The document discusses advanced sampling techniques used in Markov Chain Monte Carlo (MCMC) methods including collapsed Gibbs sampling, auxiliary variables, slice sampling, simulated tempering, and the Swendsen-Wang algorithm. Key concepts such as Rao-Blackwell theorem and Hamiltonian Monte Carlo are also explored, highlighting their applications and performance benefits in various sampling scenarios. Additionally, practical implementations and theoretical foundations of these techniques are considered to improve sampling efficiency and convergence.

1. Lecture'3
Advanced(Sampling(Techniques
Dahua%Lin
The$Chinese$University$of$Hong$Kong
1

2. Overview
• Collapsed*Gibbs*Sampling
• Sampling*with*Auxiliary*Variables
• Slice*Sampling
• Simulated*Tempering*&*Parallel*Tempering
• Swendsen?Wang*Algorithm
• Hamiltonian*Monte*Carlo
2

3. Collapsed)Gibbs)Sampling
3

4. Mo#va#ng(Example
with
We#want#to#sample#from# .
4

5. Gibbs%Sampling
Draw% %where% :
with% .
5

6. Gibbs%Sampling%(cont'd)
Draw% :
• How%well%can%this%sampler%perform%when%
?
6

7. Collapsed)Gibbs)Sampling
• Basic&idea:"replace"the"original"condi0onal"
distribu0on"with"a"condi0onal"distribu0on"of"a"
marginal(distribu.on,"o7en"called"a"reduced(
condi.onal(distribu.on.
• Consider"the"example"above,"we"consider"a"
marginal(distribu.on:
7

8. Collapsed)Gibbs)Sampling)(cont'd)
• Draw& ,&with& &marginalized&out,&as:
• Draw&
• Can&we&exchange&the&order&of&these&two&steps?&
Why?
8

9. Basic&Guidelines
• Order%of%steps%ma-ers!
• Generally,*one*can*move*components*from*"being'
sampled"*to*"being'condi0oned'on".
• replacing*outputs*with*intermediates*would*change*
the*sta:onary*distribu:on.
• A*variable*can*be*updated*mul:ple*:mes*in*an*
itera:on.
9

10. Why$do$collapsed$samplers$o/en$perform$be3er$than$
full6fledged$Gibbs$samplers?
10

11. Rao$Blackwell+Theorem
Consider)an)example) )and)we)want)to)
es1mate) .)Suppose)we)have)two)tractable)
ways)to)do)so:
(1)$draw$ ,$and$
compute
11

12. Rao$Blackwell+Theorem+(cont'd)
(2)$draw$ $where$ $is$the$
marginal$distribu4on,$and$compute
• Both&are&correct.&By&Strong&LLN,&both& &and& &
converge&to& &almost&surely.
• Which&one&is&be<er?&Can&you&jus@fy&your&answer?
12

13. Rao$Blackwell+Theorem+(cont'd)
• (Rao%Blackwell,Theorem)"Sample"variance"will"be"
reduced"when"some"components"are"marginalized"
out."With"the"se:ng"above,"we"have
• Generally,*reducing)sample)variance*would*also*lead*
to*the*reduc3on*of*autocorrela2on*of*the*chain,*
thus*improving*the*mixing*performance.
13

14. Sampling)with)Auxiliary)Variables
• The%Rao$Blackwell$Theorem%suggests%that%in%order%to%
achieve%be3er%performance,%one%should%try%to%
marginalize%out%as%many%components%as%possible.
• However,%in%many%cases,%one%may%want%to%do%the%
opposite,%that%is,%to%introduce%addi>onal%variables%
to%facilitate%the%simula>ons.
• For%example,%when%the%target$distribu6on%is%
mul6modal,%one%may%use%an%auxiliary%variable%to%
help%the%chain%escape%from%local%traps.
14

15. Use$Auxiliary$Variables
• Specify)an)auxiliary)variable) )and)the)joint)
distribu8on) )such)that)
)for)certain) .
• Design)a)chain)to)update) )using)the)M=H)
algorithm)or)the)Gibbs)sampler.
• The)samples)of) )can)then)be)obtained)through)
marginaliza)on)or)condi)oning.
15

16. Slice&Sampling
16

17. Slice&Sampler
• Sampling* *is*equivalent*to*sampling*
uniformly*from*the*area*under* :*
.
• Gibbs*sampling*based*on*the*uniform*distribu;on*
over* .*Each*itera;on*consists*of*two*steps:
• Given* ,*
• Given* ,*
17

