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

We develop a Bayesian inference method for diffusions observed discretely and with noise, which is free of discretisation bias. Unlike existing unbiased inference methods, our method does not rely on exact simulation techniques. Instead, our method uses standard time-discretised approximations of diffusions, such as the Euler-Maruyama scheme. Our approach is based on particle marginal Metropolis-Hastings, a particle filter, randomised multilevel Monte Carlo, and importance sampling type correction of approximate Markov chain Monte Carlo. The resulting estimator leads to inference without a bias from the time-discretisation as the number of Markov chain iterations increases. We give convergence results and recommend allocations for algorithm inputs. Our method admits a straightforward parallelisation, and can be computationally efficient. The user-friendly approach is illustrated on three examples, where the underlying diffusion is an Ornstein-Uhlenbeck process, a geometric Brownian motion, and a 2d non-reversible Langevin equation.

Summary Approximate Bayesian computation enables inference for complicated probabilistic models with intractable likelihoods using model simulations. The Markov chain Monte Carlo implementation of approximate Bayesian computation is often sensitive to the tolerance parameter: low tolerance leads to poor mixing and large tolerance entails excess bias. We propose an approach that involves using a relatively large tolerance for the Markov chain Monte Carlo sampler to ensure sufficient mixing and post-processing the output, leading to estimators for a range of finer tolerances. We introduce an approximate confidence interval for the related post-corrected estimators and propose an adaptive approximate Bayesian computation Markov chain Monte Carlo algorithm, which finds a balanced tolerance level automatically based on acceptance rate optimization. Our experiments show that post-processing-based estimators can perform better than direct Markov chain Monte Carlo targeting a fine tolerance, that our confidence intervals are reliable, and that our adaptive algorithm leads to reliable inference with little user specification.

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