In the economics literature, rate-distortion theory (under the name “rational inattention”) has been popular as a model of choice that depends only imprecisely on the characteristics of the options available to an individual decision maker (Sims, 2003; Woodford, 2009; Matejka and McKay, 2015; Mackowiak et al., forthcoming). In this theory, the distribution of actions taken in a given objective situation is assumed to be optimal (in the sense of maximizing expected reward), subject to a constraint on the mutual information between the objective state and the action choice. However, the assumption that a mutual-information cost is the only limit on the precision of choice has unappealing implications: for example, that conditional action probabilities should vary discontinuously with the (continuous) objective state if the rewards associated with given actions are a discontinuous function of the state. In the case of strategic interaction between multiple information-constrained decision makers, this can result in a prediction that equilibrium behavior (in which each agent’s behavior is optimally adapted to the others’ patterns of behavior) should vary discontinuously with changes in the objective state, with the discontinuous responses of each agent being justified by the discontinuous responses of the others. In the kind of example discussed, the location of the discontinuity is indeterminate, so that the assumption of mutually well-adapted behavior fails to yield definite predictions (Yang, 2015); moreover, the predicted discontinuity of equilibrium behavior does not seem to be observed in experiments (Heinemann et al., 2004, 2009; Frydman and Nunnari, 2022). We propose an alternative model of imprecise choice, in which each decision maker is modeled using a generalization of the “β-variational autoencoder” of Alemi et al. (2018), which nests the “rationally inattentive” model of choice as a limiting case. In our more general model, there are two distinct “rate-distortion” trade-offs: one between the rate of information transmission and a cross-entropy measure of distortion (as in the β-VAE of Alemi et al.), and another between the rate and the measure of distortion given by the negative of expected reward (as in rational inattention models). The generalization provides a model of how an imprecise classification of decision situations can be learned from a finite training data set, rather than assuming optimization relative to a precisely correct prior distribution; and it predicts only gradual changes in action probabilities in response to changes in the objective state, in line with experimental data.