Digital twins have become central to modern engineering by providing interconnected virtual representations that mirror the behavior of physical assets. Their effectiveness, however, depends critically on the fidelity and efficiency of the underlying computational models used for decision support. This work introduces an adaptive Bayesian framework for quantifying and optimizing the fidelity, efficiency, and overall utility of such virtual representations when approximating system responses in decision-oriented contexts. By integrating reduced-order models (ROMs) within a Bayesian inference and decision-theoretic setting, the framework identifies, at any point in time, the lowest-cost model capable of delivering the required predictive accuracy across all quantities of interest, while rigorously accounting for the uncertainty associated with each model resolution. The approach maximizes an expected-utility function that balances two competing attributes: (i) precision, reflecting the effect of over- or under-estimating target quantities on decision quality, and (ii) computational efficiency, ensuring feasibility for real-time inference. Leveraging continuously assimilated monitoring data, the framework performs this model selection recursively, enabling the virtual twin to evolve in tandem with the physical system. The resulting methodology supports automated decisions on whether to retain the current model, switch to an alternative resolution, or trigger retraining—thus establishing a pathway toward adaptive, trustworthy digital twins for engineering decision support.