18. Slice&Sampler&(Illustra0on)
18

19. Slice&Sampler&(Discussion)
• Slice&sampler"can"mix"very"rapidly,"as"it"will"not"be"
locally"trapped.
• Slice&sampler"is"o7en"nontrivial"to"implement"in"
prac8ce."Drawing" "is"
some8mes"very"difficult.
• For"distribu8ons"of"certain"forms,"which"have"an"
easy&way"to"draw" ,"
slice&sampling"is"good"strategy.
19

20. Simulated*Tempering
20

21. Gibbs%Measure
A"Gibbs%measure"is"a"probability"
measure"with"a"density"in"the"
following"form:
Here,% %is%called%the%energy&
func*on,% %is%called%the%inverse&
temperature,%and%the%normalizing%
constant% %depends%on% .
21

22. Gibbs%Measure%(cont'd)
In#literature#of#MCMC#sampling,#we#o5en#
parameterize#a#Gibbs#measure#using#the#temperature(
parameter# ,#thus#
.
22

23. Tempered'MCMC
Typical(MCMC(methods(usually(rely(on(local%moves(to(
explore(the(state(space.(What(is(the(problem?
23

24. Tempered'MCMC'(cont'd)
Local&traps&o+en&leads&to&very&poor&mixing.&Can&we&
improve&this?
24

25. Simulated*Tempering
Suppose'we'intend'to'sample'from'
Basic&idea:!Augment!the!target!distribu0on!by!
including!a!temperature(index( ,!with!joint!distribu0on!
given!by
25

26. Simulated*Tempering*(cont'd)
• We$only$collect$samples$at$the$lowest'temperature,$
.
• The$chain$mixes$much$faster$at$high$temperatures,$
but$we$want$to$collect$samples$at$the$lowest$
temperature.$So$we$have$to$constantly$switch$
between$temperatures.
26

27. Simulated*Tempering*(Algorithm)
One$itera)on$of$Simulated*Tempering$has$two$steps:
• (Base&transi+on):#update# #at#the#same#temperature,#
i.e.#holding# #fixed.
• (Temperature&switching):#with# #fixed,#propose#
#with# #such#that#
• Accept#the#change#with#probability#
.
• Any#drawbacks?
27

28. Simulated*Tempering*(Discussion)
• Set% .%Given% ,%we%should%set% %such%that%
uphill%moves%from%( )%should%have%
a%considerable%probability%of%being%accepted.
• Build%the%temperature(ladder%step%by%step%un?l%we%
have%a%sufficiently%smooth%distribu?on%at%the%top.
• The%?me%spent%on%the%base%level% %is%around%
.%If%we%have%too%many%levels,%only%a%very%
small%por?on%of%samples%can%be%used.
28

29. Simulated*Tempering*(Discussion)
• All$temperature$levels$play$an$important$role.$So$it$
is$desirable$to$spend$comparable$amount$of$8me$at$
each$level.$Se:ng$ $for$each$ ,$we$have
• The%normalzing%constants% %are%typically%unknown%
and%es8ma8ng%them%is%very%difficult%and%expensive.
29

30. Parallel&Tempering
(Basic'idea)!rather!than!jumping!between!
temperatures,!it!simultaneously!simulate!mul3ple!
chains,!each!at!a!temperature!level! ,!called!a!replica,!
and!constantly!swap!samples!between!replicas.
30

31. Parallel&Tempering&(Algorithm)
Each%itera*on%consists%of%the%following%steps:
• (Parallel'update):"simulate"each"replica"with"its"own"
transi2on"kernel
• (Replica'exchange):"propose"to"swap"states"between"
two"replicas"(say"the" 7th"and" 7th,"where"
):
31

32. Parallel&Tempering&(Algorithm)
• The%proposal%is%accepted%with%probability%
,%where
• We$collect$samples$from$the$base$replica$(the$one$
with$ ).
• Why$does$this$algorithm$produce$the$desired$
distribu;on?
32

33. Parallel&Tempering&(Jus1fica1on)
Let$ .$We$define
Obviously,+the+step+of+parallel&update+preserves+the+
invariant+distribu5on+ .
33

