Monte Lecture Hamiltonian Carlo 7 : & Ming Scribes Colin mthg : Lectures No class Monday : on Due Homework Friday Z 9 Feb : ( ! this ) start week
Summary Monte Carlo Methods : .ws /Z 17k ) jk ) = Importance Sampling £s=§§ Fifa YE , 's Ewsfksl ' xsnqcxl w :-. Very simple + general very , # [ Is ] f= constant Gives of estimate normalizing + Need proposal good qkl - high Does not well dimensional problems scale to -
Summary Monte Carlo Methods : Carlo Sequential Monte wi=tff"xIhe at . ,~ Disdain . .FI . ) .gl#xaEi..lFs=&wtfKsitixtsnqix+lx?tl = ftjsfwi Ese Generic far strategy proposals high dimensional + - " " by selective natural performing [ Is ] Also # estimate of 7 gives marginal = - " " particles die during out resamplny sample → degeneracy - lend Not to all problems themselves decamp Sequential - .
Manha Monte Carlo Chain A Manha Convergence chain to converges : density target when ( 17 ) x a p(Xs=x lying ) n(X=× ) = . . "I:I÷u III : €*t¥¥ " . which X=x in visited is , " frequency " with h(X=x ) J
Markov Monte Chain Carlo - Hastings Gibbs Sampling Metropolis " ) × ) y( ply ,x xz ) qcxlxs = ' X , ~ , pix.ly/x5 's ? " K 't ' ' min ( 9k × , ,8c× ~ I ) a i. s ) Xsz 1×5 " p( Xzly ycxs " )qK' ,X ~ , Uniform ( 0,1 ) - n ~ ' { so × " correlation Much less t ×s , samples between " xs a > u conditional Deriving Really general + - / impossible hard to updates Tune proposal optimize - accept natto
Hamiltonian Motivation Monte Carlo : MCMC Algorithms Intuition combine : ' climbing " " " hill stochastic with exploration bulh Can talk often - t.vn/@#*h.yg 2 at the computation ) ( , continuous Can Idea use : we = ? gradients P×z(×\ Mill climbing of distributor 1 : the and mode . Exploration of Characterization around mode : variance 2 .
Hamiltonian Motivation Monte Carlo : Think landscape of Idea " " density energy as ; an Sumpter " marble " moving and as around a leg Uk ) ycx ) - = Jeannie ¥ l ;tum s - . - " ⇐ 11 Proposal mechanism : Simulate the trajectory " marble " of landscape in a energy Uk ) correspond to minima in ( xl Ma f.
Hamiltonian Monte Carlo Auxilli variables → Density extended : space any on " Postttm " gexptklpl ] " " momentum - > [ KCFH ) UK yc Is ycx jcp ? ) p exp = - - = , rikkfrpnkipi - log Potential UII ) Energy jrcxs = ( f ) K ' FTM ' Kinetic 12 Energy f = \ " " A matrix Man Target density : rice nig nd , F) rtf = = Fp = rbgy|dFt¥F next k¥1 = =
Hamiltonian Monte Carlo Algorithm : Auxill variables :| Density extended : : any space on ) ) FCI UCI KCF [ F ycilylp ) exp ) = - = - , - log yixl UII ) Auxiliary variables F = : FTM ' ( f ) K . tick =nKl 1512 ,B drink ) = / kmalsowomifnnkipknknlpkl Target Density , • MCI ) 1545,15 ) tilpl F Throw = away Fix ,§ ) Is Es 's mix ) p ~ → ~ ,
Hamiltonian Monte Carlo X. =K" ' Proposal Simulate trajectory nap : § 2.ws# • • • \ ' }t= { ( ± t.pt ix. limits pfcxo , FHFIFH Fit o Gibbs sampling § nlf ) p ~ . , . ,Ft9kiiP¥iP" It } ° ° ° ← , I Is , - is , update MH I ' i. I p an - ft . § ' , : = = mn( ) 9 1) Accept reject a : a , " to proposal Ensure that close Reversible 7 is ,
" .!§f) Hamiltonian Dynamics =Is ' X. ' if I ,F ) ' Fiat dY= 2¥ oayitdqldp =o Kept ] 1- UKT ( up • = . . . Ftiiox Hk ,F ) ] ix. exist = . dF=oE ftp.#t=o text UKTTKIF Hlxipl ) := ° ° ° ← , , ,p,1 ] ) . Ft ) ] ¥146 mm ( , , a = expttlk , ( 14 ) Conservation Energy of Solution Conserves : : DI OF OH DH :-. of dt dt dt DH , 214214 - of
Hamiltonian Dynamics =Is ' X. ' if . ) . KF ) ] ' ( I. F) Fini expfuk :###.. • = . . 2 - Hlxjpl ] expl :X = Fix ' ' Klp ) Hlxipl UKI + = . ° ° ° ← , ( I ) UIEI log = y - KIFI 512 FTM . ' = PDES Trajectory Integrate Coupled ; Velocity da§ momentum :c " ' " ' ' IH F by 14 M divided ddx÷ F :-. mass = of " " downhill Going means at 0¥ Txycxi = - :-. 2I - momentum in gains ycxl
Numerical Integration Leapfrog Euler 0.3 E=a3 E : P § Numerical More Stable r \ Instability Enter than % p = pit x x gay ) b. N' ycxtts + El % Icti Ict Fit fate fctt El ' " + = + = fatten )+0×HxttteD_ M "pH+4z ) By CH ) - = FCH - + e) - xlttel xctlt - ( t { e p + = " ' " ) ✓ ' }{ ) E jklttel
Monte Hamiltonian Carlo Algorithm : iffy "÷" .IT :L " YIIY amine ) - S -1 X ' = × , It ,pIi= , ,p , ,M,e,T ) ( VIU , I LEAPFROG .it#iIi?otH Hamiltonian Uniform ( 0,1 ) conserved b ts ~ so acceptance prob It 54 " { depends only Is on ,= integration error Is " d > u
Hamiltonian Monte Tuning Parameters Carlo : ¥§) ' ' finlpl |K.=Is • . Parameters .FM . Tunable M T 2 : e. p , , ;K It ,pIi- ,T ) ixtixo ( RU ,I ,e LEAPFROG , * • . ° ° ° ← , Estimate by running g sampler £ ✓ " E E [ ;) M M ] EKITEK when : xixj = := - , # [47top Tune fun to E acceptance achieve : ( NUTS ) " doubling bach No Utwn " Sampler Step T : : when -
# It " " it Right Getting Debugging Monte Carlo Methods ÷ . style Net Assume Testing Bayes wehe : Sample xn pcxl prtw ~ likelihood Sampling " " data from " p( xly ~ y plylxnl " ) from Run In sample inference to ~ ISISMCIIYHIGBBSIHMC ) ( ::IIi±t¥tft posterior xnvpixl . " " . . . : . : . . .
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