Hammersley Clifford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 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(or complete subgraphs)of the graph.

The relationship between Markov and Gibbs random fields was initiated by

Roland Dobrushin[1]and Frank Spitzer[2]in the context of statistical

theorem is named after John Hammersley and Peter Clifford

who proved the equivalence in an unpublished paper in 1971.[3][4]Simpler

proofs using the inclusion-exclusion principle were given independently

by Geoffrey Grimmett,[5]Preston[6]and Sherman[7]in 1973,with afurther

proof by Julian Besag in 1974.[8]

Notes

^Dobrushin,P.L.(1968),

Conditio nal Probabilities and Conditions of Its Regularity

Probability and its Applications 13(2)

：

197

–

224,doi

：

10 .1137/1113026,Spitzer,Frank(1971),

Ensem bles

：

142

–

< br> 154,doi

：

10.2307/2317621, JSTOR 2317621,Hammersley,J.M.

；

Clifford,P.(1971),Markov

fields on finite graphs and lattices,Clifford,P.(1990),

fields in st atistics

；

Welsh,D.J.A.,Disord er in

Physical Systems

：

A Volume in Honour of John sley,Oxford

University Press,pp.19

–

32,ISBN ,MR 1064553,retrieved 2009-

05-04^Grimmett,G .R.(1973),

the London Mathematical Society 5(1)

：

81

–< /p>

84,doi

：

10.1112/bl ms/5.1.81,MR 0329039^Preston,C.J.(1973),

states and Markov random fields

：

242

–

261,doi

：

10.2307/1426035,JSTO R 1426035,MR

1426035,Sherman,S.(1973),

fields

：

92

< br>–

103,doi

：

10.1007/BF

02761538,MR 0321185^Besag,J.(1974),

statistical analysis of lattice systems

Statistical B(Methodological)36(2)

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192

–

236,MR

2984812 Further reading Bilmes,Jeff(Spring 2006),Handout < /p>

2

：

Hammersley

–

Clifford,course notes from University of Washington

tt,Geoffrey,Probability on Gr aphs,Chapter 7,Helge,The

Hammersley

–

Clifford Theorem and its Impact on Modern

Statistics,probability-related article is can help Wikipedia

by expanding it.

Retrieved from

–

Clifford_theorem

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 collection of

tributes showed how much of alarger-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 for[extreme]sports and

outdoors.(The stories during and after diner were of amore 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 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 interesting 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).Christophe 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

l,I am quite glad Icould take part in this memorial and

Iam grateful to both Peters for organising it as afitting tribute to

Chain Monte Carlo(MCMC)methods are currently avery active

field of methods are sampling methods,based on Markov

Chains which are ergodic with respect to the target probability

principle of adaptive methods is to optimize on the fly some

design parameters of the algorithm with respect to agiven criterion

reflecting the sampler's performance(opti mize the acceptance

rate,optimize an importance sampling function,etc…).A postdoctoral

position is opened to work on the numerical analysis of adaptive MCMC

methods

：

convergence,numerical efficiency,development and analysis of

new algorithms.A particular emphasis will be given to applications in

statistics and molecular dynamics.(Detailed description)Position funded

by the French National Research Agency(ANR)through the 2009-2012 project

position will benefit from an interdisciplinary

environment involving numerical analysts,statisticians and

probabilists,and of strong interactions between the partners of the

project ANR-08-BLAN-021 In the most recent issue of Statistical

Science,the special topic is

Quandunciacentennial

and Wing Wong on the emergence of MCMC Bayesian computation in the

1980′s,This survey is more focused and more informative than our global

history(also to appear in Stati stical Science).In particular,it

provides the authors'analysis as to why MCMC was delayed by ten years or

so(or even more when considering that aGibbs sampler as asimulation tool

appears in both Hastings'(1970)and Besag's(1974)papers).They

dismiss[our]concerns about computing power(I was running Monte Carlo

simulations on my Apple IIe by 1986 and asingle mean square error curve

evaluation for aJames-Stein type estimator would then take close to

aweekend

！

)and Markov innumeracy,rather attributing the reluctance to

alack of confidence into the perspective remains debatable

as,apart from Tony O'Hagan who was then fighting again Monte Carlo

methods as being un-Bayesian(1987,JRSS D),I do not remember any negative

attitude at the time about simulation and the immediate spread of the

MCMC methods from Alan Gelfand's and Adrian Smith's presentations of

their 1990 paper shows on the opposite that the Bayesian community was

ready for the move.

Another interesting point made in this historical survey is that

Metropolis'and other Markov chain methods were first presented outside

simulation sections of books like Hammersley and

Handscomb(1964),Rubinstein(1981)and Ripley(1987),perpetuating the

impression that such methods were mostly optimisation or niche specific

is also why Besag's earlier works(not mentioned in this

survey)did not get wider recognition,until ing Iwas not

aware is the appearance of iterative adap tive importance

sampling(tion Monte Carlo)in the Bayesian literature of the

1980′s,with proposals from Herman va

n Dijk,Adrian Smith,and

appendix about Smith et al.(1985),the 1987 special issue of JRSS D,and

the computation contents of Valencia 3(that Isadly missed for being in

the Army

！

)is also quite informative about the perception of

computational Bayesian statistics at this time.

A missing connection in this survey is Gilles Celeux and Jean Diebolt's

stochastic EM(or SEM).As early as 1981,with Michel Broniatowski,they

proposed asimulated version of EM for mixtures where the latent variable

zwas simulated from its conditional distribution rather than replaced

with its this was the first half of the Gibbs sampler for

mixtures we completed with Jean Diebolt about ten years later.(Also

found in Gelman and King,1990.)These authors did not get much

recognition from the community,though,as they focused almost exclusively

on mixtures,used simulation to produce arandomness that would escape the

local mode attraction,rather than targeting the posterior

distribution,and did not analyse the Markovian nature of their algorithm

until later with the simulated annealing EM algorithm.

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概率图模型分为有向和无向的模型。有向的概率图模型主要包括贝叶斯网络和隐马

尔 可夫模型

,

无向的概率图模型则主要包括马尔可夫随机场模型和 条件随机场模

型。

2001