Abdel Rahman Aminezza Scribes Tuan : - , , Wed Out - - PowerPoint PPT Presentation

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Abdel Rahman Aminezza Scribes Tuan : - , , Wed Out - - PowerPoint PPT Presentation

Sep 17, 2023 464 likes •710 views

Lecture Marte Markov f Chain Carlo : Abdel Rahman Aminezza Scribes Tuan : - , , Wed Out Homework 2 : on Problem * Models Motivating Hidden Manha : (f) Yt yt t & zz 7 z . , , Et Posterior Parameters al

slide-1
SLIDE 1 Lecture f : Markov Chain Marte Carlo Scribes : Abdel
  • Rahman
, Tuan , Aminezza Homework 2 : Out
  • n
Wed
slide-2
SLIDE 2 Motivating Problem : Hidden Manha Models

(f)

yt Yt & 7 , zz . z , t

*

al : Posterior an Parameters Et Pc Oly ) = fdz PCO , 71g ) t High . dimensional : 3kt K ' continuous prams T discrete variables ( KT configurations)
slide-3
SLIDE 3 Sequential Monte Carlo ( Bootstrapped Particle Filter ) Intuition : Break a high dimensional sampling problem down into a Sequence
  • f
lower . dimensional sampling problems First step : X

?~pCx

, ) Ws := pcy , ,x ,

)/pK

, ) Subsequent steps : y pc X , :t .

.ly

' :t . ' ) X !t . ,~ list . ,c× , :D ,

risk

, . . , )=§oE8xs" + Kit ' XS ~

pcxtlxi

:t . , ) Wst := ply , 1×51 itl
slide-4
SLIDE 4 Sequential Monte Carlo : Example lwi ,x , ' ) ( wi , x ? ) ( w ? ,xs , ) x ! ~

pH

wi := pcyilx ! !
slide-5
SLIDE 5 Sequential Monte Carlo : Example ( Wi ,xi ) , ... , ( w ! ,xs , ) WT

i€?=⇐ois×nx

' Method

:

x
  • risk
, , an . Discrete ( ini , ... ,I ? ) is ,=wI x= x. a FW , " |d×8×.kifkli= flxo ) Xin .tk?.xYtI.xYEY.. . ) a , ' ~ Disc ( w , ' , ... , his ) Xi~p( x. 1×9 ! ) Wi :-. piyzk , ?z )
slide-6
SLIDE 6 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-7
SLIDE 7 Degenerate Diverse set

Sequ~iaMontCaloExa#pk

near beginning near

indy

*

2 sampling step repeated In " prunes " bad particles
slide-8
SLIDE 8 Sequential Marte Carlo ( General Formulation ) ply , ,X , ) Plyiit ,×iit ) Assume : Unnmmalized Densities g. I × , ) ... . jflxt ) Sequential Decomposition (

BayesNet))

W , = p(yiii,×i:t)_ = pcy , ,x , ) the

pcytixtix

, it . , ) 91×1 : " qcx , ) ft qktlxiitnl
  • =2
  • Wi
W+ t > I
  • t
  • =j(×i:t)_
= g. K ? M 8+1×1 :t ) qlxiit ) c tea yt . ,( × , :t . , ) 964-1×1 ' .tt ) plytixtlx , :t . , ) = pcyiit.X.it ) := 2+1×1 :t ) . =
  • plyiit
. i. Xiiki ) Jt . ,/Xi:t . i )
slide-9
SLIDE 9 Sequential Marte Carlo ( General Formulation ) Assume : Unnmmalized Densities g. K , ) ... . ytkt ) First step : Importance Sampling xis ~ q ( × , ) ws , :-. ycxsilqkil Subsequent steps : Propose from previous samples at . , ~ Discrete ( WT . ' , ' " int ' )

}

× , it . ,~ ins !×" + . , )

xst~qcxi.ly#i7x!t:=x!.x.aiiIilxt~qcx+ix.:t.i

) 8+4 slit ) Incremental weiswl wI÷ a

.sn#sqcxilx9?I

, ' ft . ,K 1
slide-10
SLIDE 10 Running Example : Gaussian Mixture Model ( HWZ ) Grapical Model Iris Dataset 0€ +00++0 bn & 7n µ Generative Model
  • mean

