A research frontier has emerged in scientific computation, wherein discretisation error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly deterministic) computational work-flow, in order to assess the impact of discretisation error on the computer output. This paper examines the case for probabilistic numerical methods in routine statistical computation. Our focus is on numerical integration, where a probabilistic integrator is equipped with a full distribution over its output that reflects the fact that the integrand has been discretised. Our main technical contribution is to establish, for the first time, rates of posterior contraction for one such method. Several substantial applications are provided for illustration and critical evaluation, including examples from statistical modelling, computer graphics and a computer model for an oil reservoir. arXiv:1512.00933v6 [stat.ML] 18 Oct 2017 2 computer code to reproduce experiments reported in this paper can be downloaded from http://www. warwick.ac.uk/fxbriol/probabilistic_integration.

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Gradient information on the sampling distribution can be used to reduce the variance of Monte Carlo estimators via Stein's method. An important application is that of estimating an expectation of a test function along the sample path of a Markov chain, where gradient information enables convergence rate improvement at the cost of a linear system which must be solved. The contribution of this paper is to establish theoretical bounds on convergence rates for a class of estimators based on Stein's method. Our analysis accounts for (i) the degree of smoothness of the sampling distribution and test function, (ii) the dimension of the state space, and (iii) the case of non-independent samples arising from a Markov chain. These results provide insight into the rapid convergence of gradient-based estimators observed for low-dimensional problems, as well as clarifying a curse-of-dimension that appears inherent to such methods.

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Markov Chain Monte Carlo methods have revolutionised mathematical computation and enabled statistical inference within many previously intractable models. In this context, Hamiltonian dynamics have been proposed as an efficient way of building chains which can explore probability densities efficiently. The method emerges from physics and geometry and these links have been extensively studied by a series of authors through the last thirty years. However, there is currently a gap between the intuitions and knowledge of users of the methodology and our deep understanding of these theoretical foundations. The aim of this review is to provide a comprehensive introduction to the geometric tools used in Hamiltonian Monte Carlo at a level accessible to statisticians, machine learners and other users of the methodology with only a basic understanding of Monte Carlo methods. This will be complemented with some discussion of the most recent advances in the field which we believe will become increasingly relevant to applied scientists.