 
Summary: Discussion of the paper by Kou, Zhou and Wong
Yves F. ATCHADE1 and Jun S. LIU2
(October, 2005)
We congratulate Samuel Kou, Qing Zhou and Wing Wong (referred subsequently as KZW) for
this beautifully written paper, which opens a new direction in Monte Carlo computation. This
discussion has two parts. First, we describe a very closely related method, multicanonical sampling
(MCS), and report a simulation example that compares the EquiEnergy (EE) sampler with MCS.
Over all, we found the two algorithms to be of comparable eciency for the simulation problem
considered. In the second part, we develop some additional convergence results for the EE sampler.
1 A multicanonical sampling algorithm
Here, we take on KZW's discussion about the comparison of the EE sampler and MCS. We compare
the EE sampler with a general states space extension of MCS proposed by Atchade and Liu (2004).
We compare the two algorithms on the multimodal distribution discussed by KZW in Section 3.4.
Let (X, B, ) be the state space equipped with its algebra and appropriate measure, and let
(x) eh(x)
be the density of interest. Following the notations of KZW, we let H0 < H1 <
· · · < HKe < HKe+1 = be a sequence of energy levels and let Dj = {x X : h(x) [Hj, Hj+1)},
0 j Ke be the energy rings. For x X, dene I(x) = j if x Dj. Let 1 = T0 <
T1 < . . . < TKt be a sequence of temperatures. We use the notation k(i)
(x) = eh(x)/Ti
