Interacting Multiple Try Algorithms with Different Proposal Distributions

We propose a new class of interacting Markov chain Monte Carlo (MCMC) algorithms designed for increasing the efficiency of a modified multiple-try Metropolis (MTM) algorithm. The extension with respect to the existing MCMC literature is twofold. The sampler proposed extends the basic MTM algorithm by allowing different proposal distributions in the multiple-try generation step. We exploit the structure of the MTM algorithm with different proposal distributions to naturally introduce an interacting MTM mechanism (IMTM) that expands the class of population Monte Carlo methods. We show the validity of the algorithm and discuss the choice of the selection weights and of the different proposals. We provide numerical studies which show that the new algorithm can perform better than the basic MTM algorithm and that the interaction mechanism allows the IMTM to efficiently explore the state space.

💡 Research Summary

The paper introduces a novel class of interacting Markov chain Monte Carlo (MCMC) algorithms that extend the Multiple‑Try Metropolis (MTM) framework in two complementary directions. First, it relaxes the traditional MTM assumption that all K trial proposals are drawn from a single proposal distribution q(·|x). Instead, each trial i can be generated from its own proposal distribution q_i(·|x), allowing the sampler to combine local, small‑step proposals with global, large‑step moves within a single iteration. This flexibility is particularly valuable for high‑dimensional or multimodal target densities where a single proposal often fails to explore all relevant regions efficiently.

Second, the authors embed the multi‑proposal MTM into a population‑based setting, creating the Interacting Multiple‑Try Metropolis (IMTM) algorithm. A collection of parallel chains (or particles) runs concurrently; each chain independently generates its K proposals using its own set of q_i’s, but the candidate sets are shared among all chains. When a chain selects a trial, the reverse‑move candidates are drawn not only from its own proposals but also from those contributed by the other chains. This cross‑chain sharing introduces a natural interaction mechanism that encourages diverse exploration while preserving detailed balance.

The theoretical contribution consists of a rigorous proof of detailed balance for the generalized algorithm. In the forward move, the weight for trial i is defined as

 w_i(x, y) = π(y) q_i(y|x) /