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First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. Overview: this paper presents a fast alternative to MC methods for approximating intractable integrals. The main idea behind Bayesian quadrature is to exploit assumptions and regularities in the likelihood surface, something which pure Monte Carlo ignores. Samples are then drawn according to some criterion - in this case, samples are chosen to the location of the maximal expected posterior variance of the integrand. Intuitively, this is a location where the model knows the least about the value of the integrand, and stands to gain a lot of information.