Editorial Review:
A FIRST COURSE IN MONTE CARLO shows you how to design, perform, and analyze the results of MC experiments based on independent replications, Markov chain MC, and MC optimization. The text emphasizes the variance-reducing techniques of importance sampling, stratified sampling, Rao-Blackwellization, control variates, antithetic variates, and quasi-random numbers. For solving optimization problems it describes several MC techniques, including simulated annealing, simulated tempering, swapping, stochastic tunneling, and genetic algorithms. Examples from many areas show how these techniques perform in practice. Hands-on exercises allow you to experience challenges encountered when solving real problems. An answer key to selected problems is included.
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Customer Reviews
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A very good first introduction to Monte Carlo 2007-12-16
A first course in Monte Carlo offers an introduction to Monte Carlo sampling, a very important topic in both statistics and computer science. Monte Carlo sampling can be described as follows. Suppose one has a function f(x) that is defined over a state space X. Two common problems are to
i) determine the integeral/expectancy of f over the state space, and
ii) find a value x* in the space for which f(x*) <= f(x) for all x in X.
In this text Fishman presents how to approximate the solutions to both problems via Monte Carlo sampling; both independent Monte Carlo and Markov chain Monte Carlo sampling. Independent Monte Carlo sampling involves using a probability distribution w(x) over X to randomly generate n sample points x1,...,xn and then e.g. approximate the expectancy of f over the space by averaging the values
(f/w)(x1),...,(f/w)(xn). Statistically one can show that the average will converge to the desired value as n is allowed to increase. Chapter 2 discusses this method in good detail. I especially appreciated his discussion on the cost of generating a sample, along with the variance of the function values evaluated at the sample points, and the trade-off that ensues when one attempts to reduce variance (i.e. obtain faster convergence) at a cost of a more costly sampling technique. He then goes on to state a very useful fact with respect to two different sampling plans, in that the better plan will induce CR * VR < 1, where CR and VR are the cost and variance ratios of the two plans (I'm assuming the better plan is in the numerator of both ratios). This result applies to both problems i) and ii) above, and can be used as a means for testing the merit of a new sampling/optimization technique relative to some existing one.
Chapters 3 and 4 are very well done, and involve techniques for random-number generators and random-variable generators, the latter being essential for implementing either Monte Carlo technique on a computer. I especially was grateful for his discussion in Chapter 4 of how good generators generate values that uniformly fill a k-dimensional lattice, with each lattice component being an element from the set
{0,1/m,2/m,...(m-1)/m} which is the set of possible integer outputs of the generator. It helped me better understand why the Mersenne Twister generator is so highly regarded in practice.
Chapters 6-9 cover the Markov-Chain Monte Carlo method of sampling points in the state space. The basic idea is that now the sample points are DEPENDENT and forming a Markov chain according to some transition kernel. Good coverage is given to the Hastings-Metropolis algorithm which comes in very handy when the distribution w(x) that one needs to use for sampling is only partially known. The HM algorithm allows one to generate dependent sample paths whose elements converge to the w(x) distribution, yet without fully knowing the distribution w(x)! (usually one is only required to know the distribution up to a multiplicative constant).
In Chapter 8, the HM algorithm is applied to the theory of Simulated Annealing. And although the full implementation details seemed somewhat sketchy, the presentation succeeded in providing a good introduction to the theory behind this extremely important optimization technique.
Finally, this book has several good application examples provided in each chapter. My only minor criticism is that more basic examples would have helped in understanding some of the more advanced material. But all-in all this book will serve as a very usefull reference for years to come.
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