HammersleyClifford定理.docx

1、HammersleyClifford定理Hammersley Clifford定理Hammersley Clifford theorem The Hammersley Clifford theorem is aresult in probability theory,mathematical statistics and statistical mechanics,that gives necessary and sufficient conditions under which apositive probability distribution can be represented as

2、aMarkov network(also known as aMarkov random field).It states that aprobability distribution that has apositive mass or density satisfies one of the Markov properties with respect to an undirected graph Gif and only if it is aGibbs random field,that is,its density can be factorized over the cliques(

3、or complete subgraphs)of the graph.The relationship between Markov and Gibbs random fields was initiated by Roland Dobrushin1and Frank Spitzer2in the context of statistical mechanics.The theorem is named after John Hammersley and Peter Clifford who proved the equivalence in an unpublished paper in 1

4、971.34Simpler proofs using the inclusion-exclusion principle were given independently by Geoffrey Grimmett,5Preston6and Sherman7in 1973,with afurther proof by Julian Besag in 1974.8NotesDobrushin,P.L.(1968),The Description of aRandom Field by Means of Conditional Probabilities and Conditions of Its

5、Regularity,Theory of Probability and its Applications 13(2)：197 224,doi：10.1137/1113026,Spitzer,Frank(1971),Markov Random Fields and Gibbs Ensembles,The American Mathematical Monthly 78(2)：142 154,doi：10.2307/2317621,JSTOR 2317621,Hammersley,J.M.；Clifford,P.(1971),Markov fields on finite graphs and

6、lattices,Clifford,P.(1990),Markov random fields in statistics,in Grimmett,G.R.；Welsh,D.J.A.,Disorder in Physical Systems：A Volume in Honour of John M.Hammersley,Oxford University Press,pp.19 32,ISBN 0198532156,MR 1064553,retrieved 2009-05-04Grimmett,G.R.(1973),A theorem about random fields,Bulletin

7、of the London Mathematical Society 5(1)：81 84,doi：10.1112/blms/5.1.81,MR 0329039Preston,C.J.(1973),Generalized Gibbs states and Markov random fields,Advances in Applied Probability 5(2)：242 261,doi：10.2307/1426035,JSTOR 1426035,MR 0405645.JSTOR 1426035,Sherman,S.(1973),Markov random fields and Gibbs

8、 random fields,Israel Journal of Mathematics 14(1)：92 103,doi：10.1007/BF 02761538,MR 0321185Besag,J.(1974),Spatial interaction and the statistical analysis of lattice systems,Journal of the Royal Statistical Society.Series B(Methodological)36(2)：192 236,MR 0373208.JSTOR 2984812 Further reading Bilme

9、s,Jeff(Spring 2006),Handout 2：Hammersley Clifford,course notes from University of Washington course.Grimmett,Geoffrey,Probability on Gr aphs,Chapter 7,Helge,The Hammersley Clifford Theorem and its Impact on Modern Statistics,probability-related article is astub.You can help Wikipedia by expanding it

10、.Retrieved from Clifford_theoremFrom Wikipedia,the free encyclopediaThe first afternoon of the memorial session for Julian Besag in Bristol was an intense and at times emotional moment,where friends and colleagues of Julian shared memories and stories.This collection of tributes showed how much of a

11、larger-than-life character he was,from his long-termed and wide-ranged impact on statistics to his very high expectations,both for himself and for others,leading to atotal and uncompromising research ethics,to his passion forextremesports and outdoors.(The stories during and after diner were of amor

12、e personal nature,but at least as much enjoyable！)The talks on the second day showed how much and how deeply Julian had contributed to spatial statistics and agricultural experiments,to pseudo-likelihood,to Markov random fields and image analysis,and to MCMC methodology and practice.I hope Idid not

13、botch too much my presentation on the history of MCMC,while Ifound reading through the 1974,1986 and 1993 Read Papers and their discussions an immensely rewarding experiment(I wish Ihad done prior to completing our Statistical Science paper,but it was bound to be incomplete by nature！).Some interest

14、ing links made by the audience were the prior publication of proofs of the Hammersley-Clifford theorem in 1973(by Grimmet,Preston,and Steward,respectively),as well as the proposal of aGibbs sampler by Brian Ripley as early as 1977(even though Hastings did use Gibbs steps in one of his examples).Chri

15、stophe Andrieu also pointed out to me avery early Monte Carlo review by John Halton in the 1970 SIAM Rewiew,review that Iwill read(and commment)as soon as possible.Overall,I am quite glad Icould take part in this memorial and Iam grateful to both Peters for organising it as afitting tribute to Julia

16、n.Markov Chain Monte Carlo(MCMC)methods are currently avery active field of research.MCMC methods are sampling methods,based on Markov Chains which are ergodic with respect to the target probability measure.The principle of adaptive methods is to optimize on the fly some design parameters of the algorithm with respect to agiven criterion reflecting the samplers performance(opti mize the acceptance rate,optimize an importance sampling