trans-dimensional mcmc peter green - vcla...

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ICCV05 Tutorial: MCMC for Vision. Zhu / Dellaert / Tu October 2005

Trans-dimensional MCMC

Frank Dellaert


References

• Peter Green, Reversible jump Markov chain MonteCarlo computation and Bayesian model determination,Biometrika, 1995

• Peter Green, Trans-dimensional Markov chain MonteCarlo, in “Highly Structured Stochastic Systems”, 2003

• David Hastie, Ph.D. Thesis, 2005• Paskin & Thrun, Robotic Mapping with Polygonal

Random Fields, UAI 2005


Outline

• Model Selection• Reversible Jump MCMC• Regression Example• Rao-Blackwellized Sampling• Polygonal Random Fields


Model Selection

• Most common case: inference on p(x|z), xcontinuous parameters

• What if there are competing models ?

!

p(x1 | z),x1 " R3

!

p(x2 | z),x2 " R5

!

p(x3 | z),x3 " R7

2


Ex. 1: Regression

• Degree of polynomial ?


Ex.2: 3D Curve Fitting

• How many control points ?• How many sharp corners ?

Image courtesy of Michael Kaess


Ex.3: Tracking Multiple Targets

• With an unknown number of targetsImage courtesy of Zia Khan


Union Space

• Model indicator k• Parameter vector• State space = union space

!

"k# R

nk

!

X = Uk"K ({k}# R

nk )

!

{1}" R3

!

{2}" R5

!

{3}" R7

!

"

!

"

!

"K

3


Graphical Model

k

θk

z• Joint density factors as

p(z,θk,k)= p(z|θk,k) p(θk|k) p(k)

Model indicatorp(k)

Datap(z|θk,k)

Model parametersp(θk |k)


Outline



Trans-Dimensional MCMC

• set up Markov chain in union space X• allow trans-dimensional jumps

!

{1}" R3

!

{2}" R5

!

{3}" R7

!

"

!

"

!

"K


Regression Example

• Live Demo !

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Explicit Jump Random Variables

• Re-state MCMC to avoid measure theory• Explicit jump random variable u• Proposal density q(x,x’) replaced by

– draw u ~ g(u)– calculate x’=h(x,u)

Pictorial representationinspired by Peter Green

g(u)

u

xx’=h(x,u)


Example (single dimension)

• Random walk on SO(2)

!

q(x,x ') = N(x';x," 2)

#

u ~ N(0," 2)

x'= h(x,u) = x + u

g(u)

u

x

x’=x+u


“Jacobian”

A New Proposal Ratio

g(u)

u

xx’=h(x,u)

g(u’)

u’

=h’(x’,u’)

!

a =" (x')q(x',x)

" (x)q(x,x ')

!

a =" (x')g(u')

" (x)g(u)

#(x',u')

#(x,u)


Jacobians• Jacobian arises from change of variable• Review from probability 101:

!

x ~ g(x)

y = t(x)

f (y) ="t(x)

"x

#1

g(t#1(y)) =

"x

"yg(x)

!

x ~ g(x)

y = t(x) = x /2

f (y) ="t(x)

"x

#1

g(t#1(y)) = 2g(2y)

g(x)

f(y)

Simple example:

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Detailed Balance in RJMCMC

!

f (y)dy = g(x)dx and dx ="x

"ydy hence : f (y)dy =

"x

"yg(x)dy

• Change of variables as differential equality

!

"(x,x ')# (x)g(u)dxdu ="(x',x)g(u)# (x ')g(u')dx 'du'

dx'du'=$(x ',u')

$(x,u)dxdu

"(x,x ') =min 1,# (x ')g(u')

# (x)g(u)

$(x ',u')

$(x,u)

% & '

( ) *

Shows up in detailed balance equation:


Reversible -> Diffeomorphism

• We need to calculate both x’ and u’:(x’,u’) = t (x,u)

• Required that t is a diffeomorphism:– invertible– differentiable

• Jacobian =

!

