Decision-making with possibilistic inferential models

Abstract

Inferential models (IMs) are data-dependent, imprecise-probabilistic structures designed to quantify uncertainty about unknowns. As the name suggests, the focus has been on uncertainty quantification for inference and on its reliability properties in that context. Focusing on a likelihood-based possibilistic IM formulation, the present paper develops a corresponding framework for decision making, and investigates the decision-theoretic implications of the IM's reliability guarantees. Here we show that the possibilistic IM's assessment of an action's quality, defined by a simple Choquet integral, tends not be too optimistic compared to that of an oracle. This ensures that the IM tends not to favor actions that the oracle doesn't also favor, hence the IM is also reliable for decision making. We also establish a complementary, large-sample efficiency result that says the IM's reliability isn't achieved by being grossly conservative. In the special case of equivariant statistical models, further connections can be made between the IM's and Bayesian's recommended actions, from which certain optimality conclusions can be drawn.