Inferential models and the decision-theoretic implications of the validity property
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Inferential models (IMs) are data-dependent, probability-like structures designed to quantify uncertainty about unknowns. As the name suggests, the focus has been on uncertainty quantification for inference, and on establishing a validity property that ensures the IM is reliable in a specific sense. The present paper develops an IM framework for decision problems and, in particular, investigates the decision-theoretic implications of the aforementioned validity property. I show that a valid IM's assessment of an action's quality, defined by a Choquet integral, will not be too optimistic compared to that of an oracle. This ensures that a valid IM tends not to favor actions that the oracle doesn't also favor, hence a valid IM is reliable for decision-making too. In a certain special class of structured statistical models, further connections can be made between the valid IM's favored actions and those favored by other more familiar frameworks, from which certain optimality conclusions can be drawn. An important step in these decision-theoretic developments is a characterization of the valid IM's credal set in terms of confidence distributions, which may be of independent interest.
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