34. Parallel&Tempering&(Jus1fica1on)
Note%that%the%step%of%replica(exchange%is%symmetric,%i.e.%
the%probabili0es%of%going%up%and%down%are%equal,%then%
according%to%the%Metropolis(algorithm,%we%have%
%with
34

35. Parallel&Tempering&(Discussion)
• It$is$efficient$and$very$easy$to$implement,$especially$
in$a$parallel$compu6ng$environment.
• It$is$o9en$an$art$instead$of$a$technique$to$tune$a$
parallel$tempering$system.
• The$parallel-tempering$is$a$special$case$of$a$large$
family$of$MCMC$methods$called$Extended-Ensemble-
Monte-Carlo,$which$involves$a$collec6on$of$parallel$
Markov$chains$and$the$simula6on$switches$
between$these$them.
35

36. Swendsen'Wang+Algorithm
The$Swendsen'Wang$algorithm$(R.-Swendsen$and$J.-
Wang,$1987)$is$an$efficient$Gibbs$sampling$algorithm$
for$sampling$from$the$extended-Ising-model.
36

37. Standard'Ising'Model
The$standard$Ising&model$is$defined$as
where% %for%each% %is%called%a%spin,%and%
.
• Gibbs&sampling&is&extremely&slow,&especially&when&
the&temperature&is&low.
37

38. Extended'Ising'Model
• We$extend$the$model$by$introducing$addi5onal$
bond%variables$ ,$each$for$an$edge.$Each$bond$
has$two$states:$ $indica5ng$connected$and$ $
indica5ng$disconnected.
• We$define$a$joint$distribu5on$that$couples$the$spins$
and$bonds,
38

39. Extended'Ising'Model'(cont'd)
Here,% %is%described%as%below:
• When& ,& &for&every&
se.ng&of&
• When& ,&
39

40. Extended'Ising'Model'(cont'd)
With%this%se(ng,% %can%be%wri1en%as:
where% :
• when& ,& &must&be&
• when& ,& &is&set&to&zero&with&probability&
.
40

41. Swendsen'Wang+Algorithm
!Each!itera*on!consists!of!two!steps:
• (Clustering):"condi(oned"on"the"spins" ,"draw"the"
bonds" "independenly."For"an"edge"
:
• If" ,"set"
• If" ,"set" "with"probability"
"or" "otherwise.
41

42. Swendsen'Wang+Algorithm
• (Swapping):"condi(oned"on"the"bonds" ,"draw"the"
spins" .
• For"each"connected"component,"draw" "or" "
with"equal"chance,"and"assign"the"resultant"value"
to"all"nodes"in"the"component.
42

43. Swendsen'Wang+Algorithm+
(Illustra7on)In the case of a rectangular grid, this Gibbs sampling algorithm mixes very rapidly.
The following figures illustrate Gibbs sampling. Spin states up and down are
shown by filled and empty circles. Bond states 1 and 0 are shown by thick lines and
thin dotted lines. We start from a state with five connected components. (Remember
that isolated spins count as connected components, albeit of size 1.)
First, let’s update the bonds The forbidden bonds are highlighted
Bonds are forbidden from forming wherever the two adjacent spins are in opposite
states. The bonds that are not forbidden are set to the 1 state with probability p.
After updating the bonds Now we update spins Update bonds again
1.2 Other properties of the extended model
We already mentioned that the partition function Z is the same as that of the Ising
model.
In the case of a rectangular grid, this Gibbs sampling algorithm mixes very rapidly.
The following figures illustrate Gibbs sampling. Spin states up and down are
shown by filled and empty circles. Bond states 1 and 0 are shown by thick lines and
thin dotted lines. We start from a state with five connected components. (Remember
that isolated spins count as connected components, albeit of size 1.)
First, let’s update the bonds The forbidden bonds are highlighted
Bonds are forbidden from forming wherever the two adjacent spins are in opposite
states. The bonds that are not forbidden are set to the 1 state with probability p.
After updating the bonds Now we update spins Update bonds again
1.2 Other properties of the extended model
We already mentioned that the partition function Z is the same as that of the Ising
model.
43

44. Swendsen'Wang+Algorithm+
(Discussion)
• When& &is&large,& &has&a&high&probability&of&
being&set&to&one,&i.e.& &and& &are&likely&to&be&
connected.
• Experiments&show&that&the&Swendsen)Wang&
algorithm&mixes&very&rapidly,&especially&for&
rectangular&grids.
• Can&you&provide&an&intui?ve&explana?on?
44

