Knowledge of animal diets provides essential insights into their life history and ecology, although diet estimation is challenging and remains an active area of research. Quantitative fatty acid signature analysis (QFASA) has become a popular method of estimating diet composition, especially for marine species. A primary assumption of QFASA is that constants called calibration coefficients, which account for the differential metabolism of individual fatty acids, are known. In practice, however, calibration coefficients are not known, but rather have been estimated in feeding trials with captive animals of a limited number of model species. The impossibility of verifying the accuracy of feeding trial derived calibration coefficients to estimate the diets of wild animals is a foundational problem with QFASA that has generated considerable criticism. We present a new model that allows simultaneous estimation of diet composition and calibration coefficients based only on fatty acid signature samples from wild predators and potential prey. Our model performed almost flawlessly in four tests with constructed examples, estimating both diet proportions and calibration coefficients with essentially no error. We also applied the model to data from Chukchi Sea polar bears, obtaining diet estimates that were more diverse than estimates conditioned on feeding trial calibration coefficients. Our model avoids bias in diet estimates caused by conditioning on inaccurate calibration coefficients, invalidates the primary criticism of QFASA, eliminates the need to conduct feeding trials solely for diet estimation, and consequently expands the utility of fatty acid data to investigate aspects of ecology linked to animal diets.