Probabilistic Integration and Intractable Distributions
Chris J. Oates, François‐Xavier Briol, Mark Girolami · arXiv (Cornell University) · 2016
This paper considers numerical approximation for integrals of the form $$\int f(x) p(\mathrm{d}x)$$ in the case where $f(x)$ is an expensive black-box function and $p(\mathrm{d}x)$ is an intractable distribution (meaning that it is accessible only through a finite collection of samples). Our proposal extends previous work that treats numerical integration as a problem of statistical inference, in that we model both $f$ as an a priori unknown random function and $p$ as an a priori unknown random distribution. The result is a posterior distribution over the value of the integral that accounts for these dual sources of approximation error. This construction is designed to enable the principled quantification and propagation of epistemic uncertainty due to numerical error through a computational pipeline. The work is motivated by such problems that occur in the Bayesian calibration of computer models.