Lecture 8 Monte Sequential ( cont ) Carlo : Sampling Gibbs Alesia Scribes Chernihova : Ankur Bambharoliya
Problem * Models Motivating Hidden Manha : fT#↳ Yt yt t & zz Z z . , , Et Posterior Parameters al : an fdz PCO Oly ) Fly ) Pc = , t " " Guess from prior - likelihood " ? " Will likelihood weighty Chen work using . ) Z I , 17 :t,t plyiit Ws 0 19 ) PCO ) i= , PCZ ~ ~ it ,
Sequential Monte Carlo Example : ' ) lwi ,x , pH ( wi x ? ) , w ? ,xs , ) ( x ! wi ! ! pcyilx ~ :=
Sequential Monte Carlo Example : 1×9 ! E x : wi ai ) , his ) Disc ( wi CK lynx n p =p i n , , . . . ,
Sequential Monte Carlo Example : ' ) ply l wi X , xp - - \ Hit ,xIl ( wi - . \ cwse.xsesxf.it 1×9 ! E ai XI ) , his ) Was Disc ( wi pcyaix ~ n . , . . . . ,
* Diverse Degenerate set beginning near near indy Sequ~iaMontCaloExa#pk bad " " 2 particles In sampling repeated step prunes
Sequential Importance Sampling Express Idea over : weights product incremental as weight I ft weight . , Hi i ) Kiit , ) Vt ( hit ) ft ) Kiit Tt it - importance , . t.lt/Xiit-i)plyiit-ii4it-i :c . ) ) ply ply Xi it , , - - - = ( I ) X 9 g I IX. ) gcxiit it it Xt , - i i - t Jt Wt wt ) I Xiit M Vt = = t . , - ✓ t illicit i ) t 's qlxtlxy.it , ) q - - , . is weight Incoming Incremental
Importance Resaurpling Perform " by Idea selection natural " selecting : probability proportional their weights to particles with THI wk xk h K qlx ) I n = = . . , . , " " 9 I ( a.k.a ) " Particle " sample . WI ink " ink ) ( Discrete hi ' a n - - . . . , , wh ' I le ' - ¥ h The I E Eh Wh xa = = - & I ,w~h=E I High weight samples often selected more
Resaurplingi Importance Example , li xhrqcx ; , , " The # Law ahndicfws.in " Eh=xa " ) ! : i rcxiwh.mg#Iu 2 4 2 - o 1.5 ) k " C- 1.2 0.5 0.2 1.2 × = - , , , , - ( Is lik lik , II. Is ) Is ( I , 8.0 , 12.0 ) in , 3.0 w - o = zoo . , , , lik = ( 3 ) 5,5 5,4 a , , = ( 1.5 " % ( 5,5 ) " VT " 0.2 ) I 5. 1.2 1.5 1.5 5,5 , , , , , I
Importance Interpretation Resampling : Auxiliary Variable Trick Sample single : xh # a particle " treat auxiliary and x as qW÷ ITa=h ] , ) qcxh ) ( 's ) In " " a ) " qlalx qcx - = , ' u M¥914 jfx k¥1 = ) ( ¥ ) ) ( ldxh-tad.TK ' " f ( x a a ) " at ,x = , jcxa ) = II. "k FIX a ) " I , w w = . - Tix ,
w ! ( General Formulation ) Sequential Monte Carlo Initialize Assume of Sequence targets film ) ) yfx.it : . , . . , sampling i ) , ) ,9t( proposals 9. Cx , ) 92 ( and xi-lx.it Xzlx . , - . . . , M'×' x ! ( te importance I ) with , ) qcx ; ~ = h ) ( In q x , , . ,÷¥ at ! , ) FIE Disc ( q4¥ it it at > : - . . , . , nqkc.la?Ihi)resaupliug Zt , != ttkiith ) × ! wagigeht yt.fx.it?ti)9kthlxaait ' w " I for sequential \ Weight sampling importance
Gibbs Sampling Propose Idea 1 time at variable : a , holding other variables constant ) C ply y x pix ,Xa ) ) Ix Xz = , , . I y X , / ' pix ) Xz X. ply Xa ) ply Xi r = , , peaty :X , , , 's xi - Acceptance Ratio Car accept prob I with : ) ) = i ) × P' " KHz Pl 9. ) min ( bi ply ,xz ' on , = , ply ,xi,Xz7 X , ,×z ) ) pay ,xz pcy , I =
