Jonathan P. Williams

Jonathan P Williams

Conformal prediction (CP) was developed to provide finite-sample probabilistic prediction guarantees. While CP algorithms are a relatively general-purpose approach to uncertainty quantification, with finite-sample guarantees, they lack versatility. Namely, the CP approach does not {\em prescribe} how to quantify the degree to which a data set provides evidence in support of (or against) an arbitrary event from a general class of events. In this paper, tools are offered from imprecise probability theory to build a formal connection between CP and generalized fiducial (GF) inference. These new insights establish a more general inferential lens from which CP can be understood, and demonstrate the pragmatism of fiducial ideas. The formal connection establishes a context in which epistemically-derived GF probability matches aleatoric/frequentist probability. Beyond this fact, it is illustrated how tools from imprecise probability theory, namely lower and upper probability functions, can be applied in the context of the imprecise GF distribution to provide posterior-like, prescriptive inference that is not possible within the CP framework alone. In addition to the primary CP generalization that is contributed, fundamental connections are synthesized between this new model-free GF and three other areas of contemporary research: nonparametric predictive inference (NPI), conformal predictive systems/distributions, and inferential models (IMs).

Neil Dey, Jonathan P. Williams

The parameters of a machine learning model are typically learned by minimizing a loss function on a set of training data. However, this can come with the risk of overtraining; in order for the model to generalize well, it is of great importance that we are able to find the optimal parameter for the model on the entire population -- not only on the given training sample. In this paper, we construct valid confidence sets for this optimal parameter of a machine learning model, which can be generated using only the training data without any knowledge of the population. We then show that studying the distribution of this confidence set allows us to assign a notion of confidence to arbitrary regions of the parameter space, and we demonstrate that this distribution can be well-approximated using bootstrapping techniques.

Anders Hjort, Gudmund Horn Hermansen, Johan Pensar et al.

Non-parametric machine learning models, such as random forests and gradient boosted trees, are frequently used to estimate house prices due to their predictive accuracy, but a main drawback of such methods is their limited ability to quantify prediction uncertainty. Conformal prediction (CP) is a model-agnostic framework for constructing confidence sets around predictions of machine learning models with minimal assumptions. However, due to the spatial dependencies observed in house prices, direct application of CP leads to confidence sets that are not calibrated everywhere, i.e., the confidence sets will be too large in certain geographical regions and too small in others. We survey various approaches to adjust the CP confidence set to account for this and demonstrate their performance on a data set from the housing market in Oslo, Norway. Our findings indicate that calibrating the confidence sets on a spatially weighted version of the non-conformity scores makes the coverage more consistently calibrated across geographical regions. We also perform a simulation study on synthetically generated sale prices to empirically explore the performance of CP on housing market data under idealized conditions with known data-generating mechanisms.