Sampling. Very simple. high dimensional. Summary : Monte Carlo Methods. normalizing. problems. Importance. very general. good proposal qkl.

Transcription

1 Lecture 7 : Hamiltonian Monte Carlo Scribes : Ming mthg & Colin Lectures : No class on Monday Homework Z : Due Friday 9 Feb ( start this week! )

2 Need Does Summary : Monte Carlo Methods Importance Sampling 17k ) jk ) /Z xsnqcxl + Very simple s w : YE s Ewsfksl very general normalizing wsfifa + Gives estimate of good proposal qkl constant f # [ Is ] not scale well to high dimensional problems

3 Also Summary : Monte Carlo Methods Sequential Monte Carlo at ~ Disdain FI ) gl#xaeilfs&wtfksitixtsnqix+lx?tl Ese + Generic by performing witffxihe ftjsfwi strategy far high natural selective gives estimate of # marginal 7 particles die out during resamplny dimensional proposals [ Is ] sample degeneracy Not all problems lend themselves to Sequential decamp

4 J Manha Chain Monte Carlo Convergence : A Manha chain converges to a target density 17 ( x ) when lying p(xsx ) n(x ) *t III : I:I u in which Xx is visited with frequency h(xx )

5 Markov Chain Monte Carlo Metropolis Hastings Gibbs Sampling X ~ qcxlxs ) y( a i min ( 8c ycxs ) ply x xz ) 9k K t ~ pixly/x5? s )qk 1 5 I) Xsz ~ p( Xzly X s ) n ~ Uniform ( 01 ) s { so t Much less correlation xs u > a between samples + Tune Really general proposal to natto Deriving conditional optimize updates hard / impossible

6 Mill Exploration Hamiltonian Monte Carlo : Motivation hill Intuition : MCMC Algorithms combine with climbing stochastic exploration Can often talk bulh 2 tvn/@#*hyg at the computation ) continuous ( Idea : Can we use gradients P z( \? 1 climbing : and the mode of distributor 2 : Characterization of variance around mode

7 Hamiltonian Monte Carlo : Motivation Idea ; Think of density as an and Sumpter as a marble energy landscape moving around Uk ) leg ycx ) l s Jeannie ;tum Proposal mechanism : 11 Simulate of a marble in the trajectory energy landscape Ma in f ( xl correspond to minima Uk )

8 Hamiltonian Monte Carlo Auxilli any variables : Density on extended Postttm momentum gexptklpl ] > ycx p ) exp [ ) UK KCFH yc Is jcp? UII ) log jrcxs Potential rikkfrpnkipi Energy K ( f ) FTM f 12 Kinetic Energy \ A matrix Man : Target density rice F) nig rtf nd Fp space rbgy dft F k 1 next

9 Hamiltonian Monte Carlo : Algorithm Auxill : any variables : Density on extended ) exp [ FCI space F UCI ) KCF ) ) ycilylp : yixl Auxiliary variables : F tick 1512 drink B ) nkl kmalsowomifnnkipknknlpkl ) MCI ) tilpl Throw away F Is Fix ~ ) Es mix ~ ) UII ) log K ( f ) FTṀ Target Density / s p

10 an Hamiltonian Monte Carlo X K \ nap Proposal : 2ws# { ( ± tpt }t Simulate trajectory Fit ix p o ~ nlf ) Gibbs sampling FHFIFH Ft9kiiP ip I is Is } MH I i It update I p : ft Accept a reject : a mn( pfcxo limits) 9 1) Ensure that is close to 7 Reversible proposal

11 Hk Kept Hamiltonian Dynamics X Is Fiat if ( I F ) up 1 UKT ]! f) exist F ) ] ix text Ftiiox dfoe Hlxipl UKTTKIF ) ftp#to : 146 ) ] a mm ( ) expttlk p1 ] Conservation of : : Energy Solution ( Conserves 14 ) DH OF OH : DI dt of dt dt DH of dy 2 oayitdqldp o

12 at Hamiltonian Dynamics 2 Fini X Is if ( I F) expfuk ) KF ) ] :### expl Hlxjpl ] Fix :X Hlxipl UIEI KIFI Trajectory ; Integrate Coupled PDES ddx IH : M of 0 : F Txycxi 2I ycxl UKI + Klp ) log y ( I ) FTM 512 Velocity da F Going gains divided in downhill :c momentum by mass 14 momentum means

13 Numerical Integration Euler E : 03 Leapfrog Ea3 P Numerical r p More Stable \ Instability than Enter % x El Icti + N Ict x b ) gay% ycxtts pit + fatten )+0 HxttteD_ + fate El p ( t + e) Fit CH) FCH By + { xlttel xctlt M ph+4z ) e ) fctt }{ ) jklttel E

14 Hamiltonian Monte Carlo : Algorithm iffy S 1 X amine ~ Hamiltonian ts conserved b Uniform ( 01 ) so acceptance prob depends only on Is { IT :L YIIY ) it#iii?oth It pii LEAPFROG ( VIU I p MeT ) It 54 integration error Is u > d

15 Hamiltonian Monte Carlo : Tuning Parameters 2 finlpl p Tunable Parameters : M FM KIs e T ) e T * ;K ixtixo M : M E ) It pii LEAPFROG ( RU I g Estimate by running sampler when : E [ xixj ] E : Tune fun to achieve acceptance # [47top EKITEK ;) T : No Utwn Sampler ( NUTS ) : Step when doubling bach

16 : : Debugging Monte Carlo Methods Getting it Right y wehe style It : Assume Bayes Net Sampling data from likelihood posterior ( ISISMCIIYHIGBBSIHMC ) Testing ~ pcxl xn Sample#prtw ~ In ~ xnvpixl plylxnl p( xly ) Run inference to sample from ::IIi±t tft