  The Equi-Energy Sampler (EES) introduced by Kou et al [2006] is based on a
population of chains which are updated by local moves and global moves, also
called equi-energy jumps. The state space is partitioned into energy rings, and
the current state of a chain can jump to a past state of an adjacent chain that
has energy level close to its level. This algorithm has been developed to
facilitate global moves between different chains, resulting in a good
exploration of the state space by the target chain. This method seems to be
more efficient than the classical Parallel Tempering (PT) algorithm. However it
is difficult to use in combination with a Gibbs sampler and it necessitates
increased storage. In this paper we propose an adaptation of this EES that
combines PT with the principle of swapping between chains with same levels of
energy. This adaptation, that we shall call Parallel Tempering with Equi-Energy
Moves (PTEEM), keeps the original idea of the EES method while ensuring good
theoretical properties, and practical implementation even if combined with a
Gibbs sampler. Performances of the PTEEM algorithm are compared with those of
the EES and of the standard PT algorithms in the context of mixture models, and
in a problem of identification of gene regulatory binding motifs.
