The paper investigates inference in non-linear and non-Gaussian models with moderately time-varying parameters. We show that for many decision problems, the sample information about the parameter path can be summarized by an artificial linear and Gaussian model, at least asymptotically. The approximation allows for computationally convenient path estimators and parameter stability tests. Also, in contrast to standard Bayesian techniques, the artificial model can be robustified so that in misspecified models, decisions about the path of the (pseudo-true) parameter remain as good as in a corresponding correctly specified model.