The more parameters in a rheological constitutive model, the better it tends to reproduce available data, though this does not mean that it is necessarily better justified. Good fits to data are only part of model selection. We develop a Bayesian inference approach that rigorously balances closeness to data against both the number of model parameters and their a priori uncertainty. The analysis reflects a basic principle: Models grounded in physics will enjoy greater generality and perform better away from where they are calibrated. In contrast, relatively empirical models can provide comparable fits, but their a priori uncertainty is penalized. We demonstrate the approach by computing the best-justified number of modes for a multimode Maxwell model (MMM) to describe the dynamic shear moduli G'(omega), G"(omega) of a synthetic polymer network with transient crosslinks (polyvinyl alcohol with sodium tetra-borate). It is shown that a corresponding array of spring-pots, arranged as a parallel array of fractional-Maxwell model elements, is less credible. In contrast, for a biopolymer gluten dough we show that the MMM, irrespective of number of modes, is far less credible than a critical-gel/Rouse model (CGRM), which with its firmer physical basis provides a more credible model. This is true even though the MMM provides a closer fit to the data than the CGRM for the gluten system. Though quantitative, this formulation does not fully supplant user judgment. However, unlike most model fitting/selection approaches, it requires specific, quantifiable, and potentially debatable quantification of this judgment, and thus it provides a rigorous, repeatable assessment of model viability. Models are supported (or not) given numerical input, not vague assertions. (C) 2015 The Society of Rheology.