We merge in this note our two discussions about the Read Paper "Particle Markov chain Monte Carlo" (Andrieu, Doucet, and Holenstein, 2010) presented on October 16th 2009 at the Royal Statistical Society, appearing in the Journal of the Royal Statistical Society Series B. We also present a more detailed version of the ABC extension.
Deep Dive into Comments on "Particle Markov Chain Monte Carlo" by C. Andrieu, A. Doucet and R. Hollenstein.
We merge in this note our two discussions about the Read Paper “Particle Markov chain Monte Carlo” (Andrieu, Doucet, and Holenstein, 2010) presented on October 16th 2009 at the Royal Statistical Society, appearing in the Journal of the Royal Statistical Society Series B. We also present a more detailed version of the ABC extension.
The article Andrieu et al. (2010) is clearly going to have significant impact on scientific disciplines with a strong interface with computational statistics and non-linear state space models. Our comments are based on practical experience with PMCMC implementation in latent process multifactor SDE models for commodities (Peters et al., 2009), wireless communications (Nevat et al., 2009) and population dynamics (Hayes et al., 2009), using Rao-Blackwellised particle filters (Doucet et al., 2000) and adaptive MCMC (Roberts and Rosenthal, 2009).
• From our implementations, ideal use cases consist of highly non-linear dynamic equations for a small dimension d x of the state-space, large dimension d θ of the static parameter, and potentially large length T of the time series. In our cases d x was 2 or 3, d θ up to 20, and T between 100 and 400.
• In PMH, non-adaptive MCMC proposals for θ (e.g. tuned according to pre-simulation chains or burn-in iterations) would be costly for large T , and requires to keep N fixed over the whole run of the Markov chain. Adaptive MCMC proposals such as the Adaptive Metropolis sampler (Roberts and Rosenthal, 2009), avoid such issues and proved particularly relevant for large d θ and T , as can be seen in Figure 2.
• The Particle Gibbs (PG) could potentially stay frozen on a state x 1:T (i). Consider a state space model with state transition function almost linear in x n for some range of θ, from which y 1:T is considered to result, and strongly non-linear elsewhere. If the PG samples θ(i) in those regions of strong non-linearity, the particle tree would likely coalesce on the trajectory preserved by the conditional SMC, leaving it with a high importance weight, maintaining (θ(i + 1), x 1:T (i + 1)) = (θ(i), x 1:T (i)) over several iterations. Using PMH within PG would help escape this region, especially using PRC and adaptive SMC kernels, outlined in another comment, to fight the degeneracy of the filter and the high variance of pθ (y 1:T ).
Our comments on adaptive SMC relate to Particle marginal Metropolis-Hastings (PMMH) which has acceptance probability given in Equation ( 13) of the read paper for proposed state (θ * , X * 1:T ), relying on the estimate pθ * (y
w n x * ,k 1:n . Although a small N suffices to approximate the mode of joint path space distribution, producing a reasonable proposal for x 1:T , it results in high variance estimates of pθ * (y 1:T ). We study a population dynamics example from (Hayes et al., 2009, Model 3 excerpt), involving a log-transformed theta-logistic state space model, see (Wang, 2007, Equation 3(a), 3(b)) for parameter settings. PMCMC performance depends on the trade-off between degeneracy of the filter, N , and design of the SMC mutation kernel. Regarding the latter:
• A Rao-Blackwellised filter (Doucet et al., 2000) can improve acceptance rates, Nevat et al. (see 2009).
• Adaptive mutation kernels, which in PMCMC, can be considered as adaptive SMC proposals, can reduce degeneracy on the path space, allowing for higher dimensional state vectors x n . Adaption can be local (within filter) or global (sampled Markov chain history). Though currently particularly designed for ABC methods, the work of Peters et al. (2008) incorporates into the mutation kernel of SMC Samplers (Del Moral et al., 2006) the Partial Rejection Control (PRC) mechanism of Liu (2001), which is also beneficial for PMCMC. PRC adaption reduces degeneracy by rejecting a particle mutation when its incremental importance weight is below a threshold c n . The PRC mutation kernel
can also be used in PMH, where q θ (x n |y n , x n-1 ) is the standard SMC proposal, and
(2)
As presented in Peters et al. (2008), algorithmic choices for q * θ (x n |y n , x n-1 ) can avoid evaluation of r(c n , x n-1 ). Cornebise (2009b) extend this work, developing PRC for Auxiliary SMC samplers, also useful in PMH. Threshold c n can be set adaptively: locally either at each SMC mutation or Markov chain iteration; or globally based on chain acceptance rates. Additionally, c n can be set adaptively via quantile estimates of pre-PRC incremental weights, see Peters et al. (2009).
• Cornebise et al. (2008) Population dynamics state space model with log transformed theta-logistic latent process.
Figure 1: Sequence of simulated states and observations for the population dynamic log-transformed theta logistic model from Wang (2007), with static parame- x 2
Sample path for latent state value x 2 in the PMH via an SIR filter.
Markov chain iteration i 0 1 2 3 4 5 6 7 8 9 10
x 10 4 6.7
6.75
6.8
6.85
6.9
x 37
Sample path for latent state value x 37 in the PMH via an SIR filter. Sample path for parameter r in the PMH via an adaptive Metropolis proposal. , over 100, 000 PMH iterations based on N = 200 particles using a simple SIR filter -the one dimensional state did not call for Rao-Blackwellisation. Note also the effect the Adaptive MCMC proposal for θ, set-up to start at iteration 5, 000,
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