This paper concerns the modeling of stochastic processes by means of dynamic factor models. In such models the observed process is decomposed into a structured part called the latent process and a remainder that is called noise. The observed variables are treated in a symmetric way so that no distinction beta een inputs and outputs is required. This motivates the additional condition that the prior assumptions on the noise are symmetric in nature. One of the central questions in this paper is how uncertainty about the noise structure translates into nonuniqueness of the possible underlying latent processes. We investigate several possible noise specifications and analyze properties of the resulting class of observationally equivalent factor models. This concerns in particular the characterization of optimal models and properties of continuity and consistency.