Bayesian multiple instance classification based on hierarchical probit regression

Abstract

In multiple instance learning (MIL), the response variable is predicted by features (or covariates) of one or more instances, which are collectively denoted as a bag. Learning the relationship between bags and instances is challenging because of the unknown and possibly complicated data generating mechanism regarding how instances contribute to the bag label. MIL has been applied to solve a variety of real-world problems, which have been mostly focused on supervised tasks, such as molecule activity prediction, protein binding affinities prediction, object detection, and computer-aided diagnosis. However, to date, the majority of the off-the-shelf MIL methods are developed in the computer science domain, and they focus on improving the prediction performance while spending little effort on explainability of the algorithm. In this article a Bayesian multiple instance learning model, based on probit regression (MICProB), is proposed, which contributes a significant portion to the suite of statistical methodologies for MIL. MICProB is composed of two nested probit regression models, where the inner model is estimated for predicting primary instances, which are considered as the “important” ones that determine the bag label, and the outer model is for predicting bag-level responses based on the primary instances estimated by the inner model. The posterior distribution of MICProB can be conveniently approximated using a Gibbs sampler, and the prediction for new bags can be performed in a fully integrated Bayesian way. We evaluate the performance of MICProB against 15 benchmark methods and demonstrate its competitiveness in simulation and real-data examples. In addition to its capability of identifying primary instances, as compared to existing optimization-based approaches, MICProB also enjoys great advantages in providing a transparent model structure, straightforward statistical inference of quantities related to model parameters, and favorable interpretability of covariate effects on the bag-level response.