Comment 3: I was skeptical that a rejection sampler would work as written in a space of even moderately high22 dimension ... does the hyperplane... still intersect a with reasonable probability? Thisleadstolongerruntimes. ForD =85,26 we are able to conduct inference with the rejection sampling, indicating that the interaction is still possible at this27 dimensionality. We will discuss or report on this new method in the31 camerareadycopy,shouldthisworkbeaccepted.32 Comment 5: What iflotsofthetestdata isoutside thecollection ofconvexpolytopes?When weform convexhulls,33 we consider a version of the training data that includes the predictors of the testing data. Testing data lying outside34 of the convex hulls formed by the training data will be'snapped to the nearest' polytope.

We thank the reviewers for their comments and suggestions. Our point-by-point response to the reviewers' major comments follows, with their comments italicized. We have added this discussion to the manuscript. The experiments were run on an Intel Xeon CPU E5-2683v4@2.10GHz. While the runtimes are not the focus of the paper, we will include a complete version of these tables in the supplement.