  Recently, many Markov chain Monte Carlo methods have been developed with
deterministic reversible transform proposals inspired by the Hamiltonian Monte
Carlo method. The deterministic transform is relatively easy to reconcile with
the local information (gradient etc.) of the target distribution. However, as
the ergodic theory suggests, these deterministic proposal methods seem to be
incompatible with robustness and lead to poor convergence, especially in the
case of target distributions with heavy tails. On the other hand, the Markov
kernel using the Haar measure is relatively robust since it learns global
information about the target distribution introducing global parameters.
However, it requires a density preserving condition, and many deterministic
proposals break this condition. In this paper, we carefully select
deterministic transforms that preserve the structure and create a Markov
kernel, the Weave-Metropolis kernel, using the deterministic transforms. By
combining with the Haar measure, we also introduce the Haar-Weave-Metropolis
kernel. In this way, the Markov kernel can employ the local information of the
target distribution using the deterministic proposal, and thanks to the Haar
measure, it can employ the global information of the target distribution.
Finally, we show through numerical experiments that the performance of the
proposed method is superior to other methods in terms of effective sample size
and mean square jump distance per second.
