“The only good Monte Carlos are the dead Monte Carlos” (Trotter and Tukey, quoted by Halton)

When I presented my [partial] history of MCM methods in Bristol two months ago, at the Julian Besag memorial, Christophe Andrieu mentioned a 1970 SIAM survey by John Halton on A retrospective and prospective survey of the Monte Carlo method. This is a huge paper (62 pages, 251 references) and it brings a useful light on the advances in the 60′s (the paper was written in 1968). From the reference list, it seems John Halton was planning two books on the Monte Carlo method, but a search on google did not show anything. I also discovered in this list that there was a 1954 RSS symposium (Read Paper?) on Monte Carlo methods. Note that there were at least two books published at the time, Hammersley and Handscomb’s 1964 Monte Carlo Methods and Scheider’s 1966 Monte Carlo Method. There is a lot of material in this long review and most of the standard methods are listed: control variate, importance sampling, self-normalised simportance sampling, stratified sampling, antithetic variables, simulation by inversion, rejection or demarginalisation. Variance reduction is presented as the motivation for the alternative methods. Very little is said about the use of Monte Carlo methods in statistics (“many of  [the applications] are primitive and artless“)  I was first very surprised to find sequential Monte Carlomentioned as well, but it later appeared this was Monte Carlo methods for sequential problems, in the spirit of Abraham Wald. While the now forgotten EZH method is mentioned as a promising new method (p.11), the survey also contains an introduction to the conditional Monte Carlo method of Trotter and Tukey (1956) that relates to the averaging techniques of Kong, McCullagh, Meng, Nicolae and Tan as found in their 2003 Read Paper….

“The search for randomness is evidently futile” (Halton)

A large part of the review is taken by the study of uniform random generators and by the distinction between random, pseudo-random and quasi-random versions. Halton insists very much on the lack of justification in using non-random generators, even though they work well. He even goes further as to warn about bias because even the truly random generators are discrete. The book covers the pseudo-random generators, starting with the original version of von Neumann, Metropolis, and Ulam, continuing with Lehmer’s well-known congruencial generator, and the Fibonacci generalisation. For testing those generators by statistical tests (again with little theoretical ground), Marsaglia is mentioned.  The paper also covers in great detail the quasi-random sequences, covering low discrepancy requirements, van der Corput, Halton’s and Hammersley’s sequences. Halton considers quasi-Monte Carlo as “a branch of numerical analysis”.

The paper concludes with a list of 24 future developments I will hopefully soon cover in another post…