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

SLIDE 1 Lecture 7 : Annealed Importance Sampling Sequential Monte Carlo Scribes : Daniel Zeitung Xiong yi Zhang

SLIDE 2 Last Lecture : MCMC vs . Importance Sampling

• 17

Cx

) =

j(x7/Z

= play ) YC x ) = ply , X ) 2- = ply )

• importance

Sampling i

rcxs

, Ws =

• x'

n

gcx

) FIL w ' I = ply )

91×5

) Markov Chain Monte Carlo S s

• I

X n K ( X I X ) M C xslkcxs

• '

Ix ' ) = ' Mas . ' ) kcxslxsy

SLIDE 3 Last Lecture : Metropolis

• Hastings

M ( Xl = jcxllz Idea : Propose x ' ~ qcxixs ) z and set Xs " = x ' with press

• x

:*

.in/iii:::::isiil

n keep xst ' sxs with prob I

• Metropolis
• Hastings

: Implied Transition kernel x ' # x k I x' IX ) =/ x'=x

SLIDE 4 etropolis

• Hastings

: Detailed Balance Detailed Balance : MCX )

KIX

' Ix ) = Mix ' )

klxcx

't MH Kernel :

14×11×7=1094

' " ' x' Fx qcxlx ) tfdx " ( I

• d

' )qk' ' KI

X'

Ix MHI KC x Ix ) = 171×7 KCXIX ) x

'=x

next kcx ' 1×1 = a q ( x ' IX ) Mk ) x ' I x = = I

SLIDE 5 Computing Marginal Likelihoods Motivation : Model

comparison

Question : How many clusters ' K ? ' * Low ply 109 High ply if ) Fewer bad Lots

• f

bad Bayesian Approach : Compare marginal likelihood * K = angmax plylk ) = angmax / do ply 107 PCOIK) ke { I , . . . . km " } k " Best average fit "

SLIDE 6 Annealed Importance Sampling Idea I : Sample from target yco ) by way

• f

intermediate

distributions yn I 01 = yµ( G) =

r.io

"

y

Easy to generate Hard to generate good proposals good proposals Idea 2 : Use MLMC to generate proposals

SLIDE 7 Annealed Importance Sampling @ .

O

Initialize High quality proposals samples Initialization was =

• Oingo (

. ) Transition Whs =

• Onsnkn

. ,( On 10h ! ) )

SLIDE 8 Understanding Annealed Importance Sampling Ideas i Intermediate Densities y , :X , → Rt 17,1×1--8,1×117 ,

II.

ix. → at } " . . . 17µL x ) =fµkl/7µ Use density X n Mm , ( x ) as a proposal for fuk ) w = Tnk ) = Yuki n

• 7ns ,

X ~ Mm , 't ) Mn . , ( × ) Yuuki = , = Wm x

• murk

)

SLIDE 9 Understanding Annealed Importance Sampling Idea 2 : Using MCMC transitions w = X ~ qcx ) x ' n K ( x' Ix ) 91×1 Assume : JK ) 14×4×1-2 ( X ' ) KKK ' ) w ' ?

• =

= Jlxle Treat x as an auxiliary variable x ' , x ~ ⇒ x' n xn TT Hix ) = I = = =

SLIDE 10 Understanding Annealed Importance Sampling Assumption :

We

hate an importance sampler with proposal x

• q

C x ) and weight w

• ylxllqlx

) Corro long I ' . We can target a new y ' Cx ) with w ' = w w

• rg ,

xnqlxi jcxl Corral any 2 ! For any kernel kcx ' IX ) that leaves ✓ Cx ) invariant , we can propose x ' with W ' = W X ' ~ K ( x ' Ix ) x ~ 91×1

SLIDE 11 Annealed Importance Sampling Use as proposal for next step Initialize Use MC MC High quality proposals to move around samples Initialization w

!

• tq!%}-

Oi

• got

. ) Transition wins

• towns

. . Oink . f On 10ns . . ) ' The , ( Ons ) (

update

weight) ( preserves weight )

SLIDE 12

Motivating

Problem : Hidden Manha Models

fT#↳

yt Yt & Z , zz . z , t

*

al :

Posterior

an Parameters Et Pc Oly ) =

fdz

PCO , Fly ) t " Guess " from prior Will likelihood weighty work ? " Chen " using likelihood

• .

~ PCO) Z I , ~ PCZ , it 19 ) Ws i= plyiit 17 , :t,t )

SLIDE 13

Sequential

Monte Carlo

( Bootstrapped Particle Filter )

Intuition : Break a high dimensional sampling problem down into a sequence

• f

lower

• dimensional

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

• den

WE :-.

SLIDE 14 Sequential Monte Carlo : Example lwi ,x , ' ) ( wi , x ? ) ( w ? ,xs , ) x ! ~

pH

wi := pcyilx ! !

SLIDE 15 Sequential Monte Carlo : Example a , ' ~ Disc ( w , ' , ... , his ) Xi ~ p ( × .

1×9 :[

) Wi :-. plyzlx , ?z )

SLIDE 16 Sequential Monte Carlo : Example

• lwt

' ,×k ' ) Xiit \

• (

w ; ,×Il Hit

\

( use ,x ! ) ×{ it

ah

~ Disc (

51

, , ... ,

WTI

)

t.s~pcx.tl#IDwii=piyzlx.?.t..

)

SLIDE 17 Degenerate Diverse set

Sequ~iaMontCaloExa#pk

near beginning near

indy

*

2 sampling step repeated In " prunes " bad particles

SLIDE 18 Sequential Marte Carlo C General Formulation ) Assume : Unnwmalized Densities yr I x , ) . . . . Atx ) X , c. KE . . . EXT Decomposition ( Bayes Net ) wt = pcyg.io?.iiI.--

• FI

.

• T

= fCXi. =

• M
• q

C X , it ) t=z I

SLIDE 19 Sequential Marte Carlo C General Formulation ) Assume : Unnwmalized Densities yr I x , ) . . . . flat First step : Importance Sampling as ~ 9 C Xi ) Ws , :-. ycxilqk.SI Subsequent steps : Propose from previous samples at , ~ Discrete ( WI , , . . . ,w- ! , )

xsenqcxi.IM?Ii7xsnc.:--xi.xiiIti

' 8¥ Sit ) w : ÷ a :

sqcxiixa.tt

: ' ft . ik I