The parameter space of dynamic stochastic general equilibrium (DSGE) mod-els typically includes non-identification regions over which the likelihood is flat. Use of informative priors makes it difficult to diagnose identification problems, since posteriors can look very different from priors when the model is only par-tially identified, as is often the case. The Bayes factor is then misleading since it shows that there is information in the marginal likelihood while actually there isn't. Moreover, when flat priors on the structural parameters are used, the pos-terior piles up in the non-identification region, generating spurious inference. We propose a solution to the above pathologies that is based on using pri-ors/posteriors on the structural parameters that are implied by priors/posteriors on the parameters of an embedding linear model, such as a reduced-form VAR. Since the reduced-form parameters are well-identified, this approach does not lead to any a posteriori favor for the non-identification regions. An example of such a prior is the Jeffreys prior which is particularly appealing due to its invari-ance and uninformativeness properties. We provide a straigtforward rejection sampling algorithm with good convergence properties to sample from the priors and posteriors. We illustrate our analysis using the new Keynesian Phillips curve estimated using US data, and find that this model is rather poorly identified.