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

= pix , y ) y ( x ) = ply , X ) 2- = ply )

• importance

Sampling i

rcxs

, Ws =

• x'

n

gcx

) FIL w ' I = ply ) , 9C Xs I g Cheol Guess % Gives estimate

• f

marginal Man how Chain Monte Carlo S s

• I

X n K ( X I X ) M ( xslkcxs

• '

Ix s ) = ' Mas . ' ) kcxslxsy T Transition kernel Detailed balance Can de " hill climbing " but no estimate

• f

marginal

SLIDE 3 Last Lecture :

Metropolis

• Hastings

Mkt = jcxllz Idea : Propose x ' ~ qcxixs ) z and set Xs "

• _

x ' with press a-

mint

. " n

• therwise

Creep xst ' =×s Metropolis

• Hastings

: Implied Transition kernel kfxllx ) =/

OH

! " 91×4×3 x' ¥ × qcxlx ) t

fdx " ( I

• a- ( x

" ,x ) ) qlx ' 'M X' =x

SLIDE 4 etropolis

• Hastings

: Detailed Balance Detailed Balance : MCX )

KIX

' Ix ) = Mix ' I

klxcx

't MH Kernel :

14×11×7=1094

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

• d

' )qk' ' KI

X'

IX 171×1 KC x Ix ) = 171×7 KCXIX ) x

'=x

next kcx ' IN = a q ( X' 1×3171×1 x ' I x = min ( 17 Cx ) KCX ' IX ) ,

171×1114×1×1

) ) = 9 91×1×111741 = Hk ' ) Kkk 's

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 g

• f

Bnf I

To

l O) = pH ) yn I 01 = pcylo } " pco ) yµ( G) = ply , 07

a

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

SLIDE 7 Annealed Importance Sampling Use as proposal for next step ←

• 541¥

Initialize Use MCMC High quality proposals to move around samples Initialization w

:-.

4%0%4

Oi

• got

. ) f Pheu weight Transition wins

• town

!

Oink

.

f

On 10ns . . ) . ru . i ( Ons ) y Mcmc kernel

SLIDE 8 Understanding Annealed Importance Sampling Ideas i Intermediate Densities r , :X , → Rt

II

:# → a . }H=K= . . =% . " " " " t . 17µL x ) =fµkl/7µ Use density X n Mm , ( x ) as a proposal for fuk ) w =

Tnk )

= Yuki n

• 7ns ,

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

• murk

)

SLIDE 9 Understanding Annealed Importance Sampling Idea 2 :

Using

MCMC transitions w = X ~ qcx ) x ' n K ( x' Ix ) 9 I X ) I ( x ' ,X7 Assume : JK )

14×4×1-1

y ( X ' ) KKK ' ) w , ?

Vlx'KCx

I = = w q I x , JC x ) = fix ' ) KCXIX ' ) kcxilxlqcx )

• f ( x :X )

KC x' 1×1 Treat x as an auxiliary variable X ' , x ~ IT ( x ' , x ) ⇒ x' n

171×1

Xr K ( XIX ' ) A ( xix ) = Jk ! x ) ! I I =

fdx

' dxjcxllklxixl = I = y ( x 't K Cx IX ' I IF

SLIDE 10 Understanding Annealed Importance Sampling Assumption :

We

hate an importance sampler with proposal x

• q

C x ) and weight w

• ylxllqlx

) Corno long I ' . We can target a new y ' Cx ) with w ' = w w = 4¥ , xnqlx , 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 n 9 IX )

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

. )

rules )

s Transition wins =

• Wn

. , Ons n

kn

.

,(

On 1 On ! ) ' The , ( Ons ) Ii updates weight 2 : 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 ; y Low dimensional

• HMM

: Xs , ~

plx

, ) Ws :-. pcy , ,x ,

)/q(×

, ) ×i={ 7 , ,0} Subsequent steps : y pc X , :t .

.ly

' :t . ' ) Hitt ~ tiftx , :+ . )

}q,×

, ... ,

Rfkia

. , )= §

• f

8xs.n.K.it ' XS ~

PCXTIX

's :t . , ) n.pl/itlx.:+..)pcxi:t.ilyi:t)Wsti=plyilXs:tI~pcyt,xi:t1yi:t.i)/plxiki:t.i)pki:t.i1yi:t.D

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 ( General Formulation ) Assume : Unnmmalired Densities g. I × , ) ... . ytlxt ) Sequential Decomposition ( Bayes Net )

WtiP4gyI.xfh-pbgyx@M.p4txilxntqlXtlXi.l

.÷

W i wt t > I

• .
• =

raisins

, =

rafts

!

.mx#i_

f. 1×1 :t . ' )9(×tl×iitu ) ' " " nnY÷i '

III

, " .mx#Jt.i(Xi:t.i )

SLIDE 19 Sequential Mate Carlo ( General Formulation ) Assume : Unnmmalired Densities g. 1 × , ) ... . ytlxt ) First step : Importance Sampling x. EX , , ... x+eX+ x. s ~ 9 ( × , ) ws , :-. ycxsilqkil ( can be any spuul Subsequent steps : Propose from previous samples at . , ~ Discrete ( WI . , , . . . , his . , )

xst~9cxi.IM#ilxs...=*

, #

¥

,}×%t~9kt' × it . ' ' '¥' Kiki jftsiit ) , Zt 17+1×9.± " + ÷

h.de#ssqixil.xa?inzt..qkslxIIIlne.txYII