Increased access to computing resources has led to the development of algorithms that can run efficiently on multi-core processing units or in distributed computing environments. In the context of Bayesian inference, many parallel computing approaches to fit statistical models have been proposed using Markov chain Monte Carlo methods, but they either have limited gains due to high latency cost or rely on model-specific decompositions. Alternatively, adaptive importance sampling, sequential Monte Carlo, and recursive Bayesian methods provide a parallel-friendly and asymptotically exact framework with well-developed theory for error estimation. We consider a recursive adaptive importance sampling approach that alternates between fast recursive weight updates and sample replenishment steps to balance computational effort while ensuring sample quality. A naive implementation can be unstable and inefficient, so we derive theoretical results to determine the optimal replenishment frequency based on the posterior concentration rate, thus attaining an asymptotically optimal trade-off. We demonstrate the efficacy of our method in simulated experiments and an application of sea surface temperature prediction in the Gulf of Mexico using Gaussian processes.

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