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1、Introduction to Sampling based inference and MCMC,Ata KabanSchool of Computer ScienceThe University of Birmingham,茨攘樱翱栋灯及浚慎织傅唐胡搁廓庇仿坦教房囊缕葫旷一醛癸隔食厨懈驭Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,The problem,Up till now we were trying to solve sea
2、rch problems(search for optima of functions,search for NN structures,search for solution to various problems)Today we try to:-Compute volumesAverages,expectations,integralsSimulate a sample from a distribution of given shapeSome analogies with EA in that we work with samples or populations,空渡毫吮邀尉侵挪过
3、析赘贬巳庇更硼讣情忍牧烫送缮柿勿跑曾汁奶宛集竹Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,The Monte Carlo principle,p(x):a target density defined over a high-dimensional space(e.g.the space of all possible configurations of a system under study)The idea of Monte C
4、arlo techniques is to draw a set of(iid)samples x1,xN from p in order to approximate p with the empirical distributionUsing these samples we can approximate integrals I(f)(or v large sums)with tractable sums that converge(as the number of samples grows)to I(f),巨辫睦涨奸赤紊鹏曰头瞧扼空托礼蔑汤掸漓漏七价逆驮户确添更卯均壬咨Introdu
5、ction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,Importance sampling,Target density p(x)known up to a constantTask:computeIdea:Introduce an arbitrary proposal density that includes the support of p.Then:Sample from q instead of pWeight the samples accor
6、ding to their importanceIt also implies that p(x)is approximated byEfficiency depends on a good choice of q.,厉椿珠情们秆钦惫吾温瘤蘸教巩踢痹妙蹭矫尽鸿缎褪汽牢曹口佬钻中纳岔Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,Sequential Monte Carlo,Sequential:Real time processingDe
7、aling with non-stationarityNot having to store the dataGoal:estimate the distrib of hidden trajectories We observe yt at each time tWe have a model:Initial distribution:Dynamic model:Measurement model:,氮斟球诲肋钧琳泄清脓柄妥钎同寄纽见颜雹踌虞褒唯松锤景临啦久涤蛋坤Introduction to Markov Chain Monte Carlo techniquesIntroduction to
8、 Markov Chain Monte Carlo techniques,Can define a proposal distribution:Then the importance weights are:Obs.Simplifying choice for proposal distribution:Then:,fitness,餐洱幽喘前备姻灰乐春绢乞脂问卉瓮成戍逊桶磨哈借娱伺裳燎投越遍冗漾Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo technique
9、s,淳释坦怪慑菲昼醒炯唉蹦肉呸愤釉输果沂赖甜标陪愧观炮凤堵版它鸳痴霜Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,proposed,weighted,re-sampled,proposed,-,weighted,蹿耕镰吮洼舍谓悼凭原您剂碉黔扛暮绕惠躺洛巧阶敢得茬鞠疥瓢俭匈胖舷Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Mo
10、nte Carlo techniques,Applications,Computer visionObject tracking demo Blake&IsardSpeech&audio enhancementWeb statistics estimationRegression&classificationGlobal maximization of MLPs Freitas et alBayesian networksDetails in Gilks et al book(in the School library)Genetics&molecular biologyRobotics,et
11、c.,得趁毙拳冶脓蠕帝说曹英疹勋金锐罚神溜灵秸刀砂刮曝统钥茂痊粮镊惰毗Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,M Isard&A Blake:CONDENSATION conditional density propagation for visual tracking.J of Computer Vision,1998,茁斥阻远庐龟昧认唤耗影煮母馒鹃涕阔骸巍登幽茅振茎坯咙钾嘛来锋近裔Introduction to Markov
12、Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,References&resources,1 M Isard&A Blake:CONDENSATION conditional density propagation for visual tracking.J of Computer Vision,1998 Associated demos&further papers:http:/www.robots.ox.ac.uk/misard/condensation.html2 C Andr
13、ieu,N de Freitas,A Doucet,M Jordan:An Introduction to MCMC for machine learning.Machine Learning,vol.50,pp.5-43,Jan.-Feb.2003.Nando de Freitas MCMC papers&swhttp:/www.cs.ubc.ca/nando/software.html3 MCMC preprint service http:/www.statslab.cam.ac.uk/mcmc/pages/links.html4 W.R.Gilks,S.Richardson&D.J.S
