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]