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Slides of the discussion given at the RSS on October 14, 2009, about Andrieu-Doucet-Holenstein Read Paper
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PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Particle Markov chain Monte Carlo:A discussion
Christian P. Robert
Universite Paris Dauphine & CREST, INSEEhttp://www.ceremade.dauphine.fr/~xian
Joint work with Nicolas Chopin and Pierre Jacob
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
An impressive “tour de force”!
That a weighted approximation to the smoothing densitypθ(x1:T |y1:T ) leads to an exact MCMC algorithm...takes severaliterations to settle in!Especially when considering that
pθ(x?1:T |y1:T )/pθ(x1:T (i− 1)|y1:T ) (11)
is not unbiased![Beaumont, Cornuet, Marin & CPR, 2009]
Conditioning on the lineage [in PG] is an awesome resolution tothe problem!
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
An impressive “tour de force”!
That a weighted approximation to the smoothing densitypθ(x1:T |y1:T ) leads to an exact MCMC algorithm...takes severaliterations to settle in!Especially when considering that
pθ(x?1:T |y1:T )/pθ(x1:T (i− 1)|y1:T ) (11)
is not unbiased![Beaumont, Cornuet, Marin & CPR, 2009]
Conditioning on the lineage [in PG] is an awesome resolution tothe problem!
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
An impressive “tour de force”!
That a weighted approximation to the smoothing densitypθ(x1:T |y1:T ) leads to an exact MCMC algorithm...takes severaliterations to settle in!Especially when considering that
pθ(x?1:T |y1:T )/pθ(x1:T (i− 1)|y1:T ) (11)
is not unbiased![Beaumont, Cornuet, Marin & CPR, 2009]
Conditioning on the lineage [in PG] is an awesome resolution tothe problem!
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
An impressive “tour de force”!
That a weighted approximation to the smoothing densitypθ(x1:T |y1:T ) leads to an exact MCMC algorithm...takes severaliterations to settle in!Especially when considering that
pθ(x?1:T |y1:T )/pθ(x1:T (i− 1)|y1:T ) (11)
is not unbiased![Beaumont, Cornuet, Marin & CPR, 2009]
Conditioning on the lineage [in PG] is an awesome resolution tothe problem!
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
A nearly automated implementation
Example of a stochastic volatility model
yt ∼ N (0, ext) xt = µ+ ρ(xt−1 − µ) + σεt
with 102 particles and 104 Metropolis–Hastings iterations,based on 100 simulated observations, with parameter moves
µ∗ ∼ N (µ, 10−2)
ρ∗ ∼ N (ρ, 10−2)
log σ∗ ∼ N (σ, 10−2)
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Automated outcome!
Figure: Parameter values for µ, ρ and σ, plotted against iterationindices.
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Automated outcome!
Figure: Autocorrelations of µ, ρ and σ series.
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Automated outcome!
Figure: Acceptation ratio of the Metropolis-Hastings algorithm.
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Automated outcome!
Figure: Correlations between pairs of variables.
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Nitpicking!
In Algorithm PIMH,
what is the use of cumulating SMC and MCMC for fixedθ’s? Any hint of respective strength for selecting NSMC
versus NMCMC?
since all simulated Xk1:T are from pθ(x1:T |y1:T ), why fail to
recycle the entire simulation story at all steps?
why isn’t the distribution of X1:T (i) at any fixed timepθ(x1:T |y1:T ) as in PMC?
[Cappe et al., 2008]
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Nitpicking!
In Algorithm PIMH,
what is the use of cumulating SMC and MCMC for fixedθ’s? Any hint of respective strength for selecting NSMC
versus NMCMC?
since all simulated Xk1:T are from pθ(x1:T |y1:T ), why fail to
recycle the entire simulation story at all steps?
why isn’t the distribution of X1:T (i) at any fixed timepθ(x1:T |y1:T ) as in PMC?
[Cappe et al., 2008]
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Nitpicking!
In Algorithm PIMH,
what is the use of cumulating SMC and MCMC for fixedθ’s? Any hint of respective strength for selecting NSMC
versus NMCMC?
since all simulated Xk1:T are from pθ(x1:T |y1:T ), why fail to
recycle the entire simulation story at all steps?
why isn’t the distribution of X1:T (i) at any fixed timepθ(x1:T |y1:T ) as in PMC?
[Cappe et al., 2008]
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Nitpicking!
In Algorithm PIMH,
what is the use of cumulating SMC and MCMC for fixedθ’s? Any hint of respective strength for selecting NSMC
versus NMCMC?
since all simulated Xk1:T are from pθ(x1:T |y1:T ), why fail to
recycle the entire simulation story at all steps?
why isn’t the distribution of X1:T (i) at any fixed timepθ(x1:T |y1:T ) as in PMC?
[Cappe et al., 2008]
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Improving upon the approximation
Given the additional noise brought by the [whatever]resampling mechanism, what about recycling
in the individual weights ωn(X1:n) byRao–Blackwellisation of the denominator in eqn. (7)?
past iterations with better reweighting schemes like AMIS?
[Cornuet, Marin, Mira & CPR, 2009]
Danger Uncontrolled adaptation?
for deciding upon future N ’s
for designing better SMC’s
[Andrieu & CPR, 2005]
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Improving upon the approximation
Given the additional noise brought by the [whatever]resampling mechanism, what about recycling
in the individual weights ωn(X1:n) byRao–Blackwellisation of the denominator in eqn. (7)?
past iterations with better reweighting schemes like AMIS?
[Cornuet, Marin, Mira & CPR, 2009]
Danger Uncontrolled adaptation?
for deciding upon future N ’s
for designing better SMC’s
[Andrieu & CPR, 2005]
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Improving upon the approximation
Given the additional noise brought by the [whatever]resampling mechanism, what about recycling
in the individual weights ωn(X1:n) byRao–Blackwellisation of the denominator in eqn. (7)?
past iterations with better reweighting schemes like AMIS?
[Cornuet, Marin, Mira & CPR, 2009]
Danger Uncontrolled adaptation?
for deciding upon future N ’s
for designing better SMC’s
[Andrieu & CPR, 2005]
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Implication for model choice
That
pθ(y1:T ) = pθ(y1)T∏n=2
pθ(yn|y1:n−1)
is an unbiased estimator of pθ(y1:T is a major propertysupporting the PMCMC
Also suggests immediate applications for Bayesian modelchoice, as in sequential Monte Carlo techniques such as PMC
[Kilbinger, Wraith, CPR & Benabed, 2009]
PMCMC: adiscussion
CP Robert
Introduction
PMCMC
Model choice
Implication for model choice
That
pθ(y1:T ) = pθ(y1)T∏n=2
pθ(yn|y1:n−1)
is an unbiased estimator of pθ(y1:T is a major propertysupporting the PMCMC
Also suggests immediate applications for Bayesian modelchoice, as in sequential Monte Carlo techniques such as PMC
[Kilbinger, Wraith, CPR & Benabed, 2009]