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 ) j(x7/Z Cx ) ply 2- ) YC -17 play ) x = = = , Sampling importance i - ' I 91×5 rcxs gcx x' , FIL Ws ) w ) = ply - n = ) Markov Carlo Monte Chain S I xslkcxs Ix s ' ) K ( - ' ' - ) ) I C Mas kcxslxsy M ' X X . X n =

- Hastings Metropolis Last Lecture : Propose Idea ) ' qcxixs ( jcxllz x ~ M Xl : = " Xs and set ' with press z x = ox :* .in/iii:::::isiil n xst ' sxs with keep prob o I - - Hastings kernel Implied Metropolis Transition : ' # IX ) =/ x x k I x' x' =x

- Hastings klxcx KIX Balance Detailed etropolis : Balance Detailed ' ) X' d ' Ix ) MCX ) 't Mix : = x' Fx ' ' " MH Kernel 14×11×7=1094 : qcxlx ) tfdx " ( I ' KI ' )qk' Ix - '=x 171×7 MHI KC Ix ) KCXIX ) x = x ' 1×1 ' ' ( x I Mk kcx ) next ) a IX x q x = = = I

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 ) r.io I : way distributions yµ( G) I 01 " yn = = y generate Hard to to generate Easy proposals proposals good good Idea Use MLMC to generate proposals 2 :

Annealed Importance Sampling O @ . Initialize High quality proposals samples . ) was Oingo ( Initialization = - Onsnkn On 10h ! ,( ) Whs Transition = - . )

Understanding Annealed Sampling Importance Densities Intermediate Ideas i Rt :X 17,1×1--8,1×117 , → y , at } , " . . . II. ix. =fµkl/7µ 17µL x ) → for proposal Use fuk density , ( x ) ) X Mm as n a Tnk ) Yuki w 7ns , 't Mm , ) X ~ = - = n - × ) , ( Mn Yuuki . murk ) Wm = = - x ,

Understanding Annealed Sampling Importance Idea Using transitions 2 MCMC : ' K ( x' Ix ) X qcx ) w = ~ x n 91×1 Assume JK ) 14×4×1-2 : ' ) ' ) KKK ' ? ( X w = = - Jlxle Treat variable auxiliary x as an ' ⇒ x' xn x x ~ n , I Hix ) TT = = = =

Understanding Annealed Sampling Importance Assumption We hate importance sampler an : ylxllqlx and weight with ) proposal ) C x w x q - - - Corro long target We ' I with y ' Cx ) new a can . - rg , ' xnqlxi w w w - = jcxl that Corral IX ) For kernel kcx ' 2 ! any any leaves ' with ✓ Cx ) invariant x propose we can , ' Ix ) ' ( ' 91×1 K W W x x X ~ = ~

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 - - towns Oink On 10ns f . ) wins Transition - . - . . . ) , ( Ons The ' update weight ) ( weight ) preserves (

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 - First step : HMM : Xs Ws Xi = n , Subsequent steps : X ! :c is . , ) Kiit ~ . , . - - den WE :-.

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 C Formulation Carlo General Assume Unnwmalized Densities Atx , ) ) I x yr : . . . . KE X EXT c. , . . . ( Bayes Net ) Decomposition = pcyg.io?.iiI.-- FI wt - - . T M fCXi. - = = - t=z it ) X C q , I

Sequential ) Marte C Formulation Carlo General Assume Unnwmalized Densities flat , ) I x yr : . . . . Importance Sampling First step : as Ws ) ycxilqk.SI 9 Xi C ~ :-. , Propose Subsequent from steps samples previous : . ,w- ! ( WI Discrete at , ) ~ . . , , , xsenqcxi.IM?Ii7xsnc.:--xi.xiiIti ' 8¥ Sit ) w : sqcxiixa.tt a : ÷ ' . ik ft : I