Bayesian Analysis (2016) 11, Number 4, pp. 1285–1293 Contributed Discussion on Article by Chkrebtii, Campbell, Calderhead, and Girolami ∗ Comment 1 by Fran¸cois-Xavier Briol 2, 3 , Jon Cockayne 4 , and Onur Teymur 5 Abstract. We commend the authors for an exciting paper which provides a strong contribution to the emerging field of probabilistic numerics. Below, we discuss as- pects of prior modelling for differential equations which will need to be considered thoroughly in future work. Keywords: probabilistic numerics, uncertainty quantification, numerical analysis. Introduction The majority of probabilistic numerics (PN) solvers, including the present paper, take a Bayesian viewpoint and hence require several modelling choices including prior spec- ification. As with any inference problem, there exists a trade-off between representing prior beliefs and choosing a prior which is convenient and/or readily interpretable math- ematically. We believe that the consequences of these assumptions are often discussed in too little detail and therefore highlight below several issues to consider. Computational Complexity Of interest was the discussion into reduction of the computational complexity by ex- ploiting compactly supported covariance function. The authors note in Section 3.2 that while such a choice will yield a method involving inversion of a sparse matrix, this is not explored further – though this will have an effect on the rate of convergence of the estimator. We believe that a study of the extent of this effect is of some importance, as there is a clear trade-off here between steps desired to achieve a required tolerance, and the computational cost of each step. * Main article DOI: 10.1214/16-BA1017. 1 This work was completed within the Probabilistic Numerics working group which is part of the 2016–2017 SAMSI programme on Optimization. 2 FXB was supported by EPSRC [EP/L016710/1]. 3 Department of Statistics, University of Warwick, Coventry, CV4 7AL, United Kingdom, f-x.briol@warwick.ac.uk 4 Department of Statistics, University of Warwick, Coventry, CV4 7AL, United Kingdom, j.cockayne@warwick.ac.uk 5 Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom, o@teymur.uk c  2016 International Society for Bayesian Analysis DOI: 10.1214/16-BA1017A