Importance sampling type estimators based on approximate marginal Markov chain Monte Carlo

Abstract

We consider importance sampling (IS) type weighted estimators based on Markov chain Monte Carlo (MCMC) targeting an approximate marginal of the target distribution. In the context of Bayesian latent variable models, the MCMC typically operates on the hyperparameters, and the subsequent weighting may be based on IS or sequential Monte Carlo (SMC), but allows for multilevel techniques as well. The IS approach provides a natural alternative to delayed acceptance (DA) pseudo-marginal/particle MCMC, and has many advantages over DA, including a straightforward parallelization and additional flexibility in MCMC implementation. We detail minimal conditions which ensure strong consistency of the suggested estimators, and provide central limit theorems with expressions for asymptotic variances. We demonstrate how our method can make use of SMC in the state space models context, using Laplace approximations and time-discretized diffusions. Our experimental results are promising and show that the IS-type approach can provide substantial

Unbiased Inference for Discretely Observed Hidden Markov Model Diffusions

“…Proposition 2 is a restatement of Theorem 7.4.2 of [8] in case A (i) t−1 are sampled independently ('multinomial resampling'). The extension to the general unbiased case is straightforward; see [33].…”

“…The above result follows directly from the results of Section 4. It can also be seen as a multilevel version of Proposition 23 of [33], with straightforward estimators for σ 2 . See Section 5 for suggested choices for p and number of particles run at each level.…”

“…It is generally safer to choose > 0 [33], which ensures that the IS type correction in phase (P2) will yield consistent inference for the ideal model 12). Setting > 0 may be helpful otherwise in terms of improved mixing, as the PMMH will target marginally an averaged probability between a 'flat' prior and a 'multimodal' = 0 marginal posterior.…”

“…The unbiased and computationally efficient inference method suggested in this paper is built firstly on PMMH, using Euler type discretisations, but using a PMMH targeting a coarse-level model, which is less computationally expensive. This does not yield unbiased inference yet, but it can be achieved by an importance sampling (IS) type correction; see [33].…”

“…Besides being unbiased and efficient, our method is user-friendly, as it is a combination of well-known and relatively straightforward components: PMMH, Euler approximations, PF, rMLMC, and an IS type estimator. For more about the strengths of the method, see Remark 11 later, as well as [15,33] for more discussion about IS (type) estimators based on approximate Markov chain Monte Carlo (MCMC).…”