Monte Carlo Methods and Importance Sampling

History and definition: The term “Monte Carlo” was apparently first used by Ulam and von Neumann as a Los Alamos code word for the stochastic simulations they applied to building better atomic bombs. Their methods, involving the laws of chance, were aptly named after the international gaming destination; the moniker stuck and soon after the War a wide range of sticky problems yielded to the new techniques. Despite the widespread use of the methods, and numerous descriptions of them in articles and monographs, it is virtually impossible to find a succint definition of “Monte Carlo method” in the literature. Perhaps this is owing to the intuitive nature of the topic which spawns many definitions by way of specific examples. Some authors prefer to use the term “stochastic simulation” for almost everything, reserving “Monte Carlo” only for Monte Carlo Integration and Monte Carlo Tests (cf. Ripley 1987). Others seem less concerned about blurring the distinction between simulation studies and Monte Carlo methods. Be that as it may, a suitable definition can be good to have, if for nothing other than to avoid the awkwardness of trying to define the Monte Carlo method by appealing to a whole bevy of examples of it. Since I am (so Elizabeth claims!) unduly influenced by my advisor’s ways of thinking, I like to define Monte Carlo in the spirit of definitions she has used before. In particular, I use: