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Problem Formulation general state space model/DBN with hidden variables and observed variables. is a Markov process of initial distribution Transition equation Observations Estimate recursion not analytically, numerical approximation scheme
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Rao-Blackwellised Particle Filtering for Dynamic Bayesian Network
Arnaud DoucetNando de Freitas
Kevin MurphyStuart Russell
Introduction
Sampling in high dimension Model has tractable substructure
Analytically marginalized out, conditional on certain other nodes being imputed
Using Kalman filter, HMM filter, junction tree algorithm for general DBNs
Reduce size of the space over we need to sample Rao-Blackwellisation
marginalize out some of the variables
Problem Formulation
general state space model/DBN with hidden variables and observed variables . is a Markov process of initial distribution Transition equation Observations
Estimate recursion
not analytically, numerical approximation scheme
tzty
tz )( 0zp)|( 1tt zzp
},,{ 1:1 tt yyy )|( :1:0 tt yzp
)|()|()|()|()|(
1:1
11:11:0:1:0
tt
tttttttt yyp
zzpzypyzpyzp
Cont’d
Divide hidden variables into two groups and
Conditional posterior distribution is analytically tractable.
Focus on estimating (reduced dimension) Decomposition of posterior from chain rule
Marginal distribution
tz tr tx)|()|()|( 11,;11 tttttttt rrpxrxpzzp
)|()|()|( 11,;11 tttttttt rrpxrxpzzp)|( :1:0 tt yrp
)|(),|()|,( :1:0:0:1:0:1:0:0 tttttttt yrpryxpyxrp
)|()|()|()|(
)|(1:1
1:11:01:0,1:1:1:0
tt
ttttttttt yyp
yrprrpryypyrp
Importance sampling and RAO-Blackwellisation
Sample N i.i.d. random samples(particles) according to Empirical estimate
Expected value of any function of hidden variables
},,1);,{( )(:0
)(:0 Nixr i
tit )|,( :1:0:0 ttt yxrp
N
ittxrtttN dxdr
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tit
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:0)(
:0
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),(1
)|,(),()(
)(:0
)(:0
1
:0:0:1:0:0:0:0
it
it
N
it
tttttNttttN
xrfN
dxdryxrpxrffI
Cont’d
Strong law of large numbers converges almost surely towards as
Central limit theorem
)( tN fI )( tN fI N
),0()]()([ 2tfNttN fIfIN
Importance Sampling
impossible to sample efficiently from target Importance distribution q
Easy to sample p>0 implies q>0
)),(()),(),((
)(:0:0)|,(
:0:0:0:0)|,(
:1:0:0
:1:0:0
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ttt
ttt
)|,()|,(),(,
:1:0:0
:1:0:0:0:0
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ttttt yxrq
yxrpxrwwhere
),(~
)),((
)),(),((
)(ˆ)(ˆ
)(ˆ
)(:0
)(:0
1
)(:0
1
)(:0
)(:0
1
)(:0
)(:0
)(:0
)(:0
1
11
it
itt
N
i
it
N
i
it
it
N
i
it
it
it
itt
tN
tNtN
xrfw
xrw
xrwxrf
fBfAfI
Cont’d
The case where one can marginalize out analytically, propose alternative estimate for
Alternative importance sampling estimate of
To reach a given precision, will require a reduced number N of samples over
tx :0
)( tfI
)( tfI
N
i
it
N
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itt
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1
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1
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)(:0),|(
2
22
)((
)(),((
)(ˆ)(ˆ
)(ˆ )(:0:1:0
tttttt
tt
ttt
dxyxrqyrq
yrqyrprwwhere
:0:1:0:0:1:0
:1:0
:1:0:0
)|,()|(
)|()|()(
)(ˆ2tN fI
)(ˆ1tN fI
Rao-Blackwellised particle filters
Sequential importance sampling step For i=1,…,N sample:
and set:
For i=1,…,N evaluate importance weights up to a normalizing constant:
For i=1,…,N normalize importance weights:
Selection step multiply/suppress samples with high/low importance weights to obtain random samples approximately distributed
MCMC step Apply a markov transition kernel with invariant distribution given by
to obtain to obtain
),|(~)ˆ( :1)(
1:0)(
titt
it yrrqr
)ˆ( )(:0itr
)ˆ|ˆ()ˆ( )(1:0
)()(:0
it
it
it rrqr
)|ˆ(),|ˆ()|ˆ(
1:1)(
1:0:1)(
1:0)(
:1)(
:0)(
t
itt
it
it
titi
t yrpyrrqyrpw
1
1
)()()( ][~
N
j
jt
it
it www
)(~ itw
)~( )(:0itr )|~( :1
)(:0 tit yrp
)|( :1)(
:0 tit yrp )( )(
:0itr
Robot localization and MAP building
Problem of concurrent localization and map learning Location Color of each grid cell Observation
Basic idea of algorithm Sample with PF Marginalize out since they are conditionally independent given
},...,1{ Lt NL
LCt NiNiM ,...,1},,...,1{)( ))(( ttt LMfY
tL :1
)(iM t
tL :1
