1 Introduction 5,1 1 Statistical Models 6,2 Random Variable Generation 11. 2 1 Basic Methods 12,2 1 1 Desiderata and Limitations 12. 2 2 Transformation Methods 13,2 3 Accept Reject Methods 14. 3 Monte Carlo Integration 21,3 1 Importance Sampling 22. 4 Markov Chains 27,4 1 Basic notions 27,4 2 Ergodicity and convergence 29. 4 3 Limit theorems 31,5 Monte Carlo Optimization 33. 5 1 Introduction 33,6 The Metropolis Hastings Algo. 6 1 Monte Carlo Methods based on,Markov Chains 43,4 CONTENTS 0 0. 6 2 The Metropolis Hastings algorithm 44,7 The Gibbs Sampler 49. 7 1 General Principles 49,8 Diagnosing Convergence 53. 8 1 Stopping the Chain 53,8 2 Monitoring Convergence to the. Stationary Distribution 55,8 3 Monitoring Convergence of Aver. 9 Implementation in Missing Data,9 1 Introduction 61. 9 2 Finite mixtures of distributions 68,Introduction. Experimenters choice before fast computers,Describe an accurate model which would. usually preclude the computation of explicit,or choose a standard model which would. allow this computation but may not be a,close representation of a realistic model. Such problems contributed to the develop,ment of simulation based inference. 6 INTRODUCTION 1 1,1 1 Statistical Models,Example 1 1 1 Censored data models. are missing data models where densities are,not sampled directly. In a typical simple statistical model we would,The distribution of the sample would then be. given by the product,Inference about would then be based on this. distribution,With censored random variables the actual. observations are,Yi min Yi u,where u is censoring point. As a particular example if,X N 2 andY N 2,the variable. Z X Y min X Y,is distributed as,1 1 STATISTICAL MODELS 7. where and are the density and cdf of the,normal N 0 1 distribution. Similarly if,with density,f x x 1 exp x,the censored variable. Z X constant,has the density,f z z e IIz x e dx z,where a is the Dirac mass at a k. 8 INTRODUCTION 1 1,Example 1 1 2 Mixture models,Models of mixtures of distributions are based. on the assumption,X fj with probability pj,for j 1 2 k with overall density. X p1f1 x pk fk x,If we observe a sample of independent random. variables X1 Xn the sample density is,p1f1 xi pk fk xi. Expanding this product shows that it involves,k n elementary terms which is prohibitive to. compute in large samples k,1 1 STATISTICAL MODELS 9. Example 1 1 3 Student s t distribution,An reasonable alternative to normal errors is. the Student s t distribution denoted by T p,which is often more robust against possible. modeling errors and others The density of,T p is proportional to. If p is known and the parameters and are,unknown the likelihood is. This polynomial of degree 2n may have n local,minima each of which needs to be calculated. to determine the global maximum,10 INTRODUCTION 1 1. Illustration of the multiplicity of modes of the,likelihood from a Cauchy distribution C 1. p 1 when n 3 and X1 0 X2 5, 5 5in5 5in work short mcmcv22 figures bmp cauchy bmp. Figure 1 1 1 Likelihood of the sample 0 5 9 from the distribution C 1. Random Variable Generation,We rely on the possibility of producing with. a computer a supposedly endless flow of ran,dom variables usually iid for well known. distributions,We look at a uniform random number gener. ator and illustrate methods for using these,uniform random variables to produce ran. dom variables from both standard and non,standard distributions. 12 RANDOM VARIABLE GENERATION 2 2,2 1 Basic Methods. 2 1 1 Desiderata and Limitations, Any one who considers arithmetical methods of reproduc. ing random digits is of course in a state of sin As has been. pointed out several times there is no such thing as a ran. dom number there are only methods of producing random. numbers and a strict arithmetic procedure of course is not. such a method John Von Neumann 1951,The problem is to produce a deterministic. sequence of values in 0 1 which imitates,a sequence of iid uniform random variables. Can t use the physical imitation of a ran,dom draw no guarantee of uniformity no. reproducibility,random sequence in the following sense Hav. ing generated X1 Xn knowledge of Xn,or of X1 Xn imparts no discernible. knowledge of the value of Xn 1,Of course given the initial value X0 the sam. ple X1 Xn is always the same,the validity of a random number generator is. based on a single sample X1 Xn when n,tends to and not on replications X11 X1n. X21 X2n Xk1 Xkn where n,is fixed and k tends to infinity. In fact the distribution of these n tuples de,pends on the manner in which the initial val. ues Xr1 1 r k were generated,2 2 TRANSFORMATION METHODS 13. 2 2 Transformation Methods,The case where a distribution f is linked in a. relatively simple way to another distribution,that is easy to simulate. Example 2 2 1 Exponential variables,If U U 0 1 the random variable. has distribution,P X x P log U x,the exponential distribution Exp k. Other random variables that can be gener,ated starting from an exponential include. Y 2 log Uj 22,Y log Uj Ga a,j 1 log Uj,Y Pa b Be a b. j 1 log Uj,14 RANDOM VARIABLE GENERATION 2 3,2 3 Accept Reject Methods. There are many distributions from which it,is difficult or even impossible to directly. We now turn to another class of methods that,only requires us to know the functional form. of the density f of interest up to a multiplica,tive constant. The key to this method is to use a simpler,simulation wise density g from which the. simulation is actually done,For a given density g,the instrumental density. there are many densities f,the target densities,which can be simulated this way. 2 3 ACCEPT REJECT METHODS 15,We first look at the Accept Reject method. Given a density of interest f,find a density g and a constant M such that. on the support of f,Algorithm A 1 Accept Reject Method. 1 Generate X g U U 0 1,2 Accept Y X if U f X M g X. 3 Return to 1 otherwise,This produces a variable Y distributed.

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