Appendices

In Appendix A, we provide proofs of Proposition 1, Proposition 2, and Theorem 1 in the main text. In Appendix B, we provide more details of the Bayesian variable selection (BVS) and stochastic block model (SBM) in Section 4 as well as a detailed simulation study on the spatial clustering model (SCM). In addition, we study the performance of multiple-try Metropolis for the case with multimodal target distributions, following the BVS simulation setting of [54]. In Appendix D, we add a more detailed discussion on parallelization, state space of interest, and the behavior of MTM on continuous state space. Finally, we provide additional tables on the real data analysis results in Appendix E. A.1 Proof of Proposition 1 This section aims to provide a summary of the existing results on proving mixing time bound via path methods.