to di liely idepedet vecto the di vecto we get e till idepedet
7 Pove B B B e liely idepedet.
8 xil iely Idepedet Suet Def :Give vecto et i e T it uet liely idepede t; tifie iithe vecto of T c e liely epeeted y Fo y i T e liely depedet. the we y tht uet of T. Siply i xilly liely idepedet xilly idepedet et. Thee e two cutio of xilly idepedet et:.idepedece;.xil. Note:.The xilly idepedet et of et which i liely idepedet i itelf ;. The vecto et i equivlet to it xilly idepedet et ;. The xilly idepedet et of vecto et i ot uique ut ll the xilly idepedet et e equivlet.
9 e.g. Fid the xilly idepedet et < t. e liely idepede xilly idepedet et. i e idepedet But idepedet et. e lo xilly e idepedet too d
10 Popetie of xilly Idepedet Set Theoy:Give two -di vecto et I II If et I i liely idepedet d it c e liely epeeted y et II the. Pove et B C O O O C
11 Cutio: If the vecto et c e liely epeeted y d et i liely depedet. > the the Cutio: The ue of vecto which e fo y two liely idepedet d equivlet vecto et e equl. Theoy : The ue of y two xilly idepedet et of vecto et e equl. Defiitio : the ue of of vecto et Rk of Vecto Set vecto i xilly idepedet et i clled the k of the vecto et. ked. Note:the k of idepedet vecto etthe ue of it vecto. The vecto et i idepedet it k equl to it vecto ue. Theoy : If the c e liely epeeted y
12 Cutio: Equivlet vecto et hve the e k. But two vecto et which hve the e k y e ot equivlet.. fid c e liely epeeted y et e.g.: et vecto e e e e.g.:give two -di et 与 if K e liely idepedet. : if Pove K. to e equivlet C you give couteexple? d e liely idepedet
13 The Wy to Copute the Rk of Vecto Set Theoy :The k of vecto et i equl to the k of the tix which fo fo the vecto. Row k: the k of the tix foig fo ow vecto. Colu k: iil the defiitio of Row k ove. Coolly: The ow d colu k of tix e equl. Thi i ethod to fid the k of vecto et: ue thee vecto to fo tix fit d figue out the k of the tix d the the vlue i the k of the vecto et. e.g.. Fid the k Solutio:
14
15 The Wy to Fid xilly Idepedet Set Chge colu ito ow. e.g.. Fid the k d xilly idepedet et of the followig. t. idepede i xilly
16 We c tfo the tix ito echelo tix y chgig colu ito ow d thi c help u fid the k of tix. The we oly eed to pick out oe vecto t evey lie d tht c fo xilly idepedet et. the ove{ } i lo xilly idepedet et. e.g.. Wht i λ c 5 λ e liely depedet d fid the k d xilly idepedet et of the. Whe λ < eliely depedet { } i xilly idepedet et. Couteexple i xilly idepedet { } et. But we c ot chge ow ito colu! Cotdictio
17 We ee: How to ue tix to olve the pole of vecto et d i fct we lo c ue vecto et to olve the pole of tix. exple: { } B i B Give two - di vecto if i liely the et d idepede t d K e liely idepedet K Thi i ipott foul. et B E pove i liely idepede t. tht the colu ve Thi i ipott iequlity! cto et of B
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