Discussion of “Equi-energy sampler” by Kou, Zhou and Wong. (English) Zbl 1246.82051

Atchadé, Yves F.; Liu, Jun S.

Ann. Stat. 34, No. 4, 1620-1628 (2006).

Summary: We congratulate S. Kou, Q. Zhou and W. Wong [ibid. 34, No. 4, 1581–1619 (2006; Zbl 1246.82054)] 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 equi-energy (EE) sampler with MCS. Overall, we found the two algorithms to be of comparable efficiency for the simulation problem considered. In the second part, we develop some additional convergence results for the EE sampler.

MSC:

82B80	Numerical methods in equilibrium statistical mechanics (MSC2010)
65C05	Monte Carlo methods
65C40	Numerical analysis or methods applied to Markov chains
94A20	Sampling theory in information and communication theory
62F15	Bayesian inference
62D05	Sampling theory, sample surveys

Citations:

Zbl 1246.82054

References:

[1]	Atchadé, Y. F. and Liu, J. S. (2004). The Wang–Landau algorithm for Monte Carlo computation in general state spaces. Technical report.
[2]	Atchadé, Y. F. and Rosenthal, J. S. (2005). On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11 815–828. · Zbl 1085.62097 · doi:10.3150/bj/1130077595
[3]	Azuma, K. (1967). Weighted sums of certain dependent random variables. Tôhoku Math. J. ( 2 ) 19 357–367. · Zbl 0178.21103 · doi:10.2748/tmj/1178243286
[4]	Berg, B. A. and Neuhaus, T. (1992). Multicanonical ensemble: A new approach to simulate first-order phase transitions. Phys. Rev. Lett. 68 9–12.
[5]	Isaacson, D. L. and Madsen, R. W. (1976). Markov Chains : Theory and Applications . Wiley, New York. · Zbl 0332.60043
[6]	Marinari, E. and Parisi, G. (1992). Simulated tempering: A new Monte Carlo scheme. Europhys. Lett. 19 451–458.
[7]	Royden, H. L. (1963). Real Analysis . Collier-Macmillan, London. · Zbl 0121.05501
[8]	Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. 22 1701–1762. · Zbl 0829.62080 · doi:10.1214/aos/1176325750
[9]	Wang, F. and Landau, D. P. (2001). Efficient, multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett. 86 2050–2053.