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- 1. PMCMC: a discussion CP Robert Introduction PMCMC Model choice Particle Markov chain Monte Carlo: A discussion Christian P. Robert Universit´e Paris Dauphine & CREST, INSEE http://www.ceremade.dauphine.fr/~xian Joint work with Nicolas Chopin and Pierre Jacob
- 2. PMCMC: a discussion CP Robert Introduction PMCMC Model choice An impressive “tour de force”! That a weighted approximation to the smoothing density pθ(x1:T |y1:T ) leads to an exact MCMC algorithm...takes several 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!
- 3. PMCMC: a discussion CP Robert Introduction PMCMC Model choice An impressive “tour de force”! That a weighted approximation to the smoothing density pθ(x1:T |y1:T ) leads to an exact MCMC algorithm...takes several 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!
- 4. PMCMC: a discussion CP Robert Introduction PMCMC Model choice An impressive “tour de force”! That a weighted approximation to the smoothing density pθ(x1:T |y1:T ) leads to an exact MCMC algorithm...takes several 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!
- 5. PMCMC: a discussion CP Robert Introduction PMCMC Model choice An impressive “tour de force”! That a weighted approximation to the smoothing density pθ(x1:T |y1:T ) leads to an exact MCMC algorithm...takes several 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!
- 6. PMCMC: a discussion 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 )
- 7. PMCMC: a discussion CP Robert Introduction PMCMC Model choice Automated outcome! Figure: Parameter values for µ, ρ and σ, plotted against iteration indices.
- 8. PMCMC: a discussion CP Robert Introduction PMCMC Model choice Automated outcome! Figure: Autocorrelations of µ, ρ and σ series.
- 9. PMCMC: a discussion CP Robert Introduction PMCMC Model choice Automated outcome! Figure: Acceptation ratio of the Metropolis-Hastings algorithm.
- 10. PMCMC: a discussion CP Robert Introduction PMCMC Model choice Automated outcome! Figure: Correlations between pairs of variables.
- 11. PMCMC: a discussion CP Robert Introduction PMCMC Model choice Nitpicking! In Algorithm PIMH, what is the use of cumulating SMC and MCMC for ﬁxed θ’s? Any hint of respective strength for selecting NSMC versus NMCMC? since all simulated Xk 1: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´e et al., 2008]
- 12. PMCMC: a discussion CP Robert Introduction PMCMC Model choice Nitpicking! In Algorithm PIMH, what is the use of cumulating SMC and MCMC for ﬁxed θ’s? Any hint of respective strength for selecting NSMC versus NMCMC? since all simulated Xk 1: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´e et al., 2008]
- 13. PMCMC: a discussion CP Robert Introduction PMCMC Model choice Nitpicking! In Algorithm PIMH, what is the use of cumulating SMC and MCMC for ﬁxed θ’s? Any hint of respective strength for selecting NSMC versus NMCMC? since all simulated Xk 1: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´e et al., 2008]
- 14. PMCMC: a discussion CP Robert Introduction PMCMC Model choice Nitpicking! In Algorithm PIMH, what is the use of cumulating SMC and MCMC for ﬁxed θ’s? Any hint of respective strength for selecting NSMC versus NMCMC? since all simulated Xk 1: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´e et al., 2008]
- 15. PMCMC: a discussion 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) by 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]
- 16. PMCMC: a discussion 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) by 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]
- 17. PMCMC: a discussion 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) by 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]
- 18. PMCMC: a discussion 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 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]
- 19. PMCMC: a discussion 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 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]

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