Probabilistic Graphical Models

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1 Pros Grh Moes 0708 Lerg Coeey Oserve Uree Grh Moes Er Xg Leure O Reg: MJCh. 990 Re: or Bs I we ssue he reers or eh CPD re goy eee oes re uy oserve he he kehoo uo eooses o su o o ers oe er oe: D D π π ML k k ' k ' k

2 or uree grh oes For ree grh oes he kehoo eooses o su o ers oe er y oe us res. For uree grh oes he kehoo oes o eoose euse he orzo os s uo o he reers P K C C I geer we w ee o o eree.e. rgzo o er reers or uree oes eve he uy oserve se. K Log Lkehoo or UGMs wh ur ue oes Sue sss: or UGM VE he uer o es h oguro.e. X V s oserve se D{ } e reresee s oows: e e δ o ou ue ou V \ I ers o he ous he kehoo s gve y: D D δ There s sy he kehoo δ δ

3 Dervve o Lkehoo Logkehoo: Frs er: Seo er: δ δ δ Se he vue o vres o Coos o Cue Mrgs Dervve o kehoo Hee or he u kehoo reers we kow h: I oher wors he u kehoo seg o he reers or eh ue he oe rgs us e eu o he oserve rgs er ous. Ths oes e us how o ge he ML reers us gves us oo h us e sse whe we hve he. e *

4 or uree grh oes Is he grh eoose rgue? Are he ue oes ee o ues o suues? e.g. 4 o X X X X X X 4 X X 4 Are he ue oes u es or Gusss or reerze ore oy e.g. e k k? Deoose? M ue? Tur? Meho Dre IPF Gre GIF or eoose uree oes Deoose oes: G s eoose G s rgue G hs uo ree Poe se rereseo: ϕ Coser h X X X. The ues re X X X X he seror s X The er rgs us eu he oe rgs. s s s Le us guess h We very h suh guess sses he oos: sry 4

5 5 or eoose uree oes o. Le us guess h To oue he ue oes us eue he o he er rgs or oos.e. he seror us e ve o oe o s eghors. The. Oe ore ee: X X 4 X X oeoose /or wh o ue oes I he grh s oeoose or he oes re ee o o ues e.g. 4 we ou o eue er rgs or oos o o ues oes. X X 4 X X X X 4 X X } { 4 / / s.. Hoework! Hoework!

6 Ierve Prooro Fg IPF Fro he ervve o he kehoo: we erve oher reosh: whh ers y he oe rg. Ths s hereore eo euo or. Sovg oseor s hr euse ers o oh ses o hs oer euo. The e o IPF s o ho e o he rgh h se oh he ueror eoor sove or o he e h se. We ye hrough ues he ere: ee o o eree here Proeres o IPF Ues IPF eres se o eo euos. However we rove s so oore se gorh oores reers o ue oes. Hee eh se w rese he kehoo w overge o go u. Iroeo: g sruo wh he orre rgs h hs he eroy 6

7 7 Dvergee Vew IPF e see s oore se he kehoo usg he wy o eressg kehoos usg vergees. Re h we hve show zg he kehoo s euve o zg he vergee ross eroy ro he oserve sruo o he oe sruo: Usg roery o vergee se o he oo h rue: : IPF zes vergee Pug hgs ogeher we hve I e show h hgg he ue oe hs o ee o he oo sruo so he seo er uee. To ze he rs er we se he rg o he oserve rg us s IPF. We erre IPF ues s reg he o oo roes whe reg he o rg roy wh he oserve rg.