µu,£~pcµfycov

. zn ~ Discrete ( M , , ... ,hk )

ynhn.tn/Vorm4uu.S#

slide-11
SLIDE 11 Manhov Chain Monte Carlo 5
  • 1
Idea : Use previous sample x to propose the next sample x ' Maher Chain : A sequence
  • f
random variables Xl , . . . ,Xs is a 1 discrete
  • time
) Markov chain when xslxs " DX ! .
  • , ×s
" Mmhov pnoprtn p ( xslx ' is ' ' ) = p ( Xsixs " ) A Morwov Chain is homogenous when At each steps p( X '=xs 1×5 "=xs . ' ) = p( X '=×s1X=×←i ) we use the some trans . dist .
slide-12
SLIDE 12 Manha Chain Monte Carlo Convergence : A Manha chain converges to a target density 17 ( x ) when lying . p(Xs=x ) = n(X=× )

.

€*t¥¥

,

III

"

:

"I:I÷u

. in which X=x is visited with " frequency " h(X=x ) J
slide-13
SLIDE 13 Manha Chain Monte Carlo Detailed Balance : A homogenous Manha chain satisfies detailed balance when MC × ) plx 'l× ) = MK ' ) PK '× "

)

Implication ; pcx ' 1×1 leaves Mcx ) invariant 17 ( × ) = 17 ( × ) | dx ' pcx ' IX ) = |dx ' Mlxlplx 'l× ) . = |dx ' 17k ' ) 1>1×1×1 ) Invariance : If you have × '~M( x ) and then your Semple × ~ plxlx ' ) then X ~ MK )
slide-14
SLIDE 14 Metropolis
  • Hastings
Idea : Starting from the current sample xs generate a proposal × ' ~ qcxixs ) and accept xst '=× ' with probability Perfect proposal : ( detailed balance ) MCX ' ) 91×1×1 ) x 't 9 ( XIX ' ) = ncx ) qlx 'l× ) A =

mmin

( l s n( a) qcx 'l× )

↳

Ratio is I with probability ( 1
  • a
) reject the proposal and retain the previous sample xs "=xs
slide-15
SLIDE 15 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-16
SLIDE 16 Forward Acceptance etropolis
  • Hastings
: Detailed Balance a ( xlix ) Detailed Balance : A homogenous Manha chain satisfies detailed balance when my , p ,× , , . , n . , ,p . . , , g.

msn.am#

Acceptance Proposal Prob → 6 Metropolis
  • Hastings
: Define p( x.

1×1=991×11×7

Always accept underrepresented ] Sometimes accept
  • verrepresented
d nlx ) pcx ' 1×1 = 171×1 min ( 1 ,

Ty

' #

9gY×l

,

#

)

9C × '1× ) Reverse acceptance ; min ( n ( × ) 91×11×1 , 171×491×1×1 )) XCX 1×1 ) rate = . myn ( 171×191×11×12 1 ) ncx ' )

91×1×1

  • r
m ( × ' ) 91×1×4 1 X ' ¥ × = 17 ( x ' ) PCXIX ' )
slide-17
SLIDE 17 etropolis
  • Hastings
: Unrormalired Densities Nice property : Can calculate acceptance prob from unhormaliud densities ycx ' ) and YCX ) . 17 ( x ' ) q( xlx ' ) 17 ( × ) = ykl 17 A =

mmin

( l s n( × , qkilx ) ) 171×11=81×1/7
  • rcx
' ' 91×1×1 ' "n' I , =

#

F¥

, =

mmin

( 1 s y( × , qcxllx ) ) pcy ,× ' 7 91×1×1 ) y ( × ) = pcy ,× ) =

mmin

( 1 ' pcy

,x)9K'