45. Swendsen'Wang+Algorithm+
(Discussion)
• The%Swendsen'Wang%algorithm%can%be%generalized%
to%Po4s%models%(nodes%can%take%values%from%a%finite%
set).
• The%Swendsen'Wang%algorithm%has%been%widely%
used%in%image%analysis%applicaAons,%e.g.%image%
segmentaAon%(in%this%case,%it%is%called%Swendsen'
Wang,cut).
45

46. Hamiltonian)Monte)Carlo
• An$MCMC$method$based$on$Hamiltonian)Dynamics.$
It$was$originally$devised$for$molecular)simula1on
• In$1987,$a$seminal$paper$by$Duane$et)al$unifies$
MCMC$and$molecular$dynamics.$They$called$it$
Hybrid)Monte)Carlo,$which$abbreviates$to$HMC
• In$many$arEcles,$people$call$it$Hamiltonian)Monte)
Carlo,$as$this$name$is$considered$to$be$more$
specific$and$informaEve,$and$it$retains$the$same$
abbreviaEon$"HMC".
46

47. Mo#va#ng(Example:(Free(Fall
47

48. Mo#va#ng(Example:(Free(Fall
• The%change%of%momentum% %is%caused%by%the%
accumula5on/release%of%the%poten(al+energy:
• The%change%of%loca-on% %is%caused%by%velocity,%the%
deriva-ve%of%kinema-c.energy%w.r.t.%the%momentum:
48

49. Hamiltonian)Dynamics
• Hamiltonian)Dynamics"is"a"generalized"theory"of"the"
classical)mechanics,"which"provides"a"elegant"and"
flexible"abstrac:on"of"a"dynamic"system"in"physics.
• In"Hamiltonian"Dynamics,"a"physical"system"is"
described"by" ,"where" "and" "are"
respec:vely"the"posi1on"and"momentum"of"the" @th"
en:ty.
49

50. Hamilton's+Equa/ons
The$dynamics$of$the$system$is$characterized$by$the$
Hamilton's+Equa/ons:
Here,% %is%called%the%Hamiltonian,%which%can%be%
interpreted%as%the%total)energy%of%the%system.
50

51. Hamilton's+Equa/ons+(cont'd)
• The%Hamiltonian% %is%o)en%formulated%as%the%sum%
of%the%poten+al,energy% %and%the%kine+c,energy% :
• With&this&se)ng,&the&Hamilton's+Equa/ons&become:
51

52. Conserva)on*of*Hamiltonian
The$Hamiltonian$is$conserved,$i.e.,$it$is$invariant$over$
,me:
Intui&vely,,this,reflects,the,law$of$energy$conserva/on.
52

53. Hamiltonian)Reversibility
• The%Hamiltonian)dynamics%is%reversible
• Let%the%ini+al%states%be% %and%the%states%at%
+me% %be% .%Then,%it%we%reverse%the%process,%
star+ng%at% ,%then%the%states%at%+me% %
would%be% .
• In%the%context%of%MCMC,%this%leads%to%the%
reversibility%of%the%underlying%chain.
53

54. Simula'on*of*Hamiltonian*Dynamics
A"natural"idea"to"simulate"Hamiltonian)dynamics"is"to"
use"Euler's)method"over"discre1zed"1me"steps:
Is#this#a#good#method?
54

55. Leapfrog)Method
Be#er%results%can%be%obtained%with%leapfrog:
More%importantly,%the%leapfrog%update%is%reversible.
55

56. Leapfrog)Method)(cont'd)
56

57. Example
Consider)a)Hamiltonian)system:
Write&down&the&Hamilton's+Equa/ons:
Derive&the&solu-on:
57

58. Example((Simula-on)
58

59. Hamiltonian)Monte)Carlo
(Basic'idea):!Consider!the!poten&al)energy!as!the!Gibbs)
energy,!and!introduce!the!"momentums"!as!auxiliary)
variables!to!control!the!dynamics.
59

60. Hamiltonian)Monte)Carlo)(cont'd)
Suppose'the'target&distribu,on'is'
,'then'we'form'an'augmented&
distribu,on'as
Here,%the%loca%ons% %represent%the%variables%of%
interest,%and%the%momentums% %control%the%dynamics%
of%simula7on.
60

