Probabilistic numerics casts numerical tasks, such the numerical solution of differential equations, as inference problems to be solved. One approach is to model the unknown quantity of interest as a random variable, and to constrain this variable using data generated during the course of a traditional numerical method. However, data may be nonlinearly related to the quantity of interest, rendering the proper conditioning of random variables difficult and limiting the range of numerical tasks that can be addressed. Instead, this paper proposes to construct probabilistic numerical methods based only on the final output from a traditional method. A convergent sequence of approximations to the quantity of interest constitute a dataset, from which the limiting quantity of interest can be extrapolated, in a probabilistic analogue of Richardson’s deferred approach to the limit. This black box approach (1) massively expands the range of tasks to which probabilistic numerics can be applied, (2) inherits the features and performance of state-of-the-art numerical methods, and (3) enables provably higher orders of convergence to be achieved. Applications are presented for nonlinear ordinary and partial differential equations, as well as for eigenvalue problems—a setting for which no probabilistic numerical methods have yet been developed.

MOTIVATION: Mendelian randomization is an epidemiological technique that uses genetic variants as instrumental variables to estimate the causal effect of a risk factor on an outcome. We consider a scenario in which causal estimates based on each variant in turn differ more strongly than expected by chance alone, but the variants can be divided into distinct clusters, such that all variants in the cluster have similar causal estimates. This scenario is likely to occur when there are several distinct causal mechanisms by which a risk factor influences an outcome with different magnitudes of causal effect. We have developed an algorithm MR-Clust that finds such clusters of variants, and so can identify variants that reflect distinct causal mechanisms. Two features of our clustering algorithm are that it accounts for differential uncertainty in the causal estimates, and it includes 'null' and 'junk' clusters, to provide protection against the detection of spurious clusters. RESULTS: Our algorithm correctly detected the number of clusters in a simulation analysis, outperforming methods that either do not account for uncertainty or do not include null and junk clusters. In an applied example considering the effect of blood pressure on coronary artery disease risk, the method detected four clusters of genetic variants. A post hoc hypothesis-generating search suggested that variants in the cluster with a negative effect of blood pressure on coronary artery disease risk were more strongly related to trunk fat percentage and other adiposity measures than variants not in this cluster. AVAILABILITY AND IMPLEMENTATION: MR-Clust can be downloaded from https://github.com/cnfoley/mrclust. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.