This paper identifies Bayesian probabilistic numerical methods as a particular class of noiseless Bayesian inversion problems. It is shown that, while many PN methods incorporate Bayesian ideas, the resulting posterior distribution is seldom fully Bayesian by this definition. Nevertheless, under remarkably general conditions there always exists an essentially unique Bayesian probabilistic numerical method for a particular problem, although the posterior distribution is often intractable.
Secondly, analogues are drawn with the fields of information-based complexity and average-case analysis, in a discussion of performance analysis for Bayesian PNM. It is shown that, where an average-case optimal numerical method is known for a particular problem, the posterior mean of an equivalent Bayesian PNM will naturally coincide with this point estimate, a result often commented on in other works (see the integration and PDE sections). However, it is also shown that there are situations in which optimal information for Bayesian PNM and average-case optimal information do not coincide.
Thirdly, the numerical disintegration algorithm is constructed to allow exploration of the intractable posterior distributions which often arise from Bayesian PNM. Numerical disintegration is an algorithm which approximates the true Bayesian distribution with samples from a tractable distribution close to the truth in an appropriate probability metric. Theoretical results are presented showing convergence of the numerical disintegration scheme in a limiting sense. This approach is applied to several challenging numerical problems, including Painleve’s first transcendental.
Lastly, the composition of PN methods into pipelines is discussed. Many numerical problems require the solution of multiple interdependent systems. Conditions are established for the output of such pipelines to be Bayesian, when the composite numerical methods are Bayesian PNM. This methodology is then applied to a prototypical pipeline, given by a challenging problem in industrial process monitoring.
Reference: Cockayne J, Oates CJ, Sullivan T, Girolami M. Bayesian Probabilistic Numerical Methods, arXiv:1702.03673.