Zhang Last Lecture MCMC Importance Sampling : vs . = ply ) X - PowerPoint PPT Presentation

Lecture Annealed 7 Sampling Importance : Monte Sequential Carlo Scribes Daniel Zeitung : Xiong yi Zhang

Last Lecture MCMC Importance Sampling : vs . = ply ) X ) Cx ) ply 2- ) y ( -17 pix , y ) x = = = , j(x7/Z Sampling importance i - ' I gcx rcxs x' , FIL Ws ) w ) = ply - n = 9C Xs I g Guess % , of Gives estimate marginal Cheol Man Carlo Monte Chain how S I xslkcxs Ix ' ) s s K ( - ' - ) ( ) Mas kcxslxsy I M ' X X . X n = T balance Detailed kernel Transition " but " estimate of marginal climbing hill Can de no

- Hastings Metropolis Last Lecture : OH Propose Idea ) ' qcxixs Mkt jcxllz x ~ : = " Xs and set ' with press z x -_ . " mint a- n xst ' =×s otherwise Creep - Hastings kernel Implied Metropolis Transition : x' ¥ ) =/ ! " 91×4×3 × kfxllx fdx " ( I ' 'M X' ,x ) ) qlx - a- ( x " qcxlx ) =x t

- Hastings klxcx KIX Balance Detailed etropolis : Balance Detailed ' I X' d ' Ix ) MCX ) 't Mix : = x' Fx ' ' " MH Kernel 14×11×7=1094 : qcxlx ) tfdx " ( I ' KI ' )qk' IX - '=x 171×7 171×1 KC Ix ) KCXIX ) x = x 171×1114×1×1 ' ( X' I 1×3171×1 kcx ' IN next a x q x = ) ) ( 17 KCX Cx ) ' IX ) min = , ' ) 91×1×111741 Kkk 's 9 Hk = =

Marginal Likelihoods Computing Motivation Model : ' K ? Question ' How clusters : many * Low 109 High if ) ply ply comparison Fewer bad bad Lots of 0 0 Bayesian Approach : likelihood marginal Compare angmax / * PCOIK ) K plylk ) do 107 ply angmax = = " } k " " km ke { I fit Best . . . . , average

Annealed Importance Sampling intermediate Idea Sample from target by of yco ) To I : way distributions Bnf I of g pcylo } " yµ( G) pH ) I 01 ply , 07 l O ) pco ) yn = = = a generate Hard to to generate Easy proposals proposals good good Idea Use MLMC to generate proposals 2 :

Annealed Importance Sampling -541¥ Use proposal for next step as ← Initialize High quality MCMC Use proposals to samples around move :-. 4%0%4 . ) - got Oi Initialization w Pheu weight f town Oink On 10ns f . ) ! wins Transition - . - . ) i ( Ons y ru kernel . . Mcmc

Understanding Annealed Sampling Importance Densities Intermediate Ideas i Rt :X " " " " t → r . , , . }H=K= =% II . . . a :# =fµkl/7µ 17µL x ) → for proposal Use fuk density , ( x ) ) X Mm as n a Tnk ) Yuki w 7ns , 't Mm , ) X ~ = - = n - × ) , ( Mn Vu . .CN . murk ) Wm = = - x ,

Understanding Annealed Sampling Importance Idea Using transitions 2 MCMC : ' K ( x' Ix ) X qcx ) w = ~ x n 9 I X ) ' ,X7 Vlx'KCx Assume JK ) I ( 14×4×1-1 x : ' ) 171×1 ' ) KKK , ? ( X y I w w = = ' ) ' ) KCXIX fix JC x ) I = x , q kcxilxlqcx ) - KC x' 1×1 of ( x :X ) Treat variable auxiliary as x an ' ) IT ( K ( ' ⇒ x ) x' XIX ' Xr X x ~ x n , , fdx I ! Jk I ! I ( xix ) A dxjcxllklxixl x ) ' = = = ' I IF 't IX ( Cx K = x y

Understanding Annealed Sampling Importance Assumption We hate importance sampler an : ylxllqlx and weight with ) proposal ) C x w x q - - - Corno long target We ' I with y ' Cx ) new a can . = 4¥ ' xnqlx w w , w = , jcxl that Corral IX ) For kernel kcx ' 2 ! any any leaves ' with ✓ Cx ) invariant x propose we can , ' Ix ) ' ( ' K IX ) W x 9 w X x n ~ =

Annealed Importance Sampling Use proposal for next step as Initialize High quality Use MC MC proposals to samples around move ! tq!%%- . ) - got Oi Initialization w - - rules ) 1 On ! kn ,( ) s On wins Transition Ons Wn = n , - . . ) , ( Ons The ' weight weight preserves 2 : updates Ii

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 ( Bootstrapped Particle Filter ) Carlo Break dimensional Intuition high sampling a : problem into of down Sequence a dimensional sampling lower problems . dimensional Low First step y n.pl/itlx.:+..)pcxi:t.ilyi:t)Wsti=plyilXs:tI~pcyt,xi:t1yi:t.i)/plxiki:t.i)pki:t.i1yi:t.D ; plx )/q(× HMM : - , ,0} Xs Ws ×i={ 7 , ) ) ,x pcy ~ :-. , , , , Subsequent steps PCXTIX : ' ) .ly Rfkia X pc :t :t ' y , . . Hitt tiftx ) . , )= § 8xs.n.K.it of ~ ' }q,× :+ , ... , . , XS 's , ) :t ~ .

Sequential Monte Carlo Example : ' ) lwi ,x , pH ( wi x ? ) , w ? ,xs , ) ( x ! wi ! ! pcyilx ~ :=

Sequential Monte Carlo Example : 1×9 :[ Wi ' Xi ) , his ) Disc ( ' w ( ?z × plyzlx ) a ~ p :-. ~ , . , ... , ,

Sequential Monte Carlo Example : ' ) Xiit ' lwt ,×k - - \ Hit w ; ,×Il - ( \ ×{ ,x ! ( use ) it t.s~pcx.tl#IDwii=piyzlx.?.t.. ah WTI 51 ) Disc ( ) ~ ... , , ,

* Diverse Degenerate set beginning near near indy Sequ~iaMontCaloExa#pk bad " " 2 particles In sampling repeated step prunes

Sequential ) ( Marte Formulation Carlo General Assume Unnmmalired Densities ytlxt , ) ) I × g. : ... . ( Bayes Net ) Sequential Decomposition WtiP4gyI.xfh-pbgyx@M.p4txilxntqlXtlXi.l .÷ t W I > wt i - - rafts .mx#i_ . ! raisins = = ) )9(×tl×iitu 1×1 , f. :t . ' III " ' " " .mx#Jt.i(Xi:t.i ) nnY÷i ' ,

Sequential ) ( Mate Formulation Carlo General Assume Unnmmalired Densities ytlxt , ) ) 1 × g. : ... . Importance Sampling First EX x+eX+ step : x. ... , , s ws ( , ) ycxsilqkil be 9 ( × spuul x. ~ :-. can any , h.de#ssqixil.xa?inzt..qkslxIIIlne.txYII Propose Subsequent from steps samples previous : ( WI , his Discrete at ) ~ . . . , , . , ' ' '¥' Kiki . , . , }×%t~9kt' × it . xst~9cxi.IM#ilxs...=* ¥ # , jftsiit 17+1×9 .± ) " Zt , ÷ +