Jay : Seaman Iris Last Lecture : Importance Sampling Xsnqcx - - PowerPoint PPT Presentation

SLIDE 1 Lecture to : Markov Chain Monte Carlo Scribes : Jay De Young Iris Seaman

SLIDE 2 Last Lecture : Importance Sampling Idea : Generate samples Xsnqcx ) from a proposal distribution that is similar to Mk ) MCN y high weight : gcxi underrepresented

""iµq%*+

E :-. use . ✓

• verrepresented

by proposals x Elftlx) ) = fdxtrcxlfki = = = = xsngcxl

SLIDE 3 Default Choice fan

Proposal

: Likelihood Weighting Assume Bayes Net ycx ) = = 2- = = ply ) Set Proposal to Prior q l x ) = Importance Weights : Likelihood F- s = & Is , f Ks ) us = = S

• i

[ W s ' = )

SLIDE 4 Motivating Problem : Hidden Markov Models

§→€→

yi Yt & 7 , Ze . z , t ← Goal : Posterior an Parameters It PC O I y ) = t Will libel ihocd weighty wah ? O I w ' =

SLIDE 5 Markov Chain Monte Carlo S

• I

Idea : Use previous sample x to propose the neat sample x ' Masher Chain : A sequence

• f

random variables X " , . . . ,Xs is a I discrete-time ) Markov chain when xslxs

• '

Ix ' , . . . ,x " p ( x ' I x' is

• '

) = p 1×51×5-1 ) A Markov Chain is homogenous when pcxs.es/Xs-!xs-i)=pCX'=xslX-xs . ' )

SLIDE 6 Manha Chain Monte Carlo Convergence : A Manha chain converges to a target density 17 ( x ) when lying . p(Xs=x ) = n(X=× ) 2

*¥*y¥¥±

,

III :

"II÷u

. in which X=x is visited with " frequency " h(X=x ) J

SLIDE 7 Markov Chain Monte Carlo Detailed Balance : A homogenous Markov chain satisfies detailed balance when MCX ) p I X' Ix ) = Mix ' ) plxcx 't Implication i pcx ' 1×1 leaves Mix ) invariant 17 C x ) = = = If you start with a sample x ' n next and then sample XIX ' ~ pl XIX ' ) then x n Mex )

SLIDE 8 Metropolis

• Hastings

Idea : Starting from the current sample xs generate a proposal x ' n qcxixs ) and accept xst ' = × ' with probability . a =nain ( I , ) with probability C I

• )

reject the proposal and retain the previous sample xs "=xs

SLIDE 9 Metropolis

• Hastings

Idea : Starting from the current sample xs generate a proposal × ' ~ qcxixs ) and accept xst '=x ' with probability . M ( X ' ) q( XIX ' ) a =mmin ( l i n ( × ) qkllx ) ) with probability ( l

• d

) reject the proposal and retain the previous sample xs "=xs Exercise : Show that the Markov chain x ' . . . xs satisfies detailed balance

SLIDE 10 etropolis

• Hastings

: Detailed Balance Detailed Balance : M ( X ) p I X ' Ix ) = Mix ' ) pcxcx 't Metropolis

• Hastings

: Define

Kix

' IN = a = minfl ,

7457,44272¥

kcx

' Ix ) MCA = + min ( , ) =

SLIDE 11 etropolis

• Hastings

: Unrormalired Densities Nice property : Can calculate acceptance prob from unhormaliced densities jcx ' ) and ✓ C x ) . 17 ( x ' I 9 ( XIX ' ) a =

mmin

( I i mix , qcx ' Ix ) ) =

nain

( I ,

• )

=

main

( I , ) 81×1 =

SLIDE 12 Metropolis

• Hastings

i Choosing proposals Independent

• Mtt

: Sample proposers from pretor 9 C X ' IX ) = pcx ' ) ( Independent from prau . sample ) a = mm

in

( I , ) = mm

in

( I , ) =

mind

,

• )

Straightforward , but low acceptance prob ( )

SLIDE 13 Metropolis

• Hastings

: Choosing proposals dark

• f

" 2 Continuous variables : Gaussian

(

qcx ' Ix ) = Norm ( x ' ; x , 82 )

• .

Trade

• ff

far proposal van ,

and

small

• r

' Symmetric proposal

• 82

too small ; good acceptance prob A , but high correlation between samples 91×4×1=941×1 )

• 82

too large : less correlation , but ← min ( I I ) lower acceptance prob A = min ( I ,

• )

Rule

• f

thumb , time

• f

' to make 9 I 0.234

SLIDE 14 Gibbs Sampling ( Next Lecture ) Idea : Propose 1 variable at a time , holding

• ther

variables constant y C x ) = ply Ix , ,Xa ) pix . , Xz ) X , ' n = ply , X. , Xa ) / ply , xi xi

• Acceptance

Ratio : Car accept with prob I = A = min ( I , ,

)

= I

SLIDE 15 MCMC vs . Importance Sampling NCO ) = 8107 IZ y ( O ) = ply , O ) 2- = ply )

• importance

Sampling i ws = P'{j% ,

• n

geo ) Elws )

• pigs

y a I Guess and check Gives estimate

• f

marginal Metropolis

• Hastings

ps = { Os " u > a u

• Unit ( 0,1 )

Anglo

'1o " , O " n ca g = min ( , , 8105910541014 HO " ) 9104054 ,) Can do " hill climbing " but no estimate

• f

marginal

SLIDE 16 Computing Marginal Likelihoods Motivation : Model

comparison

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

SLIDE 17 Computing Marginal Likelihoods Motivation : Model

comparison

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

• f

bad O

SLIDE 18 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 likelihoods * K = angmax plylk ) = angmax / do ke { I , . . . , km " } K

SLIDE 19 Annealed Importance Sampling Idea I : Sample from target yco ) by way

• f

intermediate

distributions yn lol ya , I O)

r.mg

Idea 2 : Use MLMC to generate proposals

SLIDE 20 Annealed Importance Sampling @ .

O

. Initialization w

:-.

4%0%4

Oi

• got

. ) Transition wins =

• Onsnkn

. ,( On I On ! ) )