61. Hamiltonian)Monte)Carlo)(cont'd)
In#prac(ce,#the#kine%c#energy#is#o2en#formalized#as
61

62. Hamiltonian)Monte)Carlo)(Algorithm)
Each%itera*on%of%HMC%comprises%two%steps:
• Gibbs%update:#sample#the#momentums# #from#the#
Gaussian#prior#given#by
62

63. Hamiltonian)Monte)Carlo)(Algorithm)
• Metropolis*update:#using#Hamiltonian#dynamics#to#
propose#a#new#state.#Star8ng#from# ,#simulate#
the#dynamic#system#with#the#leapfrog#method#for# #
steps#with#step<size# ,#which#yields# .#The#
proposed#state#is#accepted#with#probability:
63

64. HMC$(Discussion)
• If$the$simula.on$is$exact,$we$will$have$
,$and$thus$the$proposed$state$
should$always$be$accepted.$
• In$prac.ce,$there$can$be$some$devia.on$due$to$
discre.za.on,$we$have$to$use$the$Metropolis$
rule$to$guarantee$the$correctness.
64

65. HMC$(Discussion)
• HMC%has%a%high%acceptance%
rate%while%allowing%large%
moves%along%less6constrained%
direc8ons%at%each%itera8on.
• This%is%a%key%advantage%as%
compared%to%random'walk%
proposals,%which,%in%order%to%
maintain%a%reasonably%high%
acceptance%rate,%has%to%keep%
a%very%small%step%size,%
resul8ng%in%substan8al%
correla8on%between%
consecu8ve%samples.
65

66. Tuning&HMC
• For%efficient%simula1on,%it%is%important%to%choose%
appropriate%values%for%both%the%leapfrog%step%size% %
and%the%number%of%leapfrog%steps%per%itera1on% .
• Tuning%HMC%(and%actually%many%generic%sampling%
methods)%oCen%requires%preliminary*runs%with%
different%trial%seGngs%and%different%ini1al%values,%as%
well%as%careful%analysis%of%the%energy%trajectories.
66

67. Tuning&HMC&(cont'd)
• For%most%cases,% %and% %can%be%tuned%independently.
• Too%small%a%stepsize%would%waste%computa8on%
8me,%while%large%stepsize%would%cause%unstable%
simula8on,%and%thus%low%acceptance%rate.
• One%should%choose% %such%that%the%energy%
trajectory%is%stable%and%the%acceptance%rate%is%
maintained%at%a%reasonably%high%level.
• One%should%choose% %such%that%back@and@forth%
movement%of%the%states%can%be%observed.
67

68. Generic'Sampling'Systems
A"number"of"so,ware"systems"are"available"for"
sampling"from"models"specified"by"the"user
• WinBUGS:*based*on*BUGS*(Bayesian*inference*
Using*Gibbs*Sampling).
• provide*a*friendly*language*for*user*to*specify*
the*model
• Running*only*on*Windows
• Note:*The*development*has*stopped*since*2007.
68

69. Generic'Sampling'Systems'(cont'd)
• JAGS:'"Just'Another'Gibbs'Sampler"
• Cross8pla9orm'support
• Use'a'dialect'of'BUGS
• Extensible:'allow'users'to'write'customized'
funcCons,'distribuCons,'and'samplers
69

70. Generic'Sampling'Systems'(cont'd)
• Stan:'"Sampling'Through'Adap5ve'Neighborhoods"
• Core'wri=en'in'C++,'and'ports'available'in'
Python,'R,'Matlab,'and'Julia
• A'user'friendly'language'for'model'specifica5on
• Use'Hamiltonian'Monte'Carlo'(HMC)'and'No'UL
Turn'Samplers'(NUTS)'as'core'algorithm
• Open'source'(GPLv3'licensed)'and'under'ac5ve'
development'on'Github
70

71. Stan%Example
data {
int<lower=0> N;
vector[N] x;
vector[N] y;
}
parameters {
real alpha;
real beta;
real<lower=0> sigma;
}
model {
for (n in 1:N)
y[n] ~ normal(alpha + beta * x[n], sigma);
}
71

72. Generic'Sampling'System'vs.'
Dedicated'Algorithms
Generic' Dedicated'
Easy%to%use% Require%knowledge%and%experience%
High%produc9vity% Time=consuming%to%develop%
Slow% O@en%remarkably%more%efficient%
Limited%flexibility% Necessary%for%many%new%models%
72