Adaptive and Recursive Monte Carlo Methods This section discusses more advanced techniques of Monte Carlo integration. As examples of the use of these techniques, we include two rather different, fairly sophisticated, multidimensional Monte Carlo codes: vegas [1,2] , and miser [3]. The techniques that we discuss all fall under the general rubric of reduction of variance (§), but are otherwise quite distinct. | 316 Chapter 7. Random Numbers and Recursive Monte Carlo Methods This section discusses more advanced techniques of Monte Carlo integration. As examples of the use of these techniques we include two rather different fairly sophisticated multidimensional Monte Carlo codes vegas 1 2 and miser 3 . The techniques that we discuss all fall under the general rubric of reduction of variance but are otherwise quite distinct. Importance Sampling The use of importance sampling was already implicit in equations and . We now return to it in a slightly more formal way. Suppose that an integrand f can be written as the product of a function h that is almost constant times another positive function g. Then its integral over a multidimensional volume V is ffdV y f g gdV J h gdV In equation we interpreted equation as suggesting a change of variable to G the indefinite integral of g. That made gdV a perfect differential. We then proceeded to use the basic theorem of Monte Carlo integration equation . A more general interpretation of equation is that we can integrate f by instead sampling h not however with uniform probability density dV but rather with nonuniform density gdV. In this second interpretation the first interpretation follows as the special case where the means of generating the nonuniform sampling of gdV is via the transformation method using the indefinite integral G see . More directly one can go back and generalize the basic theorem to the case of nonuniform sampling Suppose that points xi are chosen within the volume V with a probability density p satisfying y pdV 1 The generalized fundamental theorem is that the integral of any function f is estimated using N sample points xit. xN by I fdv f pdV f 2 p2i- hf pP J J P PI V N where angle brackets denote arithmetic means over the N points exactly as in equation . As in equation the plus-or-minus term is a one standard deviation .