Mixture Gaussian ( Gibbs Sampling 2) Homework : Model Generative Grapical Model let K , Eh I ) Mh pipe ~ . , , , . . b " . it ) Discretely N ' In ~ n = . . - . . , , , fun , fu ) Norm ×nlZn=h n g a Gibbs Sampler Updates he Local variables I ) pl N Fnl I 2- Yu , µ n n - - u , . . , . , Global variables . ) 6--1 Eun plpele.ch/yiin,Ziix , K 14h , - ,
Gibbs Independence Conditional Sampling : 114,2 I ym-tapm.eu Kith Local Variables : N M E ) ) I 71in ply ,7n1µ 114 pcyn = .im , , , h = I , @ ) Global ,Zn=h I plyn µ , @ ) , µ = - 9 I I ) § pcyu ,7u=l I pl7n=h1yn compute µ Can , 044K ) updates all in Mh Variables k : .FI/Ue-th.Ie*u/zplyiiu./u,E12-iin)--Mpyuu.Eu)MNonm/ynspu , h 2- n=h n : - I -
Gibbs Global Update Sampling : likelihood conjugate Idea Ensure that is to prion : conjugate prior likelihood far h cluster posterior - - I ) I Mpcynlzih.in , plmh.su/pCMk,EulyiiN :tn=h3 µ ) In , 2- = n h ) M P' but lnizi.az - - likelihood marginal
Families Exponential An family has form exponential distribution the Depends Only Only depends depend x on on I I and y x an m d hcx ) I YT aey ) ) text pcx y ) I exp = - n E leg ÷ : normalizer ( " , Lebesgue ) ( Canting Base measure ( only depends ) X an
Gaussian Example Uniuaniate ; exp[ yttcx ) ] hcx ) pcxiy ) acy ) = - µ Panama k¥2 ' expf ; ] Dependent " # 2) = = ( zntzj "expft(x2 . z×µ+µ2)/ . ] ( tcx ×2 ) ) × = , ( µ1r2 ) -1/262 n = , log 1262 µ2 6 acy ) + = hcxl 16 1 =
Properties of Families Exponential [ yttcx 1 y ) ] hcx ) ( exp ) ) acy p x = - Derivatives Moments Normalizer Log of . . / = |d× hkiexpfyttki ) ) lacy 1 dx exp pixiy ) → = = gqµg/d× ) ) ) ( yttcx has again exp ' 7) / _.-- # '× P hcxiexplytfki ) taxi ax = = / 1 yitk hki expel alnl ) I exp - ax ) ( hkieaplyttki = |d× . acy ) ))= Epcxiy ,[ ) ] tix tax
Properties Families of Exponential [ yttcx ) ] hcx ) exp pcxiy ) acy ) = - Moments from of computable derivatives acy ) - are a a 'Y tkih ] ) [ # = pain dqn linearly tk independent exponential When - ) are an known family minimal is as family Far minimal and acy ) is convex any - tcxi ] Epa , , ,[ c→ y µ := pcxiy )[ Hxl ] ) from there ( to # to i. 1 mapping n is a .
: plx Conjugate priors Likelihood : , ] hey petty tix ) acy exp hlx ) = - lqttcx hey i Conjugate prior : := D , .dz ) ( d is exp IT ' I " act hip payed tch ) ) = - ( y tip - acyl ) , Joint hey ) explyt ( i 9 , ) , yl rt Da ) ) aim ( aids ) i - - - exploit - augier acts ) explant ) acts ] ) hey , - , = - # . I IT ) x ) p C pcyl 7 - aids ) Fa -
Conjugate priors Joint ? : hex 51 ) ] I explored ) d alt fix , pint pcyix : pox hcx - = = + plx.nl , , I Dat I = a fdypcx.nl Marginal act ] I ' aol exp , ) = - ptx.gs/plx)=pcy1Dttcxs is g normalizer from marginal leg Com compute Posterior ! ) ) = J family Posterior here Conjugacy saz prior : as - aids ) Fa -
Homework Gibbs Sampling : - likelihood conjugate Idea Ensure that is to prion : conjugate prior likelihood far h cluster posterior - - ) I Mpcynlfihduh.su/plMh,Eul9h :tn=h3 I y µ ) In , Eh , 2- Pl Mk Iim = n M ) lnizi.az/0lYnl7n=h - - likelihood marginal . I this Derive I 9h the ( y , 7 ) ) , Ch p ( Mu homework t = in
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