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# Discussion of PMCMC

## on Oct 15, 2009

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## Discussion of PMCMCPresentation Transcript

• 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]