@michael e sullivan: re “Monte Carlo methods can’t buy you any correctness”—actually, they can. If you have an exact closed-form solution (or a rapidly-converging series, or whatever) for your numbers, you really want to use it. However not all problems have such a thing; generally, you either simplify (giving a precise, incorrect number that is readily computable and hopefully close), or you can do a numerical evaluation, which might approach the correct solution arbitrarily closely based on how much computation power you devote to it.
Quadrature (the straightforward way to do numerical integration using regularly-spaced samples) is a numeric evaluation method which is efficient for smooth, low-dimensional problems. However, for higher-dimensional problems, the number of samples becomes impractical. For such difficult problems, Monte Carlo integration actually converges faster, and can sometimes be the only feasible method.
Somewhat ironically, one field where Monte Carlo buys you correctness is numeric evaluation of Bayesian statistical models!
@michael e sullivan: re “Monte Carlo methods can’t buy you any correctness”—actually, they can. If you have an exact closed-form solution (or a rapidly-converging series, or whatever) for your numbers, you really want to use it. However not all problems have such a thing; generally, you either simplify (giving a precise, incorrect number that is readily computable and hopefully close), or you can do a numerical evaluation, which might approach the correct solution arbitrarily closely based on how much computation power you devote to it.
Quadrature (the straightforward way to do numerical integration using regularly-spaced samples) is a numeric evaluation method which is efficient for smooth, low-dimensional problems. However, for higher-dimensional problems, the number of samples becomes impractical. For such difficult problems, Monte Carlo integration actually converges faster, and can sometimes be the only feasible method.
Somewhat ironically, one field where Monte Carlo buys you correctness is numeric evaluation of Bayesian statistical models!