Probabilistic Graphical Models

Transcription

1 School o Cour Scinc robabilisic Grahical Mols Aroia Inrnc: Mon Carlo hos Eric ing Lcur 6 March Raing: S class wbsi Eric CMU

2 Aroachs o inrnc Eac inrnc algorihs Th liinaion algorih Mssag-assing algorih su-rouc bli roagaion Th juncion r algorihs Aroia inrnc chniqus Variaional algorihs Looy bli roagaion Man il aroiaion Sochasic siulaion / saling hos Markov chain Mon Carlo hos Eric CMU

3 How o rrsn a join or a arginal isribuion? Clos-or rrsnaion E.g. Sal-bas rrsnaion: Eric CMU

4 Mon Carlo hos Draw rano sals ro h sir isribuion il a sochasic rrsnaion o a col isribuion arginals an ohr cions can b aroia using sal-bas avrags N E[ ] N Asyoically ac an asy o aly o arbirary ols Challngs: how o raw sals ro a givn is. no all isribuions can b rivially sal? how o ak br us o h sals no all sal ar usul or qally usul s an al lar? how o know w'v sal nough? Eric CMU

5 Eal: naiv saling Consruc sals accoring o robabiliis givn in a BN. E0 B0 A0 M0 J Alar al: Choos h righ saling squnc Saling:B=< > suos i is als B0. Sa or E0. AB0 E0=< > suos i is als... 2 Frquncy couning: In h sals righ JA0=JA0/A0=</9 8/9>. E B0 A M J Eric CMU

6 Eal: naiv saling Consruc sals accoring o robabiliis givn in a BN. Alar al: Choos h righ saling squnc 3 wha i w wan o cou JA? w hav only on sal... JA=JA/A=<0 >. 4 wha i w wan o cou JB? No such sal availabl! JA=JB/B can no b in. For a ol wih hunrs or or variabls rar vns will b vry har o garnr vough sals vn ar a long i or saling... E0 B0 A0 M0 J E B0 A M J Eric CMU

7 Mon Carlo hos con. Dirc Saling W hav sn i. Vry iicul o oula a high-insional sa sac Rjcion Saling Cra sals lik irc saling only coun sals which is consisn wih givn vincs. Liklihoo wighing... Sal variabls an calcula vinc wigh. Only cra h sals which suor h vincs. Markov chain Mon Carlo MCMC Mroolis-Hasing Gibbs Eric CMU

8 Rjcion saling Suos w wish o sal ro is. ='/Z. is iicul o sal bu ' is asy o valua Sal ro a silr is Rjcion saling Corrcnss: iall / ' w.. acc ~ * * * * k ' ' / ' / ' k k 8 Eric CMU

9 Rjcion saling iall: Using =N 2/ q o sal =N 2/ I q cs by % an insional=000 Th oial accanc ra k= q / /20000 Big was o sals! Aaiv rjcion saling Using nvlo uncions o in Eric CMU

10 Unnoraliz ioranc saling Suos saling ro is har. Suos w can sal ro a "silr" roosal isribuion insa. I oinas i.. > 0 whnvr > 0 w can sal ro an rwigh: Wha is h robl hr? w M M ~ whr 0 Eric CMU

11 Noraliz ioranc saling Suos w can only valua ' =.g. or an MRF. W can g aroun h nasy noralizaion consan as ollows: Now r ' ' ' L r r r w w r r r r whr ~ whr ' Eric CMU

12 Noraliz vs unnoraliz ioranc saling Unoraliz ioranc saling is unbias: E w Noraliz ioranc saling is bias.g. or M = : E r r Howvr h varianc o h noraliz ioranc salr is usually lowr in racic. Also i is coon ha w can valua ' bu no.g. = ' / or Bays n or = '/Z or MRF. Eric CMU

13 Liklihoo wighing W now aly noraliz ioranc saling o a Bays n. Th roosal is gon ro h uila BN whr w cla vinc nos an cu hir incoing arcs. Call his M. Th unnoraliz osrior is ' =. So or i = i = i w g whr. i i i i w w ˆ / M w 3 Eric CMU

14 Liklihoo wighing algorih Eric CMU

15 Eicincy o liklihoo wighing Th icincy o ioranc saling ns on how clos h roosal is o h arg. Suos all h vinc is a h roos. Thn = an all sals hav wigh. Suos all h vinc is a h lavs. Thn is h rior so any sals igh g sall wigh i h vinc is unlikly. W can us arc rvrsal o ak so o h vinc nos b roos insa o lavs bu h rsuling nwork can b uch or nsly connc. Eric CMU

16 Wigh rsaling robl o ioranc saling: ns on how wll achs I is srongly varying an has a signiican roorion o is ass concnra in a sall rgion r will b oina by a w sals **** * No ha i h high-rob ass rgion o alls ino h low-rob ass rgion o h varianc o r / can b sall vn i h sals co ro low-rob rgion o an onially rronous. Soluion Us havy ail. w Wigh rsaling / l l / l Eric CMU r r 6

17 Wigh rsaling Saling ioranc rsaling SIR:. Draw N sals ro : N 2. Consrucing wighs: w w N 3. Sub-sal ro { N } w.. w w N w / l l / l r r aricular Filring A scial wigh rsalr il sals ro osrior : Also known as squnial Mon Carlo... A A A + + Eric CMU

18 Skch o aricl Filrs Th saring oin Thus : is rrsn by A squnial wigh rsalr Ti ua Masurn ua : : : : : : M w : ~ : : : w sal ro a iur ol M w : ~ rwigh 8 Eric CMU

19 F or swiching SSM Rcall ha h bli sa has O2 Gaussian os Eric CMU

20 F or swiching SSM Ky ia: i you knw h iscr sas you can aly h righ Kalan ilr a ach i s. So or ach ol aricl sal S ~ S S ro h rior aly h KF using arars or S o h ol bli sa ˆ o g an aroiaion o y : s : Usul or onlin racking aul iagnosis c. Eric CMU

21 Rao-Blackwllis saling Saling in high insional sacs causs high varianc in h sia. RB ia: sal so variabls an coniional on ha cou c valu o rs analyically: This has lowr varianc bcaus o h iniy: ~ E E E M E E var var var 2 Eric CMU

22 Eric CMU

23 Rao-Blackwllis saling Saling in high insional sacs causs high varianc in h sia. RB ia: sal so variabls an coniional on ha cou c valu o rs analyically: This has lowr varianc bcaus o h iniy: Hnc so is a lowr varianc siaor. ~ E E E M E E var var var var var E E 23 Eric CMU

24 Suary: Mon Carlo Mhos Dirc Saling Vry iicul o oula a high-insional sa sac Rjcion Saling Cra sals lik irc saling only coun sals which is consisn wih givn vincs. Liklihoo wighing... Sal variabls an calcula vinc wigh. Only cra h sals which suor h vincs. Markov chain Mon Carlo MCMC Mroolis-Hasing Gibbs Eric CMU