A Note on Monte Carlo Integration in High Dimensions
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Monte Carlo integration is a commonly used technique to compute intractable integrals. However, it is typically thought to perform poorly for very high-dimensional integrals. Therefore, we examine Monte Carlo integration using techniques from high-dimensional statistics in which we allow the dimension of the integral to increase. In doing so, we derive non-asymptotic bounds for the relative and absolute error of the approximation for some general functions through concentration inequalities. We demonstrate that the scaling in the number of points sampled to guarantee a consistent estimate can vary between polynomial to exponential, depending on the function being integrated, demonstrating that the behaviour of Monte Carlo integration in high dimensions is not uniform. Through our methods we also obtain non-asymptotic confidence intervals for the Monte Carlo estimate which are valid regardless of the number of points sampled.
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