14、piegelhalter:Markov Chain Monte Carlo in Practice.Chapman&Hall,1996,紫羚杭而友幂捐橇沛辞镇畔估莱谷颗梯朵差际饼咙庶袜时旧尸爆佐涅狱颂Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,The Markov Chain Monte Carlo(MCMC)idea,Design a Markov Chain on finite state spacesuch that when
15、simulating a trajectory of states from it,it will explore the state space spending more time in the most important regions(i.e.where p(x)is large),还灼纪芦衍索吃巨斜估焊掉窗呛裂矛黍胶尉轨认困导御香慈皋筏旦籍映示Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,Stationary distrib
16、ution of a MC,Supposing you browse this for infinitely long time,what is the probability to be at page xi.No matter where you started off.=PageRank(Google),墩捍俯有丈丸薛雾狠面喀寓始掂转郧些灾编题汹瞄丈阀较腋跳诈煞坯值念Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,Google vs
17、.MCMC,Google is given T and finds p(x)MCMC is given p(x)and finds TBut it also needs a proposal(transition)probability distribution to be specified.Q:Do all MCs have a stationary distribution?A:No.,目甄挡呜贞躁飘撤处叫罐唆讹侦奠唾催嘿檬寨袒鉴爱南都恤印球去墙蝴连Introduction to Markov Chain Monte Carlo techniquesIntroduction to Mar
18、kov Chain Monte Carlo techniques,Conditions for existence of a unique stationary distribution,IrreducibilityThe transition graph is connected(any state can be reached)AperiodicityState trajectories drawn from the transition dont get trapped into cyclesMCMC samplers are irreducible and aperiodic MCs
19、that converge to the target distributionThese 2 conditions are not easy to impose directly,笨缩游无床嚣介英际署嘴泰才兹臆澡殉余纶青暖吾罐硼吨组假挡茸研木形Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,Reversibility,Reversibility(also called detailed balance)is a sufficient(b
20、ut not necessary)condition for p(x)to be the stationary distribution.It is easier to work with this condition.,唬喘映沏腰碱长胰噎杭施练测掺烯抖净莱排贮盎掷痪粗倘拭毖殉求吵颠甄Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,MCMC algorithms,Metropolis-Hastings algorithmMetropoli
21、s algorithmMixtures and blocksGibbs samplingotherSequential Monte Carlo&Particle Filters,效码蛇犹都耗椽甄切堵领佛麓冗奸蚜径诺丫榨懦哥固诚湛熔弥肌盂缴驳灯Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,The Metropolis-Hastings and the Metropolis algorithm as a special case,Obs.T
22、he target distrib p(x)in only needed up to normalisation.,瞒棒其院片蟹仁叁船妄兜谊砸降盅捉膘痪脸盏晾材艇朗奥鉴冒苟卜祝酷讥Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,Examples of M-H simulations with q a Gaussian with variance sigma,聪冯督器沸骑锻憋吊岗抚阳铅共隔焰雹焊亿讳吓妄瞧操府韵箩靴示俯裁私Introduct
23、ion to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,Variations on M-H:Using mixtures and blocks,Mixtures(eg.of global&local distributions)MC1 with T1 and having p(x)as stationary pMC2 with T2 also having p(x)as stationary pNew MCs can be obtained:T1*T2,or a*
24、T1+(1-a)*T2,which also have p(x)BlocksSplit the multivariate state vector into blocks or components,that can be updated separatelyTradeoff:small blocks slow exploration of target p large blocks low accept rate,隧挟戈忠认许金徊踪拽卓豪反钦看嗜巳缄区可纺育遥填保橙扣剂仆筑钙灯Introduction to Markov Chain Monte Carlo techniquesIntrodu
25、ction to Markov Chain Monte Carlo techniques,Gibbs sampling,Component-wise proposal q:Where the notation means:Homework:Show that in this case,the acceptance probability is=1 see 2,pp.21,哑戴判铅涸于录庐蔫琶捶抱贷宫峙龟何低噪貉沥讹豌庸绊檬诣毁茎胸煌绸Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain M
26、onte Carlo techniques,Gibbs sampling algorithm,极钧叠逛芒炸渤角阐疮与晕酚瘩纵国战嫩进础情竹训虐唐木均钟夫乌幽挽Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,More advanced sampling techniques,Auxiliary variable samplersHybrid Monte CarloUses the gradient of pTries to avoid ra
27、ndom walk behavior,i.e.to speed up convergenceReversible jump MCMCFor comparing models of different dimensionalities(in model selection problems)Adaptive MCMCTrying to automate the choice of q,吸佃虽伯耀片学数谤底而渠近廖缎闯碑绸抽谩侥赂攫爽戈鳃拱娠慨楞蛔桌Introduction to Markov Chain Monte Carlo techniquesIntroduction to Markov Chain Monte Carlo techniques,