1 × ) ) ( Banes Net )
slide-18
SLIDE 18 largest Metropolis
  • Hastings
: Choosing proposals

y

2 Continuous variables : Gaussian E.gg#aMg@@.@ qcxllx ) = Norm ( x ' ; × , 82 )

\

How big are Trade .
  • ff
for proposal variance your jumps between samples ?
  • 82
too small ; good acceptance fwb A , but high correlation between samples
  • 82
too large : less correlation , but } Tune acceptance to 4=0.6-0,5 lower acceptance preb X
slide-19
SLIDE 19 Metropolis
  • Hastings
: Choosing proposals Independent
  • MH
: Sample proposals from pntw 9 ( X ' IX ) = pcxl ) plylx ' ) PK ' )

gpcx

) Independent f
  • f
previous plb ,× ' ) q( × ,× , , sample a =

mmin

(

i pcy

,×)qK'y#

pix ' ' =

main

( , , MY " "PH "

PH

) )

pcyixspypyx

, =

mind

,PpYy#,) Dirt simple : You can always from Ratio
  • f likelihoods
the pntcr
slide-20
SLIDE 20 Gibbs Sampling Can sunphe from this Idea : Propose 1 variable at a time , distributer holding
  • ther
variables constant y ( × ) = plylx , ,× . ) pk , ,×z 7 BnYegqhsamfyx.inpcxi1yxLpyxxn1piy@into1ovenXzr.p 1×21 Y 's Xi ' )
  • Can
exploit conditional dim probes independence ? Acceptance Ratio : Car accept with prob I a , ming , ,p,y ,# . , p

,yµ,×

. ,

ppg

, ) ) , , my

,µ×

,

p.mn#y.

slide-21
SLIDE 21 Gibbs Sampling : Gaussian Mixture ( Homework 2) Grapical Model Gibbs Sampler Steps 1. Zu I y ,µ , E ~ PC Znly ,µ,E ) b "
  • 2. µ ,[
ly ,7 ~ PCMEIY , 7 ) Q 7n i. Conditional Distributions : 7h µ PC7n=hlyn , µ ,[ ) = ply ,7n=h ,µ,C ) Generative Model PEI µu ,£~pc µ ,[ ) = pCyl7n=h,µ , E) plz=h ) t zn~ Discrete ( n , , ... .tk ) § ,kpiytZn=l , µ ,&|pl7=l ) ynlZn=k~N0rm(µu ,£ ) and 7m¥nlµi%4 / Calculate joint probs and normalize them )
slide-22
SLIDE 22 Gibbs Sampling : Gaussian Mixture ( Homework 2) Grapical Model 2. Conditional Distributions : µ , { Normal Inverse . Wishart prior bn { w ~ In ✓ Wishart ( u . , %) µh ~ Normal (µo , Eu 17 . ) & 7am Exploit Gnjugacn N N Generative Model [ h I y ,7

~Inv

Wishart ( ✓ a ,

In

) N N µu,£~pcµ,E ) µ hly ,7 ~ Normal ( µu , Eu

17h)

zn ~ Discrete ( M , , ... , 17k ) ynttn.hn/V0rm(/uu.Su)
slide-23
SLIDE 23 Gibbs Sampling : Conjugacu ( Homework 2) Conjugate Conditional gP¥n¥g Sufficient Statistics [ h I y ,7 ~ Inv Wishart (

van

, [ 'I ) Number
  • f
points in duster N n N M µ hly ,7 ~ Normal (

Mu

, Lu

Hu

) Nh :-. [ I[7n=h] . n =L 9Ynn.7otNhvY-VotNhAadudntnyu.jgu@jII7ueHynNIoMotNkYhlWeightedpmp.r .cat mean
  • f
point ' µ h =
  • average )
Io + Nh in cluster h IF = [ . + gfottthulyrlu . )( In . µ .lt " Weighted average between prior (
  • ur
+ &hI[z=h](yi5ul(yr4uT and empirical cover "
slide-24
SLIDE 24 Metropolis within Gibbs ( Homework 2) 1. Zu 1 y , µ , E ~ PC Znly ,µ ,E )
  • 2. µ ,[
ly ,7 ~ p4u,E|y , 71 Idea : Use Metropolis . Hastings to Sample 2 . µ ' , [ ' ~ ql µ ,E1µ?2 ' ) ← Use diagonal form Accept with pros :

£=µ"

  • I. ±
, ] a = m .,n( , , 3 ( 9 ' 7 ' 'M :[ ' ) 9449442 ' ) ply .tt/us.Elq4u'.E 'lµ,[ ) )