"(x ',u')

"(x,u)="t(x,u)

"(x,u)=

"tx '(x,u)

"x

"tx'(x,u)

"u"t

u'(x,u)

"x

"tu'(x,u)

"u


A Constructive MCMC Recipe (Green)

• (Conventional) MCMC with jump variables:– draw u ~ g(u)– calculate proposal x’=h(x,u)– calculate reverse jump variable u’ s.t. x=h’(x’,u’)– calculate acceptance ratio:

!

a =min 1," (x ')g(u')

" (x)g(u)

#(x ',u')

#(x,u)

$ % &

' ( )


1D Example (continued)

• Random walk on SO(2)

!

u ~ N(0," 2)

x'= x + u

!

"(x + u,#u)

"(x,u)=

"(x + u)

"x

"(x + u)

"u"(#u)

"x

"(#u)

"u

=1 1

0 #1=1 g(u)

u

x

x’=x+u

!

u'= x " x '= "u

# t(x,u) = (x + u,"u)

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Reversible-Jump MCMC

g(u)

u

xx’=h(x,u)

g(u’)

u’

=h’(x’,u’)

!

a =" (x')g(u')

" (x)g(u)

#(x',u')

#(x,u)

• Story holds if x and x’have different dimensions !


5D5D

2D 3D

Dimension Matching Constraint• Requirement that (x’,u’)=t(x,u)

remains a diffeomorphism⇒ dim(x’)+dim(u’)= dim(x)+dim(u)

u

xx’=h(x,u)

u’

=h’(x’,u’)2D

3D


Move Types

• In each space, proposal is mixture of differentmove types

• Each move has probability jm(x)

m=1m=2

m=3

x!

" a =# (x') jm (x ')gm (u')

# (x) jm (x)gm (u)

$(x ',u')

$(x,u)


Outline


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Regression Example

• k = degree of polynomial

P(k=0)=0.5 P(k=1)=0.5


Regression Example


!

{0}" R1

!

{1}" R2

!

{2}" R3!

a =" (x') jm (x ')gm (u')

" (x) jm (x)gm (u)

#(x ',u')

#(x,u)

m=01,10 m=12,21m=0 m=1 m=2 m=23,32


Regression Example


!

{0}" R1

!

{1}" R2

!

{2}" R3!

a =" (x') jm (x ')gm (u')

" (x) jm (x)gm (u)

#(x ',u')

#(x,u)

!

"(c '0,c'

1)

"(c0,u)

=1 0

0 1=1

!

x = c0[ ]

!

x'=c0

u

"

# $ %

& '

g(u)

u


Regression Example

• Ambivalent data• 10000 Samples

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Regression Example

• Less Noise


Regression Example

• More Data


Regression Example

• No ambivalence


Regression Example

• No ambivalencefor k=0 model

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Careful Move Design

• E.g., move from constantto line such that reachessame height at mean(x)

!

{0}" R1

!

{1}" R2

!

{2}" R3

!

"(c '0,c'

1)

"(c0,u)

=1 0

1/ x #1/ x =1/ x !

x = c0[ ]

!

x'

!

x'=c0

(c0" u) / x

#

$ %

&

' (

!

x


Tracking Multiple Targets

• Showing meannumber of targets

Khan et al PAMI October


Outline



Rao-Blackwellized Sampling

• Second strategy:– integrate out model parameters θk

– sample over marginal posterior

!

{1}" R3

!

{2}" R5

!

{3}" R7

!

"

!

"

!

"K!

p(k | z)" p(k) p(z |#k,k)p(#k | k)# k$

!

"k

#!

p(k | z)

1 2 3

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Rao-Blackwell Theorem

• Variance of sample approximation isreduced when part of state space isintegrated out


Curve Fitting Results

Images courtesy of Michael Kaess


Probabilistic Topological Maps

Sampling over the space of topologies


The space of topologies

Topologies ⇔ Space of set Partitions, very big !!

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Experiments


Experiments (2)

With appearance measurements

99.5% 0.25% 0.14% 0.12% 0.01%

Only odometry

43% 14% 7% 3% 2%


Outline



Images by Mark Paskin

• From field of spatial statistics• Space = colorings of R2 in window D• Uses measure theory to build a (non-

RJMCMC) sampler that satisfies detailedbalance

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PRF in Robotics

Images by Mark Paskin

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