Monte Carlo is Fundamentally Unsound

We present some fundamental objections to the Monte Carlo method of numerical integra- tion. 1 Background As Bayesian inference is applied to more and more complex and realistic models combined with more and more realistic prior distributions, we become increasingly dependent on numerical methods to explore the resulting complex, high-dimensional, posterior distributions. In particular, there has been considerable interest lately in techniques of numerical integration. The Monte Carlo method, which has long been known to numerical analysts, was brought to the attention of the Bayesian statistics community by Kloek & van Dijk (1978), although Stewart had been using it in this context several years earlier. See Stewart & Johnson (1971). There are many variations and elaborations of Monte Carlo integration, but for our purposes it is enough to study the most basic problem. Consider the one-dimensional integral 00 k= f f(x)dx.