8 Feurese Cue Poes So r we hve susse he os geer or o uree grh oe whh ues re reerze y geer oe uos. Bu or rge ues hese geer oes re eoey osy or eree hve eoe uers o reers h we us er ro e. Oe souo: hge he grh oe o ke ues ser. Bu hs hges he eeees y ore us o ke ore eeee ssuos h we wou ke. Aoher souo: kee he se grh oe u use ess geer reerzo o he ue oes. Ths s he e eh eurese oes. Feures Coser ue o ro vres UGM e.g. hree oseuve hrers srg o Egsh e. How wou we u oe o? I we use sge ue uo over he u o ue oe wou e huge: 6 reers. However we oe kow h soe rur o segs o he vres ue re ue key or ue ukey. e.g. g e o?e u? k zzz... A eure s uo whh s vuous over o segs ee ew rur oes o whh s hgh or ow. For ee we gh hve g whh s he srg s g 0 oherwse sr eures or?e e. We so ee eures whe he us re ouous. The he e o e o whh s ve sers u we gh s hve o reerzo o he eure. 8

9 Feures s Mrooes By eoeg he eh eure uo e e o rooe. We uy hese rooes ogeher o ge ue oe. Ee: ue oe ou e eresse s: e g g?e?e e K K ekk k Ths s s oe over 6 osse segs u oy uses K reers here re K eures. By hvg oe or uo er oo o we reover he sr ur oe. Cog Feures Eh eure hs wegh k whh rereses he uer sregh o he eure wheher reses or ereses he roy o he ue. The rg over he ue s geerze eoe y sruo uy GLIM: e g u? g?e?e u? zzzzzz L I geer he eures y e overg uosre ors or y uo o y suse o he ue vres: e ek k I How we oe eure o roy oe? 9

10 0 Feure Bse Moe We uy hese ue oes s usu: However geer we orge ou ssog eures wh ues us use se or: Ths s us our re he eoe y oe wh he eures s sue sss! Lerg: re h IPF we hve o ovous how o ue he weghs eures vuy I k k e e o Feure Bse UGMs Se kehoo uo Ise o ozg hs oeve rey we k s ower ou The rh hs er uer ou Ths ou hos or µ rur or Thus we hve D D / µ µ µ D

11 Geerze Ierve Sg GIS Lower ou o se kehoo Dee Re g Assue Covey o eoe: We hve: D e e e e e D e e π π e e Λ D 0 GIS Lower ou o se kehoo Tke ervve: Se o zero where s he uorze verso o Ue e e Λ D Λ e e e

12 Where oes he eoe or oe ro? Revew: Mu Lkehoo or eoe y.e. A ML ese he eeos o he sue sss uer he oe us h er eure verge. D D Mu Eroy We roh he oeg roe ro erey ere o o vew. Beg wh soe e eure eeos: Assug eeos re osse here y es y sruos whh ssy he. Whh oe shou we see? The os uer or ee oe.e. he oe wh u eroy. Ths yes ew ozo roe: α s.. H α Ths s Ths s vro vro eo o sruo! eo o sruo!

13 Souo o he ME Proe To sove he ME roe we use Lgrge uers: So eure osrs ME eoe y. Proe s sry ove w.r.. so souo s uue. L µ α e e L e se e e * * µ µ µ A ore geer ME roe h h h s.. H e α h e

14 Cosrs ro D Where o he osrs α oe ro? Jus s eore esure he er ous o he rg : α Ths so esures ossey uoy. Kow s he eho o oes... w o rge uers We hve see se o ove uy: I oe se we ssue eoe y show h ML es oe eeos us h er eeos. I he oher se we ssue oe eeos us h er eure ous show h ME es eoe y sruo. o uy g ye he se vue o he oeve Geoer erreo A eoe y sruo: E : h e A sruos ssyg oe osrs M : Pyhgore heore M M ME : s.. h M h h M M MLk : s.. E M M 4

15 Coo Ro Fes So r we hve ousse o e oes or esy eso. We so orue suh oes or sso regresso oo esy eso. y e y The oe ove s ke og s regresso o he eures. ow eures e very oe oer uos o he. Y Y Y 5 X X Coo Ro Fes y e y Aow rrry eeees o u Cue eeees o es Use roe eree or geer grhs 5

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