PMCMC: a
discussion
CP Robert
Introduction
PMCMC Particle Markov chain Monte Carlo:
Model choice
A discussion
Christian P. Robert
Universit´ Paris Dauphine & CREST, INSEE
e
http://www.ceremade.dauphine.fr/~xian
Joint work with Nicolas Chopin and Pierre Jacob

An impressive “tour de force”!
PMCMC: a
discussion
CP Robert
Introduction
That a weighted approximation to the smoothing density
PMCMC
pθ (x1:T |y1:T ) leads to an exact MCMC algorithm...takes several
Model choice iterations to settle in!
Especially when considering that
pθ (x1: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 to
the problem!

An impressive “tour de force”!
PMCMC: a
discussion
CP Robert
Introduction
That a weighted approximation to the smoothing density
PMCMC
pθ (x1:T |y1:T ) leads to an exact MCMC algorithm...takes several
Model choice iterations to settle in!
Especially when considering that
pθ (x1: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 to
the problem!

An impressive “tour de force”!
PMCMC: a
discussion
CP Robert
Introduction
That a weighted approximation to the smoothing density
PMCMC
pθ (x1:T |y1:T ) leads to an exact MCMC algorithm...takes several
Model choice iterations to settle in!
Especially when considering that
pθ (x1: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 to
the problem!

An impressive “tour de force”!
PMCMC: a
discussion
CP Robert
Introduction
That a weighted approximation to the smoothing density
PMCMC
pθ (x1:T |y1:T ) leads to an exact MCMC algorithm...takes several
Model choice iterations to settle in!
Especially when considering that
pθ (x1: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 to
the problem!

A nearly automated implementation
PMCMC: a
discussion
CP Robert Example of a stochastic volatility model
Introduction
PMCMC
yt ∼ N (0, ext ) xt = µ + ρ(xt−1 − µ) + σεt
Model choice
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 )

Automated outcome!
PMCMC: a
discussion
CP Robert
Introduction
PMCMC
Model choice
Figure: Parameter values for µ, ρ and σ, plotted against iteration
indices.

Automated outcome!
PMCMC: a
discussion
CP Robert
Introduction
PMCMC
Model choice
Figure: Autocorrelations of µ, ρ and σ series.

Automated outcome!
PMCMC: a
discussion
CP Robert
Introduction
PMCMC
Model choice
Figure: Acceptation ratio of the Metropolis-Hastings algorithm.

Automated outcome!
PMCMC: a
discussion
CP Robert
Introduction
PMCMC
Model choice
Figure: Correlations between pairs of variables.

Nitpicking!
PMCMC: a
discussion
CP Robert
In Algorithm PIMH,
Introduction
PMCMC what is the use of cumulating SMC and MCMC for fixed
Model choice θ’s? Any hint of respective strength for selecting N SMC
versus N MCMC ?
k
since all simulated X1: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 time
pθ (x1:T |y1:T ) as in PMC?
[Capp´ et al., 2008]
e

Nitpicking!
PMCMC: a
discussion
CP Robert
In Algorithm PIMH,
Introduction
PMCMC what is the use of cumulating SMC and MCMC for fixed
Model choice θ’s? Any hint of respective strength for selecting N SMC
versus N MCMC ?
k
since all simulated X1: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 time
pθ (x1:T |y1:T ) as in PMC?
[Capp´ et al., 2008]
e

Nitpicking!
PMCMC: a
discussion
CP Robert
In Algorithm PIMH,
Introduction
PMCMC what is the use of cumulating SMC and MCMC for fixed
Model choice θ’s? Any hint of respective strength for selecting N SMC
versus N MCMC ?
k
since all simulated X1: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 time
pθ (x1:T |y1:T ) as in PMC?
[Capp´ et al., 2008]
e

Nitpicking!
PMCMC: a
discussion
CP Robert
In Algorithm PIMH,
Introduction
PMCMC what is the use of cumulating SMC and MCMC for fixed
Model choice θ’s? Any hint of respective strength for selecting N SMC
versus N MCMC ?
k
since all simulated X1: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 time
pθ (x1:T |y1:T ) as in PMC?
[Capp´ et al., 2008]
e

Improving upon the approximation
PMCMC: a
discussion
CP Robert Given the additional noise brought by the [whatever]
Introduction
resampling mechanism, what about recycling
PMCMC in the individual weights ωn (X1:n ) by
Model choice Rao–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]

Improving upon the approximation
PMCMC: a
discussion
CP Robert Given the additional noise brought by the [whatever]
Introduction
resampling mechanism, what about recycling
PMCMC in the individual weights ωn (X1:n ) by
Model choice Rao–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]

Improving upon the approximation
PMCMC: a
discussion
CP Robert Given the additional noise brought by the [whatever]
Introduction
resampling mechanism, what about recycling
PMCMC in the individual weights ωn (X1:n ) by
Model choice Rao–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]

Implication for model choice
PMCMC: a
discussion
CP Robert
Introduction
That
PMCMC T
Model choice pθ (y1:T ) = pθ (y1 ) pθ (yn |y1:n−1 )
n=2
is an unbiased estimator of pθ (y1:T is a major property
supporting the PMCMC
Also suggests immediate applications for Bayesian model
choice, as in sequential Monte Carlo techniques such as PMC
[Kilbinger, Wraith, CPR & Benabed, 2009]

Implication for model choice
PMCMC: a
discussion
CP Robert
Introduction
That
PMCMC T
Model choice pθ (y1:T ) = pθ (y1 ) pθ (yn |y1:n−1 )
n=2
is an unbiased estimator of pθ (y1:T is a major property
supporting the PMCMC
Also suggests immediate applications for Bayesian model
choice, as in sequential Monte Carlo techniques such as PMC
[Kilbinger, Wraith, CPR & Benabed, 2009]