Artificial Intelhgence 56 (1992) 397-398 397 Elsevier

Addendum

Use of the Gibbs sampler in expert sy s t ems *

Jeremy York Department of Stattsttcs, GN-22, Untverstty of Washtngton, Seattle, WA 98195, USA

In my paper "Use of the Gibbs sampler in expert systems", I attempted to survey statistics and genetics literature of interest to researchers in A I I faded to stress, however, that the best reference for understanding Markov chain Monte Carlo (MCMC) methods is the paper by Hastings [3 ], which generalizes the Metropolis algorithm [7] Several recent papers, and others which I neglected to reference, are described below

In early May 1992, papers on MCMC were read by Smith and Roberts [ 10 ], Besag and Green [1 ], and Gilks et al [2] at a meeting of the Royal Statisti- cal Society. The first of these has the most general discussion, and illustrates that the Gibbs sampler IS a special case of the methods presented in [3] Anyone wishing to understand the theoretical issues involved is well ad- vised to consult the thorough and rigorous paper by Tlerney [11] or the abbreviated version [ 12 ]

In the genetics literature, Ott [ 8 ] and Ploughman and Boehnke [ 9 ] present methods for drawing independent samples from the distribution of interest if the graphical structure is simple Lange and Matthysse [4] and Lange

Correspondence to J York, Department of Statistics, Carnegie MeUon Umverslty, Pittsburgh, PA 15213-3890, USA

*Arttf Intell 56 (1) (1992) 115-130

0004-3702/92/$ 05 00 (~) 1992 - - Elsevier Science Pubhshers B V All rights reserved

398 J ~orh

and Sobel [5] use the Metropohs algorithm to perform calculations on a pe&gree

In Lln [6], a variety of Hastings algorithms for genetics are proposed and assessed In th~s paper, and in Lln's ongomg Ph D work under Thompson, the intent is to overcome the problems of reducibility and slow convergence that plague straightforward Metropolis and Gibbs sampling approaches in this context

Acknowledgement

Th~s material ts based upon work supported under a National Science Foundat:on Graduate Fellowship

References

[1] J Besag and P Green Spatml statistics and Bayesian computation, J Ro~ Star 5oc B (to appear 1993)

[2] W Gllks, D Clayton, D Splegelhalter, N Best A McNeil L Sharpies and h Kirby, Modehng complexity apphcatlons of Gibbs sampling m me&cme, J Rot Stat Soc B (to appear 1993)

[3] W Hastings Monte Carlo samphng methods using Marko~ chains and their apphcatlons Btometltka 57 ( 1 ) (1970) 97-109

[4] K Lange and S Matthysse, Simulation of pe&gree genotkpes by random walks, 4tn J Hum Genet 45 (1989) 959-970

[5] K Lange and E Sobel, ~ random walk method for computing genetic location scores 4m J Hum Genet 49 (1991) 1320-1334

[6] S Lm, On the performance of Markov chain Monte Carlo methods on pedigree data and a new algorithm, Tech Rept 231, Department of Statistics, Umvers~tv of Washington Seattle, WA (1992)

[7] N Metropohs, A Rosenbluth, M Rosenbluth and E Teller, Equations ot state calculations by fast computing machines, J Chem Phvs 21 (1953) 1087-1092

[8] J Ott, Computer slmulauon methods m human linkage analysis, P~o¢ Nat ~kad ~1 86 (1989) 4175-4178

[9] L Ploughman and M Boehnke, Estimating the power of a proposed hnkage study for a complex genetic traat, 4m J Hum Genet 44 (1989) 543-551

[10] A Smith and G Roberts, Bayesmn computatmn via the Gibbs sampler and related Markov chain Monte Carlo methods, J Roy Stat Sot B (to appear 1993)

[11 ] L Tlerney Markov chains for exploring posterior distributions Tech Rept 560 School of StatlsUcs, University of Minnesota, Mmneapohs MN ( 1991 )

[ 12] L Tlerney, Exploring postermr &strxbutlons using Markov chains, in Computmg Science and Statistics, Proceedings 23rd Symposmm on the Interlace, Seattle, W~X ( 1991 ) 563-570