计算机视觉教程答案,计算机视觉模型、推理课后习题答案

计算机视觉模型、推理课后习题答案

计算机视觉模型、推理课后习题答案AnswerbookletThisdocumentaccompaniesthebook"Computervision:models,learning,andinference'bySimonJ.d.Prince,(http://www.computervisionmodels.com).Thisdocumentcontainsanswerstoasclcctcdsubsctoftheproblcmsatthecndofcachoft...

计算机视觉模型、推理课后习题答案Answer bookletThis document accompanies the book "Computer vision: models, learning, andinference'bySimonJ.d.Prince,(http://www.computervisionmodels.com).Thisdocument contains answers to a sclcctcd subsct of the problcms at the cnd of cachof the chapters of the Imlainl book. The reMaining answers are available only toinstructors via Cabridge university pressThe included answers mostly involve derivations which would have detractedfrom the main text. I've also included the answers to any problems which containedan error in the original book and ive corrected all of these errors in the currentdocument. If you are systematically working through the problems in the book, itis hence better to work from this booklet which I will try to keep as up to date aspossibleThis document has not yet been checked very carefully I really need your helin this rcgard and I'd bc very grateful if you would mail mc at s prince@cs. ucl. ac ukif you cannot understand the text or if you think that you find a mistake. Suggestions for extra problems will also be gratefully receivedSimon princeJuly7,2012Copyright 2012 by Simon Prince. This latest version of this documcnt can bc downloaded fromhttp://www.computervisionmodels.com4Copyright (2012 by Simon Prince. This latest vcrsion of this documcnt can be downloaded fromhttp://www.computervisionmodels.comChapter 2Introduction to probabilityProblem 2.1 Give a real-world example of a joint distribution Pr(a, y)where a is discreteand y is continuousProblem 2.2 What remains if I marginalize a joint distribution Pr(a, w, a, y, x) over fivevariables with respect to variables w and y? What remains if I marginalize the resultingdistribution with respect to v:Problem 2.3 Show that the following relation is truePr(w, z)= Pr(, y)Pr(z, ,Pr(ul, y)Problem 2. 4 In my pocket there are two coins. Coin 1 is unbiased, so the likelihood(h-1c-1) of getting heads is 0.5 and the likelihood Pr(h-0c-1) of gettingtails is also 0.5. Coin 2 is biased, so the likelihood Pr(h=1c=2 of getting head0.8 and the likelihood Pr(h=oc=2)of getting tails is 0. 2. I reach into my pocket anddraw one of the coins at random. There is an equal prior probability I might have pickedeither coin. I flip the coin and observe a head. Use Bayes'rule to compute t he posteriorprobability that I chose coin 2Problem 2. 5 If variables c and y are independent and variables and z are independentdoes it follow that variables y and z are independent?nswerNo, it does not follow. Consider any general distribution Pr(y, 2) where y and z are NOTindependent. Now consider the marginal distributions Pr(y) and Pr(z). It is perfectlypossible to have a third distribution Pr(a) which does not provide any information abouty or z and hence is independent of each and Pr(, y)= Pr(a)pr(y) and Pr(a, aPIf you are unsure about this, then construct a counter example where y and a arcll discrete variables with two entries. Construct a non-independent 2x 2 distributionbetween y and z, marginalize it with respect to each and then construct 2x 2 independentoint distributions between and z and a and yCopyright 2012 by Simon Prince. This latest version of this documcnt can bc downloaded fromhttp://www.computervisionmodels.com62 Introduction to probabilityProblem 2.6 Use equation 2.3 to show that when and y are independent, the marginaldistribution Pr(a is the same as the conditional distribution Pr(aly=y) for any yProblem 2.7 The joint probability Pr(w, i, 3, a) over four variables factorizes asPr(w, a, y, a)=Pr(u)Pr(aly)Priya, w)Pr(aDemonstrate that a is independent of w by showing that Pr(, w)= Pr()Pr(Problem 2. 8 Consider a biased die where the probabilities of rolling sides 1, 2, 3, 4, 5, 6are (1/12, 1/12, 1/12, 1/12, 1/6, 1/23, respectively. What is the expected value of the die?If I roll the die twice, what is the expected value of the sum of the two rolls?Problem 2.9 Prove the four relations for manipulating expectationsE[∫c]RElfEIfc+ glaEff[+ Elg[c,Ef]g y= elf[celsly, if r: y independentAnswerRelation 1EK Pr(a)dan:/ Pr(a)daRelation 2Ekfkifa]Pr(a)d zk/f[]Pr(a)dKEIrRelation 3.EI[a]+g[all=/([a]+g[c))Pr(r)d.cjuaa Pr(a)+gla Pr(a))d.cla]Pr(a)dr+/ g[z)Pr(az)dcEIf+egalCopyright (2012 by Simon Prince. This latest vcrsion of this documcnt can be downloaded fromhttp://www.computervisionmodels.comRelation 4:ELfx]·g列//∫x·glPr(x,y)dcdy//sel. y Pr(a)Pr(v)dedyf[z]Pr(a)dx/ o[vlPr(y)EfEl9lyll,where we have used the definition of independence between the first two linesProblem 2 10 Use the relations from problem 2g to prove the follow ing relationship between the second moment around zero and the second moment about the mean(variance)E[(x-)2]=E[x2]-EECopyright 2012 by Simon Prince. This latest version of this documcnt can bc downloaded fromhttp://www.computervisionmodels.com82 Introduction to probabilityCopyright (2012 by Simon Prince. This latest vcrsion of this documcnt can be downloaded fromhttp://www.computervisionmodels.comChapter 3Common probabilitydistributionsProblem 3. 1 Consider a variable a which is Bernoulli distributed with parameter A. Showthat the mean ela is A and the variance E[(a -ea is X(1-X)Problem 3.2 Calculate an expression for the mode(position of the peak) of the betadistribution with a, B>I in terms of the parameters or and BProblem 3. 3 The mean and variance of the beta distribution are given by the expressionsE[小]=p+B(a+B)2(a++1)We may wish to choose the parameters or and B so that the distribution has a particu.rmean u and variance o. Derive suitable expressions for a and B in terms of u and oProblem 3. 4 All of the distributions in this chapter are members of the exponential familydbe writtthe forPr(ra)=a[r expbg c[a-d[elwhere ala and cla arc functions of the data and b[e and d@ arc functions of theparameters. Find the functions a, b[a, c[c and de that allow the Beta distributiono be represented in the generalized form of the exponential familAnswerThe beta distribution is given byPWe choose functionsCopyright 2012 by Simon Prince. This latest version of this documcnt can bc downloaded fromhttp://www.computervisionmodels.com3 Common probability distributionslogard e= logroll+ logrie-logr[a+PThese functions now conform to the multivariate version of the exponential familPr(a0)-ala[e cla-delIProblem 3.5 Use integration by parts to prove that if]cnI[z+1]Problem 3.6 Consider a restricted family of univariate normal distributions where thelways 1, so tIPr(alu)Show that a normal distribution over the parameter uPr(p)=Normallyhas a conjugate relationship to the restricted normal distributionAnswerTaking the product of the two distributions we ge/p[-05(x-1)2]p-0.5√2ro22xP-05(x2-2+p2To cxp-05a+o月∞p-0.5(0+12C+up+1C+0.5The constant and the first exponential terms in the third line do not depend on u so wedenote them by the constant h1 for short. The second term is a quadratic function in p.Copyright 2012 by Simon Prince. This latest vcrsion of this documcnt can be downloaded fromhttp://www.computervisionmodels.com

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