Integration with an Adaptive Harmonic Mean Algorithm

Numerically estimating the integral of functions in high dimensional spacesis a non-trivial task. A oft-encountered example is the calculation of themarginal likelihood in Bayesian inference, in a context where a samplingalgorithm such as a Markov Chain Monte Carlo provides samples of the function.We present an Adaptive Harmonic Mean Integration (AHMI) algorithm. Givensamples drawn according to a probability distribution proportional to thefunction, the algorithm will estimate the integral of the function and theuncertainty of the estimate by applying a harmonic mean estimator to adaptivelychosen regions of the parameter space. We describe the algorithm and itsmathematical properties, and report the results using it on multiple testcases.